Journal of Automation, Mobile Robotics and Intelligent Systems JAMRIS 02/2018

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VOLUME 12 N°1 2018 www.jamris.org VOLUME 13 N°2(PRINT) 2018/ eISSN www.jamris.org pISSN 1897-8649 2080-2145 (ONLINE)

Journal of Automation, Mobile Robotics & Intelligent Systems

pISSN 1897-8649

(PRINT)

/ eISSN 2080-2145

(ONLINE)

VOLUME 13, N° 2

2018

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

Publisher: Industrial Research Institute for Automation and Measurements PIAP

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

Indexed in SCOPUS Indexed in SCOPUS

Journal of Automation, mobile robotics & Intelligent Systems

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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) Dmitry A. Novikov (Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia) Duc Truong Pham (Birmingham University, UK) Lech Polkowski (Polish-Japanese Institute of Information Technology, Poland) Alain Pruski (University of Metz, France) Rita Ribeiro (UNINOVA, Instituto de Desenvolvimento de Novas Tecnologias, Caparica, Portugal) Imre Rudas (Óbuda University, Hungary) Leszek Rutkowski (Czestochowa University of Technology, Poland) Alessandro Saffiotti (Örebro University, Sweden) Klaus Schilling (Julius-Maximilians-University Wuerzburg, Germany) Vassil Sgurev (Bulgarian Academy of Sciences, Department of Intelligent Systems, Bulgaria) Helena Szczerbicka (Leibniz Universität, Hannover, Germany) Ryszard Tadeusiewicz (AGH University of Science and Technology in Cracow, Poland) Stanisław Tarasiewicz (University of Laval, Canada) Piotr Tatjewski (Warsaw University of Technology, Poland) Rene Wamkeue (University of Quebec, Canada) Janusz Zalewski (Florida Gulf Coast University, USA) Teresa Zielinska (Warsaw University of Technology, Poland)

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JOURNAL of AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS VOLUME 12, N° 2, 2018 DOI: 10.14313/JAMRIS_2-2018

CONTENTS 3

52

An Analytical Insight to Investigate the Research Patterns in the Realm of Type-2 Fuzzy Logic Sonakshi Vij, Amita Jain, Devendra Tayal, Oscar Castillo DOI: 10.14313/JAMRIS_2-2018/8

PD Terminal Sliding Mode Control Using Fuzzy Genetic Algorithm for Mobile Robot in Presence of Disturbances Walid Benaziza, Noureddine Slimane, Ali Mallem DOI: 10.14313/JAMRIS_2-2018/11

Decentralized PID Control by Using GA Optimization Applied to a Quadrotor Seif-El-Islam Hasseni: Latifa Abdou DOI: 10.14313/JAMRIS_2-2018/9

Generative Power of Reduction-based Parsable ETPR(k) Graph Grammars for Syntactic Pattern Recognition Mariusz Flasiński DOI: 10.14313/JAMRIS_2-2018/12

33

45

Calibration Low-Cost Cameras with Wide-Angle Lenses for Measurements Damian Wierzbicki DOI: 10.14313/JAMRIS_2-2018/10

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An Analytical Insight to Investigate the Research Patterns in the Realm of Type-2 Fuzzy Logic Submitted: 10th April 2018, accepted: 15th May 2018

Sonakshi Vij, Amita Jain, Devendra Tayal, Oscar Castillo

DOI: 10.14313/JAMRIS_2-2018/8 Abstract: Fuzzy logic has always been one of the key research areas in the field of computer science as it helps in dealing with the real world vagueness and uncertainty. In recent years, a variant of it, Type-2 Fuzzy Logic has gained enormous popularity for research purposes. In this paper, an analytical insight is provided into the research patterns of Type-2 Fuzzy logic. Web of Science has been used as the data source which consists of Science Citation IndexExpanded (SCI-E), SSCI, A&HCI and ESCI indexed research papers. 600 research papers were extracted from it in the field of Type-2 fuzzy logic from the year 2000 to 2016, which are analyzed both manually and in an automated manner. The performed study is Scientometric in nature and helps in answering research questions like control terms and top authors in this field, the growth pattern in research publications, top funding agencies and countries etc. The major goal of this study is to analyze the research work in type-2 fuzzy logic so as to track the growth of this discipline through the years and envision future trends in this area.

Keywords: scientometric analysis, Type 2 fuzzy logic, Type 2 fuzzy systems, Type 2 fuzzy control, Type 2 fuzzy set

1. Introduction Fuzzy logic (Type-1 Fuzzy Logic) caters to the multi-valued logic. It represents a novel concept in the sense that the “truth value” of any variable under consideration may lie from 0 to 1 [1]. Type-2 fuzzy logic is the generalization of type-1 fuzzy logic which has the capability to handle a higher level of uncertainty. This is established from the fact that since the invention of type-1 fuzzy logic, there was a speculation in researchers that it doesn’t have any uncertainty value associated with the corresponding membership function. The father of fuzzy logic, Zadeh, provided a solution to solve this issue by introducing the concept of type-2 fuzzy logic. He introduced the type-2 fuzzy sets which have a unique concept associated with it that is called the “Footprint of Uncertainty” [1]. This means that in order to classify any set as type-2 fuzzy set, an uncertainty value has to be associated with its membership function which would in turn help in dealing with the real world vagueness.

Type-2 fuzzy logic can help solving or improving solutions in many fields, as can be seen by the diversity of the papers covered in this review paper, and as such in theory it could be considered as a general form of modeling and coping with uncertainty in any area of application. However, there are limitations and challenges in this area, for example: how to optimally design the structure of the type-2 fuzzy systems, how to find the optimal parameter values for a particular application, when to apply modularity or granularity to improve results, just to mention a few. In addition, the particular form of type-2 fuzzy models could be application dependent, and if this is the case, finding these forms for particular classes of problems is a crucial task. In this sense, this paper is the first step in analyzing what has been achieved to the moment by the type-2 fuzzy research community, and then what can also be done in the future.

This paper presents an in-depth analysis to chart and map the progress of research work in the field of Type-2 fuzzy logic from the year 2000-2016 based on the research papers retrieved from web of science.

The major goal of this study is to analyze the research work in type-2 fuzzy logic so as to track the growth of this discipline through the years. The study performed in this paper helps in answering the following imperative research questions: 1) What has been the growth rate in the field of type-2 fuzzy logic in terms of research publications?

2) Which journals account for the maximum publication in the field of type-2 fuzzy logic?

3) Which countries and institutes offer higher participation in the field of type-2 fuzzy logic?

4) Which authors have contributed significantly in the research publications pertaining to the field of type-2 fuzzy logic?

5) What has the ratio been of paid vs. open access publications in the field of type-2 fuzzy logic?

6) Which funding agencies have contributed the maximum in providing grants for the concerned research project based papers in the field of type-2 fuzzy logic?

7) What are the various types of research papers available in the field of type-2 fuzzy logic? Whether they are articles or proceeding papers or do they lie in any other category?

8) Which are the most cited research papers in the field of type-2 fuzzy logic and in which research domain do they lie?

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Table 1. Details of the dataset Data Source

Search Query

Time Period

Number of research papers extracted

WOS (web of science)

TI=(Type 2 fuzzy logic OR type 2 fuzzy systems OR type 2 fuzzy control OR type 2 fuzzy pattern recognition OR type 2 fuzzy clustering OR type 2 fuzzy classification OR type 2 fuzzy set)

2000-2016

600

9) Which control terms are associated with type-2 fuzzy logic?

10) What is the scenario regarding the inter country collaboration for research in type-2 fuzzy logic?

This paper assists in answering the above mentioned research questions, which would in turn help in understanding the discipline in a more elaborate manner. This type of work in the field of type-2 fuzzy logic is one of its kinds as it statistically highlights the various aspects related to it. Section 2 of the paper describes the data source and the methodology adopted for study in this paper, while Section 3 presents the results along with a detailed analysis. The work is concluded in Section 4.

2. Data and Methodology

The analysis is performed on a set of research papers obtained by using WOS (Web of Science) as the data source. WOS (Web of Science) is a database that consists of Science Citation Index-Expanded (SCI-E), SSCI, A&HCI and ESCI indexed research papers of various types (articles, reviews, proceeding papers etc.) in several languages. The details of the dataset used are given as in Table 1.

3. Detailed Analysis

This section presents the data collected through WOS [2-601] which is analyzed both manually and in an automated manner for studying the patterns of research in type-2 fuzzy logic.

a) Year-wise publication and growth pattern analysis: The 600 research papers were analyzed to present the data about the number of research publications in each year from 2000 to 2016. It can be noted from Table 2 that recent years (2015 and 2016) have seen a boost in terms of type-2 fuzzy logic research publications. In order to track the growth in the number of research papers, we have calculated two scientometric measures namely Relative Growth Rate (RGR) and Doubling Time (DT). Figure 1 shows the number of research publications each year.

RGR = (ln N2 – ln N1)/T2 – T1………..

DT = ln2/RGR………..

4

(1) (2)

RGR is a measure that represents the relative growth in the number of research publications with respect to time. On the other hand DT highlights the time that is needed for the number of research papers Articles

Indexing SCI-EXPANDED, SSCI, A&HCI, ESCI

in a particular year to become double of its current amount. From Figure 2 it can be seen that RGR and DT are inversely proportional to each other.

Table 2. NOP (Number of Papers) each year from 2000 to 2016 with DT and RGR Year

NOP

2000

4

2001

2002

2

3

CUM 4

6

3.107

27

2009

33

2010 2011 2012 2013 2014 2015 2016

27 27 59 52 65 87 104 100

1.711

0.223

7

2008

0.405

20

2006

19

0.000

1.711

16

2007

0.000

0.405

7 4

DT

9

2004

2005

RGR

46 73 106 133 192 244 309 396 500 600

0.575 0.300 0.532 0.461 0.372 0.226 0.367 0.239 0.236 0.248 0.233 0.182

1.205 2.310 1.302 1.503 1.862 3.066 1.888 2.899 2.936 2.794 2.974 3.807

Fig. 1. Number of research papers published in type-2 fuzzy logic from 2000 to 2016 b) Country-wise contribution: The top 30 countries contributing to the research publications in type-2 fuzzy logic are presented in Table 3 and illustrated pictorially as in Figure 3. For performing this type of analysis, the search query was further filtered to extract countries where the research publication record was greater than 1. It can

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be observed that China accounts for the maximum number of research papers while USA gets the second rank. Iran and Taiwan also contribute significantly to the research publications in this field.

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120

100

80

60 RECORD COUNT 40

117

85

78 78 64 49

20

40

31 30 19 19 18 17 14 14 13 11

0

Fig. 2. RGR and DT for research papers published in type-2 fuzzy logic from 2000 to 2016 Table 3. Top 30 countries contributing to the research publication in type-2 fuzzy logic S.No.

Country

1

Peoples R China

3

Iran

5

England

2

4 6 7 8 9

10

11

12

13

14

15

16

17

USA

Taiwan Turkey

Mexico

Canada India

Australia

Saudi Arab

Italy

Singapore

Malaysia

South Korea

Poland

24 25 26 27 28 29 30

78 64 49 40 31 30

1

14

2

13

4

14

Vietnam Belgium Japan

Brazil

Lithuania Slovakia

Czech Republic Norway

Pakistan

Venezuela

3

9

5

6

7

5 4 4 3 3 3 2 2 2 2

5

4

4

3

3

3

2

2

2

Table 4. Record count of research papers published in various journals

18

17

6

c) Top journals: Table 4 and Figure 4 show the data for record count of research papers published in various journals. As far as the publication of research works for type-2 fuzzy logic is being concerned in WOS from the period of 2000 to 2016, the IEEE Transactions on fuzzy systems holds the top position. The second and third spots go to Information Sciences and Applied Soft Computing respectively. It is also worth observing that about 41.67% of the top 10 journals are from Elsevier publishers while 16.67% are from IEEE publishers.

S.No.

19

8

Fig. 3. Top 30 countries in terms of research publication count in type-2 fuzzy logic from 2000 to 2016

19

8

France

23

78

Egypt

20 22

85

11

Spain

21

117

Algeria

18 19

Record Count

9

6 8 9

10

Journal

Record Count

IEEE Transactions on Fuzzy Systems

87

Applied Soft Computing

42

Information Sciences Expert Systems with Applications Soft Computing

Neuro computing

Engineering Applications of Artificial Intelligence Journal of Intelligent Fuzzy Systems

IEEE Computational Intelligence Magazine, International Journal of Fuzzy Systems

International Journal of Innovative Computing Information And Control, International Journal of Uncertainty Fuzziness and Knowledge Based Systems

60 24 22 18 16 14 11

9

Publication house IEEE ELSEVIER ELSEVIER ELSEVIER SPRINGER ELSEVIER ELSEVIER IOS PRESS IEEE, SPRINGER KYUSHU TOKAI UNIVERSITY, WORLD SCIENTIFIC Articles

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Fig. 4. Top 10 journals contributing to research publications in type-2 fuzzy logic from 2000 to 2016

Fig. 5. Top WOS categories of research for type-2 fuzzy logic from 2000 to 2016

d) Category wise growth: Several categories are defined in the WOS Core, which help in determining the domain of the research papers. As far as the study is being concerned for type-2 fuzzy logic, it can be easily observed from Figure 5 and Table 5 that the topmost category i.e. the one having number of research papers is computer science artificial intelligence. It is well justified too since fuzzy logic itself is a part of artificial intelligence which is a branch of computer science.

e) Research Domain: All the 600 research papers that are under consideration for this study were analyzed to find in which research domain they lie. Table 6 presents the data for the same which is also visualized in Figure 6.

Table 5. WOS category wise research paper count S.No. 1 2

3 4 5 6 7 8 9

10

11

12 13

14

15

16 17

6

N° 2

18

WOS Category Computer Science Artificial Intelligence

Engineering Electrical Electronic

Computer Science Interdisciplinary Applications Automation Control Systems

Computer Science Information Systems Engineering Multidisciplinary

Operations Research Management Science Mathematics Applied

Computer Science Theory Methods

Instruments Instrumentation

Computer Science Cybernetics

Mathematics Interdisciplinary Applications

Mechanics

Engineering Mechanical

Engineering Chemical

Computer Science Software Engineering

Engineering Manufacturing

Others

Articles

Record Count 326

190 88 88 78 46 38 26 26

19

18 17

14

12

10 10 9

115

Table 6. Research area wise record count S.No.

Research Area

1

Computer Science

3

Automation Control Systems

2 4

5 6

7

8 9

10 11 12 13

Engineering

Mathematics

Operations Research Management Science Instruments Instrumentation Mechanics

Environmental Sciences Ecology, Science Technology Other Topics Energy Fuels

Physics, Robotics

Business Economics, Metallurgy Metallurgical Engineering

Education Educational Research, Materials Science, Telecommunications Medical Informatics

Record Count 439 260 88

42 38

19

14 8 7 5 4 3 2

f) Type of access: The papers under concern fall under two categories regarding the type of access: Open access and Paid access. Figure 7 helps in comprehending that 578 publications i.e. 96.3% of these research papers are having paid access while the rest 22 which constitute about 3.6% of the lot have open access.

g) Top authors according to record count: The top authors were identified according to the record count. It can be noted that J.M. Mendel is the top author with 47 research papers from the year 2000 to 2016. O. Castillo holds the second position with 34 research papers. This data is recorded in Table 7.

RECORD COUNT

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50

450 400

MENDEL JM; 47

350

45

300 250

40

200 150 100

35

50 0

CASTILLO O; 34

RECORD COUNT

30

25

MELIN P; 25

20

HAGRAS H; 20 ZARANDI MHF; 17

15 KAYNAK LAM O; 13HK;LI13HY; 13 CHEN SM; JOHN 11 RI;LIN 11 TC; 11 TURKSEN IB; 11 KAYACAN ZHAO E; 10T; 10

10

Fig. 6. Research area wise record count for research papers under consideration

5

0 0

open access

2

4

6

8

10

12

14

16

Fig. 8. Visualization of research paper record count for top authors

paid access

578

The year wise data for record count of the top 3 authors is recorded as in Table 8. Figure 9 visualizes this data, from which it can be concluded that J.M. Mendel published the maximum papers in 2007 while O. Castillo and P. Melin published their maximum papers in 2014.

96,30% 22

3,60%

1

9

2

Fig. 7. Types of access of research publications in type2 fuzzy logic from 2000 to 2016

8

Table 7. Top authors according to record count

6

Author

5

Record count

NOP

S.No.

7

Mendel JM

MENDEL JM

47

3

MELIN P

25

2

17

0

2

4

5

6

7

8

9

10

11

12

13

14

CASTILLO O

MELIN P 3

34

HAGRAS H

1

20

ZARANDI MHF KAYNAK O

2000

13

CHEN SM

11

JOHN RI

11

TURKSEN IB

11

KAYACAN E

10

ZHAO T

10

Table 8. Year wise record count of top 3 authors Year/Author Mendel CASTILLO MELIN

2006

2008

2010

2012

2014

2016

h) Most cited research papers: In this study, top 5 research publications have been identified according to the number of citations, as shown in Table 9. The research paper titled “Type-2 fuzzy sets made simple” is the most cited one (953 times). The average citation for this paper is 59.56. When analyzed, it was found that among all the papers that cited it, 355 belonged to the “Computer Science” research domain. Another research

11

LIN TC

2004

Fig. 9. Top 3 author’s research publications from 2000 to 2016

13

LI HY

2002

YEARS

13

LAM HK

CASTILLO O

4

1

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 3

0

0

0

1

1

0

8

0

3

3

6

3

3

3

2

3

0

0

0

0

2

0

0

1

1

4

1

2

2

3

5

1

3

0

0

0

0

2

0

0

1

2

3

1

2

3

3

8

5

4

Articles

7

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domain catering to the citations of this research paper is “Engineering”. This shows the popularity of this paper in various domains, as presented in Table 10.

Tables 11–14 show the research areas corresponding to papers ranked 2–5. The data from these is used to map the research papers to various research domains as shown in Figure 10.

Table 9. Citation and average citation count for top 5 research papers S.No.

Research Paper

Total Citation

Average Citation

1

Type-2 fuzzy sets made simple

953

59.56

782

43.44

647

53.92

2 3 4 5

Interval type-2 fuzzy logic systems: Theory and design Interval type-2 fuzzy logic systems made simple

Centroid of a type-2 fuzzy set A hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots

496

29.18

456

32.57

Table 10. Top 5 research areas corresponding to paper titled “Type-2 fuzzy sets made simple” S.No.

Research area

Record count

1

Computer Science

355

3

Automation Control Systems

54

2

4

5

Engineering

Mathematics

Operations Research Management Science

187

42 23

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Table 11. Top 5 research areas corresponding to paper titled “Interval type-2 fuzzy logic systems: Theory and design” S. No. 1

Research area

296

Automation Control Systems

65

2

Engineering

4

Mathematics

3 5

Record count

Computer Science

Instruments Instrumentation

185 26 18

Table 12. Top 5 research areas corresponding to paper titled “Interval type-2 fuzzy logic systems made simple” S.No. 1

Research area

277

Automation Control Systems

49

2

Engineering

4

Mathematics

3 5

Record count

Computer Science

Operations Research Management Science

154 35 33

Table 13. Top 5 research areas corresponding to paper titled “Centroid of a type-2 fuzzy set” S.No.

Research area

Record count

1

Computer Science

221

3

Automation Control Systems

34

2 4

5

Engineering

Mathematics

Operations Research Management Science

Fig. 10. Top 5 research publications from 2000 to 2016 mapped to their research areas 8

N° 2

122 25 17

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Table 14. Top 5 research areas corresponding to paper titled “A hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots” S.No.

Research area

1

Computer Science

3

Automation Control Systems

2

Engineering

4

Mathematics

5

Instruments Instrumentation

Table 15. Top 10 funding agencies

Record count 174 47

1

2

3

4

5

S.No. 1

11

Application Control

Clustering

3

Classification

5

Image Processing, Image Segmentation

4

12

Filtering

Funding agencies National Natural Science Foundation of China

138 27

18 17 9

17

Fundamental Research Funds for the Central Universities

14

National Science Council Republic Of China

11

Program For New Century Excellent Talents In University

10

Program for Liaoning Excellent Talents In University

7

8

National Nature Science Foundation of China, National Science Council Of Taiwan

9

Record count

65

National Science Council Taiwan

Australian Research Council, Bogazici University, Conacyt

10

8

Centre for Intelligent Systems Research Cisr At Deakin University, Tubitak

973 Program of China, Fok Ying Tung Education Foundation of China, Key Laboratory of Integrated Automation for The Process Industry Northeast University, Natural Science Foundation of Hebei Province of China, Program for Liaoning Innovative Research Team in University, Taiwan National Science Council, Zhujiang New Star

i) Top 10 Funding agencies: Various research papers were found to be funded by some agencies that provided a research grant to carry out that research. These are listed in the table below. The top 4 countries corresponding to these agencies were analyzed and the mapped (in yellow color) on the world map as shown in Figure 11. It can be comprehended that China grants the maximum number of research grants in this aspect.

Fig. 11. World mapping of top 4 countries with maximum number of grants from funding agencies j) Major Applications: Type-2 fuzzy logic has always been associated with research applications catering to domains like classification, clustering, image processing etc. In our

2018

Record count

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N° 2

Table 16. Major research applications of Type-2 fuzzy logic

2

110

S.No.

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study, we evaluated the major research applications of type-2 fuzzy logic which are tabulated in Table 16.

It can be inferred that control is the top most research application area, followed by clustering and classification.

k) Control terms and density plot: In order to gain better understanding of the research topics in the field of Type-2 fuzzy logic, some control terms need to be identified. These terms are the most frequently occurring author keywords that are extracted using both the title and the abstract of the research articles. In this paper, the 600 research articles from 2000 to 2016 have been analyzed and the density plot for the same has been created, as shown in Figure 12.

Figure 12 shows the density plot for all the control terms, simulated on VOSviewer. Control terms are the most frequently occurring terms. There are various clusters in this figure like the one centered on the control term “controller”. The fact that the terms like fuzzy logic controller, feedback error, filter etc are centered around it helps in establishing a fact that research papers concerned with controller also deal with filters, feedback errors, tracking error, stability condition etc. l) Inter-Country collaboration: The research publication record of these 600 research papers was manually analyzed to find out the country of each author, which assisted in knowing Articles

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Fig. 12. Density plot for identifying the control terms in the field of Type-2 fuzzy logic

Fig. 13. Inter-Country collaboration pattern

10

about the inter-country collaboration which is visualized as in Figure 13. The nodes of this graph represent the countries that participate in inter-country collaboration and the edges represent the number of publications for the same. The inter-country collaboration was found to be the maximum between UK and China, Turkey and Iran. It can be observed that Turkey and Canada have also participated actively in such collaborations. As an individual country, USA, UK, China, Turkey, Canada and Iran have been dynamArticles

ically involved in such inter country collaborations for research publications.

Regarding this inter-country collaboration that is illustrated in Figure 13, it is our believe that it will grow a lot in future years, as the type-2 fuzzy community in growing in many more countries and also it is consolidating in countries that already have research on type 2. There are other publications too in 2017 (published and ongoing papers) where other collaborations are flourishing, but they could not be repre-

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sented here due to the limitation of the time interval of 2000-2016 and the year 2017 is not complete to the moment.

4. Conclusion

This paper charted and mapped the progress of research work in the field of Type-2 fuzzy logic from the year 2000-2016 on the basis of 600 research papers retrieved from web of science. The primary aim of this study was to analyze the research work in type-2 fuzzy logic so that the growth of this discipline through the years is tracked. The study performed in this paper helps in answering various significant research questions like the growth rate of type-2 fuzzy logic in terms of research publications, top journals, authors, research grant agencies etc. The inter-country collaboration was also visualized to map the authorship patterns among various countries. All this would help the researchers in understanding the discipline in a more elaborate manner. This kind of work in the field of type2 fuzzy logic is the only one of its kind as it statistically highlights the various aspects related to it. As future work we envision updating this information about type-2 research in a few more years to verify the evolution and growth of the type-2 fuzzy area.

AUTHORS

Sonakshi Vij – Department of CSE, Indira Gandhi Delhi Technical University for Women, India.

Amita Jain – Department of CSE, Ambedkar Institute of Advanced Communication Technologies and Research, India.

Devendra Tayal – Department of CSE, Indira Gandhi Delhi Technical University for Women, India.

Oscar Castillo* – Department of Computer Science, Tijuana Institute of Technology, Mexico.

E-mail: ocastillo@hafsamx.org. * Corresponding authors

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Decentralized PID Control by Using GA Optimization Applied to a Quadrotor Submitted: 26th February 2018; accepted: 14th May 2018

Seif-El-Islam Hasseni: Latifa Abdou

DOI: 10.14313/JAMRIS_2-2018/9 Abstract: Quadrotors represent an effective class of aerial robots because of their abilities to work in small areas. We suggested in this research paper to develop an algorithm to control a quadrotor, which is a nonlinear MIMO system and strongly coupled, by a linear control technique (PID), while the parameters are tuned by the Genetic Algorithm (GA). The suggested technique allows a decentralized control by decoupling the linked interactions to effect angles on both altitude and translation position.

Moreover, the using a meta-heuristic technique enables a certain ability of the system controllers design without being limited by working on just the small angles and stabilizing just the full actuated subsystem. The simulations were implemented in MATLAB/Simulink tool to evaluate the control technique in terms of dynamic performance and stability. Although the controllers design (PID) is simple, it shows the effect of the proposed technique in terms of tracking errors and stability, even with large angles, subsequently, high velocity response and high dynamic performances with practically acceptable rotors speed.

Keywords: quadrotor, non-linear systems, decentralized control, PID, optimization, genetic algorithm.

Abbreviations: BMW: Best-Mates-Worst technique DE: Differential Evolution ISE: Integral Squared Error GA: Genetic Algorithm MIMO: Multi-Input Multi-Output system LQR: Linear Quadratic Regulator OS4: Omni-directional Stationary Flying OUtstretched Robot PID: Proportional Integral Derivative Controller PSO: Particle Swarm Optimization RD: Rotor Dynamic UAV: Unmanned Aerial Vehicle

Nomenclature: E: B: RBE : x: y: z:

Earth frame Body frame Transformation coordinates matrix from body frame (B) to earth frame (E) Translation coordinate in x axis Translation coordinate in y axis Altitude coordinate

φ, θ, ψ: Roll, Pitch and Yaw Euler-angles, respectively.

V: Vector of linear velocity Ω: Vector of angular velocity qi: Coordinate of degrees of freedom = [x, y, z, φ, θ, ψ] L: Lagrangian term E k: Kinetic Energy Ekr: Kinetic-Rotation Energy Ekt: Kinetic-Translation Energy Ep: Potential Energy Г: Vector of non-conserved forces and torques Fx, Fy, Fz: Force on x, y and z axis, respectively. τ: Vector of torques τf : Vector of forces torque τg: Vector of gyroscopic torque τφ,τθ,τψ: Torque on x, y and z axis, respectively. f i: Force of ith rotor Ti: Thrust force of ith rotor (countre-torque) J: Inertia matrix Ix, Iy, Iz: Inertia on x, y and z axis, respectively. ωi: Angular speed of ith rotor ωdi: Desired angular speed of ith rotor Ω r: ω1 – ω2 + ω3 – ω4 J r: Rotor inertia b: Thrust factor d: Drag factor l: Arm length m: Quadrotor mass g: Gravity constant Ui: The ith control input Udi: The desired ith control input ux,uy: Virtual inputs of x and y subsystems k: Adaptive decoupler factor kp, ki, kd: Proportional, Integral and Derived gains of PID, respectively.

1. Introduction

The quadrotor is an Unmanned Aerial Vehicle (UAV) that is considered as an aircraft without a pilot, and which can be driven by a human at a ground control station or can fly autonomously by known flight trajectory. Quadrotors were used at the beginning of their appearances in the military domain for monitoring or reconnaissance missions. Civil applications made their appearances later, as it is the case of the monitoring of the road traffic. The quadrotor has some advantages compared with other UAVs, as vertical take-off landing, stationary and low-speed flight, which make it highly maneuverable, with precise navigation in difficult or dangerous areas. For these rea-

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sons, the researchers were carried about this aerial robot to develop the control algorithms with successful comportments. However, the quadrotor is a characteristically nonlinear, underactuated mechanical system; with six degrees of freedom (three for rotation and three for translation) and just four inputs, its dynamic is complex because of its strong coupling between the translational and rotational subsystems, which don’t allow releasing its controllers with simple ways.

This nonlinear system motivates to design a complex nonlinear control algorithm. A large number of control methods are used to solve the attitude stabilization and trajectory tracking problems. In [1] a sliding mode control had been developed for the full system, a robust terminal sliding mode is used for the full actuated subsystem (rotational) and the simple sliding mode for the under-actuated subsystem (translational). Both integral backstepping and sliding mode have been used in thesis [2], [3] and both of them conclude that the combination between backstepping and integral action is the best for quadrotor stability against disturbances and model uncertainties. A significant number of researches were based on combination between two techniques to design the full controller; in [4], a combination of sliding mode and integral backstepping is used on both of rotational and translational subsystems. In [5] an adaptive sliding mode control is developed against the actuators failures. In [6] a sliding mode control was applied with switching gains adaptively by fuzzy logic system based on information from the sliding surfaces, but just for stabilizing the attitude subsystem. An online optimization was associated to improve the backstepping designed control [7]. The predictive control is one of the effective techniques of the nonlinear systems, it is developed in [8], [9], where the authors have used the piecewise affine systems to derive the predictive model. The optimal control is also a very effective way of controller’s synthesize of unmanned aerial vehicles [10], [11], in [12] a linearized model of the quadrotor rotational subsystem was derived for applied the LQR control. The most classical linear controller, which is the PID have been used in the unmanned aerial vehicles [13] because of its simplicity.

In case of quadrotor, it is used in [12], [14], [15], [16], the linearized model was inquired, subsequently it is used for just the stabilization of the attitude subsystem, so unfortunately only for the small range of angles and low dynamics. Some works are not based on classical sensors; GPS, accelerometer and gyroscope, but, they applied vision-based control [17], [18] where the only sensor is the camera and which use the image processing techniques to measure the velocity and make more precise movement.

In this paper the classical controller PID is proposed to control the full system. In contrast to other works in literatures, our dynamic system isn’t limited to small range of angles but can exceed to nonlinear range of angles for getting a high dynamic. In addition we are motivated to use one of effective meta-heuristic optimization approach; Genetic Algorithms [19], which are applied in a large way of off-line controlArticles

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lers’ design, like control structure design in fuzzy logic controller or PID parameters tuning. In this case, the synthesized structure is kept as classical one, but the parameters are optimized by GA. The contribution of this paper can be presented as follows; a) A decentralized control is applied; we designed each controller separately from others. b) To realize that, we need to decouple the interactions between the translation and rotation coordinates. c) For the decoupling between the altitude and the attitude, we added an adaptive online factor in the altitude input. Thereafter we designed the four controllers separately, the parameters tuning is used by Genetic Algorithm, no need to linearize the model and limit the system to work on small angles, subsequently limit on low dynamic performances. The optimization process is done with both small and big angles. d) After tuning the four controllers of the actuated subsystem, the inner loop was locked with a view to decouple the influence of the rotation on the translation position and design the controllers of underactuated subsystem, which is presented by the x and y position, their controllers were also tuned by the Genetic Algorithm. This virtual control laws are used to compute the desired roll and pitch angles and used them on the full actuated subsystem.

The paper’s structure is as follows: in the next section, the mathematical model is obtained by using Euler-Lagrange approach, including the aerodynamic effects and the rotors dynamics. In section 3, we present the proposed controller structure and the strategy of decoupling the interactions, to permit the application of decentralized control. In section 4, we introduce the use of GA as a solution to optimize some parameters in order to improve the quality of our controller. In section 5, we present and discuss the obtained results by simulating 3D trajectory tracking and the rotors speeds states, and lastly our conclusion in section 6.

2. System Modeling

Before the control development, the mathematical model for this mobile robot is needed. In this work, we propose to use Euler-Lagrange technique because of its advantages, related to the fact that it is based on potential and kinetic energies, and also dynamic equations in symbolic closed form. This technique is the best for study of dynamic properties and analysis of control schemes [20], where the robot is supposed as a rigid closed system: In the quadrotor, we can find two frames: the inertial frame (E: earth) and the mobile frame, who is fixed in the quadrotor (B: body).

The distance between the earth frame and the body frame describes the absolute position of the mass center of the quadrotor r = [x y z]T. The rotation R from the body frame to the inertial frame describes the orientation of the quadrotor. The orientation of the quadrotor, is described using roll, pitch and yaw

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Fig. 1. Reference frames of Quadrotor angles (φ, θ and ψ) which represent the orientation about the x, y and z axis respectively. The matrix of rotation is defined by [21]: RBE

R z,

R y,

Where:

1 0  R ( x,ϕ ) =  0 cϕ  0 sϕ   cθ 0  R ( y,θ ) =  0 1  −sθ 0   cψ  R ( z,ψ ) =  sψ  0 

R x,

(1)

0   −sϕ  (2) cϕ  sθ   0  (3) cθ 

−sψ cψ 0

0  0  (4) 1 

Then:

sϕ sθ cψ − cϕ sψ cϕ sθ cψ + sϕ sψ   cθ cψ   R =  cθ sψ cosϕcosψ + sϕ sθ sψ cϕ sθ sψ − sϕcψ  (5)  − sθ  sϕcθ cϕcθ   Where s and c denote sin and cos respectively.

The vector of angular velocities of quadrotor is defined in body frame Ω = [ωxω yωz ]T . It is given depending on the derived Euler angles [ϕ θ ψ ]T that are measured in the inertial frame. This transformation is derived as follows [s]: Ω – angular velocities E B

 ϕ  0 0 −1 −1 −1 Ω� ==  0  + ( R ( x ,ϕ ) )  θ  + ( R ( x ,ϕ ) ) ( R ( y ,θ ) )  0  0 0 ψ       

(6)

ϕ −ψ sinθ     � ==  θ cosϕ +ψ sinϕcosθ  (7) Ω ψ cosϕcosθ − θsinϕ   

Lagrange law is given as [20]:

d  ∂L  ∂L = Γi (8)  − dt  ∂qi  ∂qi

Where: qi – the measured variable, which represents each one of the output variables: x, y, z, φ, θ and ψ. Γi – the vector of non-conserved forces or torques (Fx, Fy, Fz, τφ, τθ and τψ).

2.1. The Lagrangian

Defined with kinetic energy and potential energy as follow:

L = E k − E p = E kt + E kr − E p (9)

Ek – the kinetics energy, Ekt – for translation kinetic energy and Ekr – for rotation kinetic energy, Ep – the potential energy, then we have: L=

m 2 1 22 V + J� Ω –− mgz mgz (10) 2 2

J – the matrix symmetric of inertia, Ix, Iy, and Iz are inertias on x, y and z axis, respectively.

V – the vector of translation velocities. m – the mass of quadrotor. g – gravitional constant.

J

Ix 0 0

0

Iy 0

0 0

Iz

(11)

By applying (7) and (11) in (10), we find: m 2 1 2 x + y 2 + z 2 + I x (ϕ −ψ sθ ) + 2 2 2 1 1 + I y θcϕ +ψ sϕcθ + I z ψ cϕcθ − θsϕ 2 2

L=

(

)

(

)

(

)

2

− mgz (12) Articles

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2.2. Non-Conserved Forces and Torques Returning to Fig. 1, the motors (M1, M3) turning the opposite direction of the motors (M2, M4) and each motor generates a thrust force fi = b.ωi and a drag torque Ti = d.ωi. Where b and d are thrust and drag coefficients respectively. ωi: Is the rotation speed of motor Mi.

2.2.1. Non-conserved forces: The thrust of quadrotor is the sum of forces fi.

4

4

i =1

i =1

F = ∑ fi = b.∑ωi2

(

)

(13)

FB

(14)

0 0

F

FB – the vector of forces in the body frame.

Because, we need the forces in earth frame, we get it by transformation matrix: FE = R.FB, then:

( (

 cϕ sθ cψ + sϕ sψ ) b ω12 + ω22 + ω32 + ω42  Fx   (   FE =  Fy  =  ( cϕ sθ sψ − sϕcψ ) b ω12 + ω22 + ω32 + ω42  F    z ( cϕcθ ) b ω12 + ω22 + ω32 + ω42 

(

)

2.2.2. Non-conserved torques: Represented by:

)  ) 

  (15)

τ = τf – τg (16)

Where: τf: Is the vector of forces torques, which contains: – Torque according to x (roll), that depends on the difference between f4 and f2.

– Torque according to y (pitch), that depends on the difference between f3 and f1.

– Torque according to z (yaw), which is obtained by the difference between the sums of torques T1 and T3, and that of T2 and T1.

And τg – the gyroscopic torque.

( (

) )

 l b ω42 − ω22  l ( f 4 − f2 )     τ f =  l ( f3 − f1 )  =  l b ω32 − ω12  T − T + T − T    1 2 3 4  d ω12 − ω22 + ω32 − ω42 

(

)

   (17)   

τg – Each rotor may be considered as a rigid disc in rotation around his own vertical axis with a rotation ωi. The vertical axis itself moves during the rotation of the quadrotor around of one of their three axes.

This action product an extra torque called gyroscopic given as follow [17]: τg

36

 00   θθ JJ ©©  44    r r r r  ==∑ ∑=1©Ω©∧∧JJr r 00 ==−−ϕϕJJr r©©r r i =i 1 ((−−11))i +i +11ωω   00  i i 

Articles

Where:

� Ωrr = ω1 − ω2 + ω3 − ω4

( (

) )

2 2    τ ϕ   l b ω4 − ω2 − θ J r � r   τ =  τ θ  =  l b ω32 − ω12 + ϕ J r � r  τ    ψ  d ω12 − ω22 + ω32 − ω42 

(18)

)

2018

(19)

And Jr is the inertia of the rotor.

We apply (17) and (18) in (16) we get:

(

⇒ F = b ω12 + ω22 + ω32 + ω42

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   (20)   

From (15) to (20) and by applying the Lagrangian law presented in (8), we obtain after simplification the following mathematical model.

1  x = ( cϕ sθ cψ + sϕ sψ )U1   m   y = 1 ( cϕ sθ sψ − sϕcψ )U 1  m  1   z = ( cϕcθ )U1 − g  m  f ( X ,U ) =  ( I y − Iz ) θψ   + l U − J r � r θ ϕ = 2 I Ix Ix x   ( I − I )   + l U + Jr �r ϕ θ = z x ϕψ 3 Iy Iy Iy   ( Ix − I y ) ϕθ  1   + U4 ψ = Iz Iz 

(21)

Where: The quadrotor is controlled by the rotation speed of motors (ω1 ω2 ω3 and ω4), then the vector of control inputs is expressed as a function of rotation speed as follows:

(

)

U1 = b ω12 + ω22 + ω32 + ω42   U2 = b ω42 − ω22   U3 = b ω32 − ω12   2 2 2 2 U 4 = d ω1 − ω2 + ω3 − ω4

( (

) )

(

(22)

)

The inputs of the system are altitude control (U1), roll control (U2), pitch control (U3) and yaw control (U4).

The states vector X: X

y, y, z, z, , , , , , x, x,

The inputs vector U:

U = [U1 ,U2 ,U3 ,U 4 ]

T

T

The equilibrium points of (21) satisfy:

f ( X eq ,U eq ) = 0

(23)

(24) (25)

Journal of Automation, Mobile Robotics & Intelligent Systems

X eq

Where:

x eq , x eq , yeq , y eq, zeq , z eq ,

U eq = U1eq ,U2eq ,U3eq ,U 4eq 

eq

, eq ,

eq

, eq ,

eq

, eq

T

Solving (25) results in the stationary points

X eq = [ x ss , 0, y ss , 0, z ss , 0, 0, 0, 0, 0,ψ ss , 0]

T

U eq = [ g m, 0, 0, 0]

T

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T

(26)

(27) (28)

(29)

Where: xss, yss, zss, ψss ∈ R, we notice ss to mean steady state.

In addition, there is a limit in roll (φ) and pitch (θ) angles: −π π ≤ ϕ ,θ ≤ 2 2

(30)

As a simulation, we use the parameters of the OS4 project (Omni-directional Stationary Flying OUtstretched Robot) [2]: Table 1. OS4 parameters Parameter

Name

Value

Unit

m

Mass

0.65

Kg

Thrust coefficient

3.13 10–5

N.s2

Drag coefficient

Inertia on x axis

7.5 10–7

Inertia on y axis

7.5 10–3

N.m.s2

7.5 10–3

Kg.m2

l b d Ix Iy Iz Jr g

Arm length

Inertia on z axis Rotor inertia

Gravity constant

3. System Control

0.23

1.3 10–2 6 10-5 9.8

m

Kg.m2 Kg.m2 Kg.m2 m.s-2

 ωd 1 =   ωd 2 =    ωd 3 =   ωd 4 = 

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1 1 1 Ud1 − Ud3 + Ud 4 4b 2b 4d 1 1 1 Ud 1 − Ud 2 − Ud 4 4b 2b 4d

(32)

1 1 1 Ud1 + Ud3 + Ud 4 4b 2b 4d

1 1 1 Ud1 + Ud 2 − Ud 4 4b 2b 4d

Unlike the altitude and orientation of the quadrotor, its x and y position can’t be directly controlled using one of the four controls laws Ud1 through Ud4. On the other hand, the x and y position can be controlled through the roll and pitch angles. Then we suppose a virtual controls ux and uy where:

x ux

y u y

1 U d1 c s c m 1 U d1 c s s m

s s

s c

(33)

The desired roll and pitch angles (φd and θd) can be computed from the translational equations of motion [3]. The expression (33) is given by: 1  ux = m U d 1 ( cosϕd sinθd cosψ + sinϕd sinψ )  u = 1 U ( cosϕ sinθ sinψ − sinϕ cosψ ) d d d  y m d 1

(34)

m  ϕd = U ( ux sinψ − u y cosψ )  d1  m θ = (u cosψ + u y sinψ )  d U d 1 x

(35)

After simplification, because we have considered that the quadrotor is operating hover flight mode, we obtain the translation to rotation converter as:

We have now two control loops. Inner loop which contains the altitude (z) and the three angles (φ, θ and ψ) and the outer loop which contains the translation position coordinates x and y (Fig. 2).

3.1. Modeling for Control

3.2. Control Using PID Technique

In fact, each rotor has a non-real time dynamic response, so we have to consider their dynamic. In OS4 project [2] the rotors have a dynamic transfer function (RD) as:

The control strategy proposed in this work is the PID technique. The factors that attracted industries to choose PID could be due to low cost, easy to maintain, as well as simplicity in control structure and easy to understand [22]. Once the set point has been changed, the error will be computed between the set point and the actual output. The error signal is used to generate the proportional, integral and derivative actions, with the resulting signals weighted and summed to form the control signal applied to the plant model [22].

The six controls laws applied to quadrotor are:

RD ( s ) =

ωi 0.936 = ωdi 0.178s + 1

(31)

Where: ωi – The actual angular velocity for each rotor. ωdi – The desired angular velocity for each rotor.

Most researchers who work in control theory, applied on quadrotor and its simulation, didn’t consider rotors dynamic in their works. We notice the desired inputs Udi which will be converted to rotors inputs as follow:

U q = k p ( qd − q ) + kd ( qd − q ) + ki ∫ ( qd − q ) dt

Where: q = [x y z φ θ ψ] and Uq = [ux uy U1 U2 U3 U4].

(36)

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Fig. 2. Control structure of quadrotor For each variable, we can design its controller independently of the others; for that, each loop of variable is being a mono-variable system. In this case, we can tune the parameters of each variable alone but, there are constraints related to the couples between outputs variables, so we use decentralized control.

3.3. Decentralized Control

The steps followed to control a MIMO system by decentralized control are: analyze the interactions, then decoupling them (for being weak) [23].

We can characterize two types of interactions: (a). The interactions between the translation coordinates (x and y) with the roll and pitch angles, as we notice in section (3.1). (b). The interactions between the variables of inner loop (Fig. 2), the altitude (z) and (φ, θ) are strongly coupled. The rest of interactions are negligible; they don’t have direct influences.

For decoupling (a) we apply the sequential design [24], we design the controllers of the inner loop (altitude and three angles, as presented in Fig. 2), then the controllers of the outer loop (x and y) as a cascade design.

On the other hand, for (b), the influence of (φ, θ) in the altitude (z) is estimated (cosφcosθ). Then, to weaken this influence the idea is to replace the input such as: Ud1 by U d1

U d1

kUd1

Where: k is the adaptation coefficient:

(37)

1 (38) cosϕ cosθ We can now, apply the genetic algorithm technique for each controller. k=

4. Genetic Algorithm

38

For the reasons that the quadrotor system is non-linear and some outputs (x and y) cannot be directly controlled (they are underactuated), we cannot tune the PID parameters with one of classical approaches. Different methods can be used to obtain optimal control of robots [25], [26] in particular, optimal parameters of the PID, among them; we can find meta-heuristic techniques of optimization such as DE Articles

[27], PSO [28] and GA [27], [29] which were used in to solve some specific problems. We propose in this work, to use GA as an optimization technique to obtain the best PID's parameters. The use of such algorithm allows the control design of the system without the need of its linearization, which is not provided by using classical methods. Likewise, there are some degrees of freedom which are underactuated; as the position according x and y that have virtual controllers (ux and uy) and not direct inputs on the system. These last, can’t be provided by classical methods also. The application of the proposed technique will improve the quality of our control system.

Genetic algorithms are group of operations which are extracted from selection evolution nature, proposed firstly by Holland [30]. They were used extensively in artificial intelligence with computer science problems. The process of GAs starts by creating a random population of solutions, which are encoded in chromosomes, then, they are evaluated according to the performance of the problem, by using particular criteria, called fitness function. In order to improve the quality of solutions, to find the best ones, there are some operations which must be applied on the population; selection, crossover and mutation.

4.1. Implementation of the GA

In this work, we create randomly an initial population, which contains fifty real encoding chromosomes.

The fitness function is chosen as the Integral of the Squared Error (ISE). τ

fitness = ISE = ∫e2 ( t ) dt 0

(39)

The selection of the parents to be crossed is applied by using BMW determinist scheme [31], [32].

In this case, the best solution, in the current population, mates the worst one and the less good mates the less worse. Because the encoding is real, a linear crossover presented by Wright in [33] is used.

In a GA, after the crossover operation, a new population of children is obtained. This population must to be slowly disturbed by applying the Gaussian mutation operator with a probability of 4%. The next gen-

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eration, is then made by taking the fifty best solutions of both populations of parents and offspring.

This algorithm that includes all these operations is applied for thirty iterations, which the number of generations.

In this work, there is no determined formula for error function, so we use (ode45.m) function because the system is evolution. Then, the optimization program sends the chromosome parameters to this function and it receives its fitness function. However, we have to discrete the evolution (sample time Ts = 0.05 s, simulation time = 10 s).

Fig. 3 presents the diagram of control optimization with GA applied for each controlled variable.

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Table 2. Optimums PID's parameters obtained by GA Variable

Kp

Ki

Kd

z

9.6135

4.0704

4.8897

0.15

0

0.15

φ, θ

0.4521

ψ x, y

5. Results

0

4.8095

0.3622

0

0.8967

After the design of the controllers, we have to test their performance for the trajectory tracking. Table 3 presents the suggested trajectories’ dynamics and their conditions.

Table 3. Trajectories’ conditions and dynamics [x, y, z, φ, θ, ψ] = [0,0,0,0,0,0]

[x, y, z, φ, θ, ψ] = [0,–2,2,0,0,0]

Actuators’ saturation

[0, 400] rad/s max = 370 rad/s

[0, 400] rad/s max = 296 rad/s

1st trajectory

Description Of Trajectory

Attitude’s saturation

The GA algorithm is applied as an optimization strategy, under MATLAB environment, to the quadrotor model OS4 [2] with parameters presented in Table.1, for each variable except the yaw subsystem.

This last, could be linear with hypothesis that the quadrotor is symmetric (Ix = Iy).

The equation from the model (21)  ( Ix − I y ) ϕθ 1    + U 4  will be: ψ = Iz Iz  

1 ψ = U 4 Iz

2nd trajectory

4.2. Optimums Parameters obtained by GA

Mode 2

Initial condition Period

Fig. 3. Diagram of applied GA

Mode 1 [0, 7]s

zd = 2 xd = 0 yd = –2 ψd = 0

[7, 60]s

zd = t  2π  x d = 2sin  t   30 

 2π π  yd = 2sin  t −   30 2   2π  ψ d = π sin  t   30 

–20° < φ, θ < 20°

Initial condition

[x, y, z, φ, θ, ψ] = [0,0,0,0,0,0]

[x, y, z, φ, θ, ψ] = [0,0,3,0,0,0,0]

Actuators’ saturation

[0, 400] rad/s max = 400 rad/s

[0, 400] rad/s max = 300 rad/s

Period

Description Of trajectory

Attitude’s saturation

[0, 7]s zd = 3 xd = 0 yd = 0 ψd = 0

[7, 37]s

zd = –0.1t  2π  x d = 4 sin  t   8 

 2π  yd = 4 sin  t   8  ψd = 0

–90° < φ, θ < 90°

5.1. First trajectory (40)

The yaw has been controlled with classical PD. As a result, we obtain a PID controller for the altitude and PDs for the others controllers, the optimum parameters are presented in Table 2. These PID's parameters are applied to simulate the control of the quadrotor under MATLAB/Simulink.

The suggested trajectory is a spiral, where the quadrotor makes a circle in 30 seconds and up (1 m/s). Fig. 4 shows the tracking of the desired trajectory for each position coordinate (z, x and y) and the rotation of the quadrotor with the yaw angle too.

Fig. 5 shows the tracking of desired trajectory in 3D plan. The rotors angular speeds are shown in Fig. 6.

Firstly, from 0 to 7 seconds the quadrotor is stabilized in starter position (x, y, z) = (0, –2, 2). Starting mode needs a big energy to up against the effect of Articles

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Fig. 4. z, x, y and yaw (ψ) responses (1st Trajectory)

Fig. 5. Desired and actual trajectories in 3D (1st Trajectory)

40

gravity. Fig. 6 indicates that the angular speed of rotors in this mode can move up to 400 rad/s.

After that, the trajectory begins; a linear movement in altitude with a speed of about 1 m/s and for translation motion, the quadrotor makes a complete circle of 2 meters of radius in 30 seconds. In the same time, the quadrotor turns on around itself, and Fig. 4 shows the tracking of the desired trajectory with a precision at the end of the movement, of all variables (altitude z, translation motion of x and y, the yaw angle). In this mode, low energy is applied, as showed in Fig. 6, where the rotors turn on around 220 rad/s.

These results show clearly the possibility of our sysArticles

tem to follow trajectories and to design x and y controllers as an underactuated case.

5.2. Second Trajectory

Unlike the first trajectory which is hovering as a circle with small roll ant pitch angles, we will show the effect of the proposed technique in the case of big angles. The suggested trajectory is as follow; after the quadrotor rises in 3 meters of altitude, on the second 7 it begins to down slowly with 10 cm/s; in the same time, it balances diagonally between –4 meters and 4 meters in x axis and y axis, the period of bal-

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Fig. 6. Rotors angular speeds (ω1, ω2, ω3 and ω4) in rad/s (1st Trajectory)

Fig. 7. z, x and y responses (2nd trajectory) anced is just 8 seconds. Fig. 7 shows the tracking in z, x and y axis, where we notice that the error in z is null but in the x and y axis is not in the peaks. Fig. 8 shows the trajectory tracking in 3D plan to make the trajec-

tory clearer. Fig. 9 shows the rotors’ angular speed, which need a big energy in the starter mode (the speed reaches 400 rad/s) but when the quadrotor stabilizes in the balanced trajectory they are around Articles

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Fig. 8. Desired and actual trajectory in 3D plan (2nd trajectory)

Fig. 9. Rotors’angular speeds ω1, ω2, ω3 and ω4 in rad/s (2nd trajectory) 220 rad/s. Fig. 10 shows the roll (φ) and pitch (θ) angles that the quadrotor makes this trajectory by them; we notice that they are between –55° and 50° which makes the system absolutely nonlinear because the linear interval is between –10° and 10°

6. Conclusion

42

In this paper, we have proposed the control of a non-linear, MIMO system, which is the quadrotor. The mathematical model of this system has been developed in details, including its aerodynamic effects and rotors dynamics. The quadrotor is a non-linear complex system, strongly coupled Articles

and under-actuated. A linear PID control technique was developed and synthesized, according to the decentralized control approach, for which, we have decoupled the interactions between the quadrotor variables. A complete simulation was implemented on MATLAB/Simulink tool relying on the derived mathematical model of the quadrotor system. The tuning of controllers parameters are done using GA, where the objective function is the dynamic response of the system in term of ISE. The response’s simulation of the system presents a path tracking in 3D plan; where we notice that the tracking trajectory stabilizes, after a few seconds, this time is what the quadrotor needs to stabilize the altitude

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Fig. 10. The roll and pitch angles responses (2nd trajectory) because a biggest energy is required in this period.

Although the simplicity of the controllers’ design (PID), the effect of proposed technique and designing the controllers by GA is shown in terms of tracking errors and stability, even with the big angles, subsequently, high velocities response and high dynamic performances, this is in contrast to works that used the PID with linearized model [12], [14], [15], [16], the variation of the angles is limited between –10° and 10° which is the linear range, so, limit the dynamics by low performance. Moreover, compared of [34], which is a similar work, when the GA was used to tune the PID’s parameters, but in the evaluation’s phase, they didn’t use a high dynamic trajectory and the angles never exceeded the linear range. Even some works, where a nonlinear control is designed in, the used trajectory to evaluate the controller doesn’t contain high dynamics with big range of angles; [1], [6], [7], and [35].

Finally, the shown rotors’ speeds are acceptable in reality; so we have the possibility to apply these controllers in real system to get a trajectory tracking with a precision as presented in 3D trajectory and we don’t need more powerful actuators.

AUTHORS

Seif-El-Islam Hasseni* – Energy Systems Modeling Laboratory (LMSE), Electrical Engineering Department; University of Biskra, BP 145 RP, 07000, Biskra, Algeria.E-mail: seif.hasseni@univ-biskra.dz.

Latifa Abdou – Identification, Command, Control and Communication Laboratory (LI3CUB), Electrical Engineering Department; University of Biskra, BP 145 RP, 07000, Biskra, Algeria. *Corresponding author

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Calibration Low-Cost Cameras with Wide-Angle Lenses for Measurements Submitted: 7th November 2017; accepted: 28th May 2018

Damian Wierzbicki

DOI: 10.14313/JAMRIS_2-2018/10 Abstract: So far too many examinations concerning the possibility of the application action cameras weren’t conducted in UAV photogrammetric studies. However in the recent time development of the action cameras production technology he let receive sensors which are light, userfriendly, and most importantly can gain images and video sequences of high resolution. The essential meaning has taking into account the camera’s influencing factors in UAV photogrammetry. Due to the fact that the action cameras are non-metric cameras the significant influence on the quality of photogrammetric studies has a stability internal orientation of elements of such cameras. Within the framework of carried out research calibration of five GoPro Hero 4 Black. The calibration of cameras was carried out on different calibration tests in the GML Camera Calibration Toolbox and Agisoft Lens software. Different calibration setups and processing are presented and discussed in this article. Additionally a repetitiveness of achieved results of the calibration was examined in five GoPro cameras Hero 4 Black. Dedicated calibration templates in the form of chessboards were used to the calibration. As a part of the research a comparative analysis of the results have been done. Based on performed examinations a repetitiveness of determined internal orientation elements was checked under different video acquisition modes.

Keywords: sensors, measurements, camera calibration, videos, accuracy

1. Introduction Increasing the availability of high-resolution (4K resolution and more) action cameras such as GoPro Hero 4 equipped with wide angle lens (fish-eye) caused the noticeable height of applying sensors of this type in UAV photogrammetry. Applying cameras of this type is much more difficult in the relationship to non-metric compact cameras. Broad field-of-view of assembled lenses in action cameras very often cause difficulty in the accomplishment of calibration of such camera and this her determination of reliable elements of internal orientation. Standard wide-angle lens (fish-eye lens) assembled in action cameras are characterized by short lengths of focal lengths and high radial distortions. So far in UAV photogrammetry dominated non-metric amateur cameras [1], [2],

[3], [4], [5] which were equipped in low cost lens low-class. Carrying out the calibration of such cameras equipped with normal angle lens very often didn’t provide the internal orientation elements for the repetitiveness of this elements. In case of action cameras with wide-angle lenses determination of internal orientation elements can be even more difficult [6].

Non-metric cameras calibration lets appoint elements of the internal orientation to the purpose of extract accurate 3D metric information from their images [7].

Parameters that is determined are: calibrated focal length (ck) of the lens, the coordinates of the center of projection of the image (xp, yp), the radial lens distortion coefficients (K1, K2, K3) [8] and tangential distortion coefficients (P1, P2). Therefore it is recommended to pre-calibration action cameras in order to ensure reliable elements of internal orientation to allow for precise photogrammetric reconstructions. The calibration of cameras and the evaluation of the high credibility of appointed elements of the internal orientation are still main and with point at issue in the area of research of the development of photogrammetry [9] including UAV photogrammetry. Unknown internal geometry is a main problem in sensors equipped with wide-angle lenses [10], [11]. The full review of camera calibration methods and models became encompassed in many the publications [9], [12], [13].

Results included in above articles constitute specific summing up experience associated with using digital cameras to the photogrammetric measurement.

It was then presented in interpretation of different configurations, parameters and analysis techniques of the cameras associated with the calibration. They also presented well-known photogrammetric systems from implemented models of the calibration of cameras and increasing 3D accuracy algorithms through the self-calibration bundle adjustment. The issues associated with the calibration of cameras in the recent time also became a research topic in Computer Vision (CV) field. Research assembles on full automatism of the process of calibration [14] on the basis of linear approaches with simplified imaging models [15]. The first works above these methods concerned pinhole camera model and included the modelling radial distortion [16], [17], [18]. At present algorithms of the calibration of cameras were broadened by libraries open source ready answers e.g. OpenCV containing already ready solutions. These algorithms are based on detecting the substantial amount of points on the flat test field of the type chessboard [19], [20]. However

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the use of flat objects for camera calibration does not provide such high accuracy as 3D test fields. However in most applications applying 2D test fields of type chessboard is acceptable [21], [22]. In the photogrammetric presentation the camera calibration can be above methods regarded acceptable. Correct designing measurements, correct photographing calibration tests and achievement image measurement and bundle adjustment let to carry out the accurate and correct calibration for the majority of compact digital cameras. Research relating to photogrammetric measurements with the wide-angle lenses application are carried out from several years [23]. The pictures obtained with the use of such lenses require preceding of processing approach. The imaging process is not compatible with a central projective model. It is necessary made of the preliminary corrections of distortions caused by the large impact of the distortion. In the other approach it is possible to implement wide-angle lens model in self-calibration. Many researchers proposed the various procedures associated with the calibration of this type of cameras [6], [23], [24] they were primarily based on the geometry of the epipolar line and equidistant rectification of distorted images to be applied to central projective. In case of calibration procedures it is possible to adopt the assumptions similar to the calibration action cameras like GoPro Hero 4 which are equipped in wide-angle (fisheye) lenses. To this time several research work related to the calibration and application of these cameras in the photogrammetric perfected has been carried out [8], [25], [26]. In above articles results relating to calibration of the GoPro Hero 3 and GoPro Hero 4 cameras and of their uses to photogrammetric goals were discussed. On the basis of research it is recommended to use the highest possible resolution and where a high accuracy of applying 3D calibration field is necessary. Procedures of the calibration were based on algorithms from OpenCV and comparative analyses with the Agisoft Lens software. 2D and 3D calibration field were applied to calibration. In another approach [26] proposed a calibration procedure for the sequential method where the approximate elements of the orientation of an internal were treated as first value. Applying of action cameras for the completion of photogrammetric studies was also discussed in [25], where the Agisoft Photoscan software and Photomodeler were used to the extraction of the thick cloud of points from video frames. The results of research assembling on the calibration procedures of five cameras GoPro hero 4 Black applied - photogrammetric tests are presented in this article. Within the framework of research various procedures of the calibration of cameras were presented in order to estimate the stability of elements of internal orientation. In the framework of the research work calibrations of five cameras were performed. Each calibration has been carried out in the five measurement series in software Agisoft Lens and Camera Calibration Toolbox. The obtained results have been analyzed for their stability. The whole of the article was divided in five parts, and at the end a list of literature was put.

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2. Camera Calibration – Mathematical Model Camera calibration is intended to reproduce the geometry of the projection of the center on the basis of photographs of this camera. The calibration of the camera are: – calibrated focal length – ck; – the position of the center of the views in relation to the pictures, determined by x0 i y0 – image coordinates of the principal point; – distortion of lens – factors of radial distortion: K1, K2, K3 and tangential: P1, P2.

In case of action cameras they have one large FOV in wide angle viewing mode. In this case the calibration process plays a very important role in order to model distortion in the lens. Model of internal orientation in the photogrammetric presentation applied in researches computer vision Agisoft Lens, Camera Calibration Toolbox and Open CV based on a modified mathematical of the Brown Calibration model [27].

Parameters of the distortion are determined in the relationship of the principal point. Character of the radial distortion depends on the structure of lens and situating the aperture in it. The value of the distortion depends on the angle α and lengths of the ray r:

∆r = r − ck ⋅ tgα (1) where: Δr – the size of the radial distortion r – radial distance; ck – calibrated focal length; α – field angle.

Course of the crooked radial distortion approximate is through the polynomial odd power lengths of the ray [28]:

∆r = K 1r 3 + K 2r 5 + K 3r 7 + ... (2)

r=

where: K1, K2, K3 – polynomial coefficients of radial distortion; r – radial distance.

( x − x0 ) + ( y − y0 ) 2

2

(3)

where: x, y – image coordinates of the point related to fiducial center; x0, y0 – image coordinates of the principal point.

Tangential distortion (otherwise non-metrical, asymmetric) it is a distortion consisting in the fact that images of straight lines going through the principal point of the image aren’t straight [27]:

( (

) )

∆x tan = P1 r 2 + 2x 2 + 2P2 xy (4) ∆ytan = P2 r 2 + 2 y 2 + 2P1 xy

where: Δxtan, Δytan – the effect of tangential distortion for image coordinates x, y ; x, y – image coordinates of point before the correction related to the principal point; P1, P2 – coefficients describe the impact of tangential distortion.

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3. The Experiment 3.1. Cameras Specification In carried out research five GoPro Hero 4 Black cameras were used (Fig. 1) equipped with wide-angle lens and rolling shutter. The CMOS sensor is reading images by rows.

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research works two different calibration tests of type 2D were used based on the pattern chessboard and equation Brown and Brown equation adopted in Agisoft Lens and Camera Calibration Toolbox. Both software packages based on similar mathematical models, but various algorithms. Agisoft Lens software uses to calibration flat test field shown on the display screen (Fig. 2) and offer fish-eye camera models, too.

Fig. 2. Agisoft Lens 2D calibration field [31] Fig. 1. Five cameras GoPro 4 Hero Black GoPro 4 camera can work in camera and video modes. In these possible arrangements is also possible the use of different dividedness and the speeds of recording of the sequence of video with the different FOV. Record videos in 4K/30fps modes in ultra-wide FOV combination up to 170°, 2.7K/50fps and Full HD/120fps. The camera has also a fast serial mode allowing for taking up to 30 pictures (12 megapixels) per second [28]. Table 1 shows technical specification.

Table 1. Technical specification GoPro 4 Hero black Item Size [mm]

Weight [g]

Optical sensors type

Description 41 × 59 × 30 88

CMOS

Digital Video Format

H.264

Image Recording Format

JPEG

Nominal focal length [mm] Max Video Resolution

Effective Photo Resolution Sensor size [mm] Pixel pitch [µm]

Video sequences were registered under different angles from the same distance. Video they recruited under five different angles: from the front, from right, from left, from above and from the ground floor. During the recording of the image a condition was preserved so that the pivot of lens of every of cameras proceeded through the focal point of the test. GML Camera Calibration Toolbox software has been developed in order to determine the elements of the internal orientation of non-metric digital cameras. It is based on algorithms of the calibration of the image from the OpenCV library. The following parameters are determined in calibration process: calibrated focal length ck, principal point coordinates x0, y0, radial distortion coefficients: K1, K2 and tangential distortion coefficients: P1, P2.

3

3840 × 2160 12.0 MP

6.16 × 4.62 1.55

Additionally at present sensors of the video are deprived of mechanical systems of the shutter for electronic rolling shutters.

3.2. Cameras Calibration

2D and 3D calibration tests are most often used for the calibration of nonmetric cameras. As part of

Fig. 3. GML Camera Calibration Toolbox chessboard calibration field – camera positions [30] Test field resembling the chessboard is applied to calibration (Fig. 3) The test consists of squares: white and black which are arranged alternately. Square size is 3-5 cm. One side should contain the even number Articles

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of squares, and second odd. Calibration patterns using odd x even (or even × odd) number of squares (i.e 5 × 6, 7 × 8, 10 × 7, etc). Calibration sheet should be in the form of a rectangle. This makes it possible to specify the orientation of the design. The chessboard on the images must be located in the all places of the camera matrix [29]. Evaluation of the results of calibration cameras was based on statistical analyses and comparative analysis between individual cameras.

As the action cameras usually have large FOV in wide viewing mode, camera calibration plays an important role to calibrate the effect of lens distortion before image matching. A black and white chess box pattern and Brown equation are adopted in camera calibration.

Once the cameras has been calibrated, the author use these action cameras to take video in an indoor environment. The videos are further converted into multiple frame images based on the frame rates.

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carried out taking into consideration accomplishment in each series the accomplishment of minimum of five frames in the different locations of the camera.

During carrying out the video sequence similar measuring conditions were ensured for acquired samples for most accurate results. The results of internal orientation for five cameras for the 2.7K and 4K Wide video mode in Agisoft Lens software were presented in Table 2 and Table 3. Results of camera calibration in GML Camera Calibration Toolbox (CCT) software were presented in Table 4 and Table 5.

Within the framework of research five action cameras calibration result were in video-mode. Obtained results in both variants of calibration for the 2.7K mode are comparable. Determined calibrated focal length values on average differs about 0.3 mm from given value by the producer.

However calibrated focal length and principal point coordinates are comparable with other test results [6]. They also observed that results of the calibration of cameras in the Agisoft Lens software were repeatable for the calibration of every of five cameras.

In case of the calibration in the GML Camera Calibration Toolbox software (CCT) such a dependence was not observed. Probably because this software is using various algorithms of the calibration leaning on modified OpenCV towards Agisoft Lens which also has algorithms adapted to the calibration fish-eye lens.

4. Calibration Results and Discussion

Images data taken by five GoPro Hero4 cameras in different acquisition modes. As part of research works a calibration of cameras was carried out for the 2.7K video mode and 4K Wide video. In this research video mode were used to record the chessboard field at different view angles and positions. Video frames are converted to single picture at 1 image per second.

For each action camera five measuring series were

Table 2. Calibration results for five cameras for 2.7K video GoPro 4 Hero Black mode based on Agisoft lens CAM 1

CAM 2

CAM 3

CAM 4

CAM 5

Parameter

Mean value

σ

Mean value

σ

Mean value

σ

Mean value

σ

Mean value

σ

ck [mm]

2,70

0,015

2,70

0,001

2,77

0,025

2,72

0,041

2,79

0,019

-0,056

0,007

0,024

0,005

0,096

0,051

0,069

0,076

0,025

0,107

x0 [mm]

y0 [mm]

0,144

0,074

0,150

0,003

0,145

0,050

0,130

0,041

-0,297

0,008

K1

4,56E-04

2,31E-06

4,59E-04

3,72E-08

4,62E-04

2,26E-06

4,61E-04

1,36E-06

4,56E-04

1,42E-06

K3

3,86E-11

3,98E-11

1,75E-11

2,73E-12

3,10E-11

4,88E-11

1,06E-10

2,85E-11

3,51E-11

2,60E-11

K2 P1 P2

2,70E-07 6,00E-05 3,46E-06

1,87E-08 3,49E-05 4,85E-06

2,89E-07

-3,46E-05 -6,34E-05

9,72E-10 1,00E-06 2,54E-06

2,65E-07 9,87E-05

-3,42E-04

1,98E-08 3,29E-05 3,09E-05

2,57E-07 2,91E-05 6,04E-05

1,21E-08 2,44E-05 5,00E-05

2,80E-07 5,68E-05

-1,85E-04

1,18E-08 1,30E-05

6,49E-05

Table 3. Calibration results for five cameras for 4K video GoPro 4 Hero Black based on Agisoft Lens CAM 1

CAM 3

CAM 4

CAM 5

Parameter

Mean value

σ

Mean value

σ

Mean value

σ

Mean value

σ

Mean value

σ

ck [mm]

2,77

0,035

2,83

0,027

2,72

0,043

2,77

0,011

2,76

0,027

0,283

0,023

0,571

0,081

0,414

0,039

0,221

0,009

x0 [mm]

y0 [mm] K1

0,825

0,068

-0,288

9,27E-04

5,01E-06

9,26E-04

-0,729

0,058

0,012

0,236

1,22E-06

9,37E-04

0,022

-0,213

3,98E-06

9,22E-04

0,005

-0,108

3,01E-06

9,40E-04

0,016

1,36E-06

K2

8,26E-07

4,38E-08

9,75E-07

6,29E-09

8,65E-07

5,96E-08

9,02E-07

1,28E-07

7,80E-07

4,16E-09

P1

3,76E-05

9,27E-05

-1,41E-04

2,27E-05

-2,24E-04

2,45E-05

-4,83E-05

2,19E-05

1,39E-04

2,46E-05

K3 P2

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CAM 2

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1,47E-09 2,41E-04

1,14E-10 9,10E-05

1,20E-09 2,74E-04

1,71E-11 3,77E-05

1,49E-09 1,79E-04

1,96E-10 1,39E-04

1,18E-09 2,93E-05

9,73E-11 3,03E-05

1,69E-09

-1,18E-06

8,43E-12 1,46E-05

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Table 4. Calibration results for five cameras for 2.7K video GoPro 4 Hero Black based on camera calibration toolbox CAM 1

CAM 2

CAM 3

CAM 4

CAM 5

Parameter

Mean value

σ

Mean value

σ

Mean value

σ

Mean value

σ

Mean value

σ

ck [mm]

2,68

0,007

2,67

0,009

2,72

0,008

2,72

0,044

2,79

0,038

x0 [mm]

0,135

0,103

-0,132

0,002

0,139

0,002

0,149

0,005

-0,269

0,003

K1

4,04E-04

6,44E-06

4,05E-04

6,71E-04

4,02E-04

1,33E-04

4,04E-04

-6,63E-04

4,10E-04

6,65E-05

K3

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

N/A

y0 [mm] K2 P1

P2

0,065

2,06E-07 6,37E-05 2,48E-06

0,290

3,59E-08 5,87E-05 3,61E-07

-0,048

0,002

3,03E-07

-1,29E-08

4,65E-05

-2,56E-05

-4,65E-05

2,85E-05

0,070

2,08E-07 3,67E-05

-1,66E-04

0,004

2,12E-08 9,95E-10 5,04E-09

0,079

2,12E-07 -2,92E-05 3,35E-04

0,009

1,46E-07

-2,18E-06 -2,78E-04

0,019

2,10E-07

7,27E-05

-8,98E-05

0,005

1,47E-07

-1,04E-07

-2,20E-07

Table 5. Calibration results for five cameras for 4K video GoPro 4 Hero Black based on camera calibration toolbox CAM 1 Parameter

Mean value

ck [mm]

2,80

x0 [mm] y0 [mm]

0,800

-0,700

CAM 2 σ

Mean value

0,177

2,81

0,014

-0,289

0,039

0,261

CAM 3 σ

Mean value

0,012

2,79

0,005

0,003

0,204

0,533

CAM 4 σ

Mean value

0,016

2,76

0,025

-0,196

0,014

K1

9,17E-04

6,39E-06

9,12E-04

6,47E-05

9,05E-04

6,48E-03

K3

N/A

N/A

N/A

N/A

N/A

N/A

K2 P1

P2

1,42E-05 3,63E-05 2,26E-04

1,88E-05 2,75E-06 7,14E-05

9,53E-07 4,44E-05 2,31E-04

1,40E-07 -6,53E-10 -1,74E-08

In the case of the results obtained for the 4K video discrepancies for determined external orientation elements are much bigger especially in the case of values assigned to the distortion also appearing differs in sign. The calibration of cameras in 4K Wide mode is showing the instability of determined elements of the internal orientation, where FOV for this mode of up to 170°. These differences can result from restrictions of the GML CCT software for wide-angle lens calibration, due to the fact that in the calibration process the radial distortion parameters K3 values are not determined, which are important in case of calibration of cameras with a large FOV like GoPro Hero 4. Therefore one should cautiously approach comparing distortion parameters, because by the applying various calibration parameters by software it is difficult. The results of other research show that the calibration of camera using OpenCV differs most compared to the others.

5. Conclusions

Results of calibration and their quality for five action cameras – GoPro Hero 4 Black of two calibration methods in Agisoft Lens Software and GML Camera Calibration Toolbox are presented in this article.

Applying different display modes and algorithms of the calibration allowed to investigate stability of elements of internal orientation of cameras with an ultra-wide FOV. Obtained results described the instability of the elements of internal orientation in the mode

8,58E-07 1,27E-04

-2,00E-04

σ

σ

0,054

2,74

0,019

0,005

-0,102

9,96E-04

-6,65E-05

9,23E-04

-6,50E-05

N/A

N/A

N/A

N/A

0,414

1,40E-05

-9,73E-07

2,97E-09

4,65E-05

2,77E-08

CAM 5 Mean value

2,78E-05

0,004

1,47E-07

-1,19E-07 -2,27E-07

0,227

9,40E-07 5,45E-05 1,71E-05

0,002

0,003

1,41E-07 1,94E-10

-3,40E-08

of wide-angle even for the principal point. Therefore one should assume that achieved results are only partly comparable with oneself.

Repeatable results of the calibration were achieved for the 2.7 K Video mode in the Agisoft Lens software.

This confirms that these cameras can be successfully used in photogrammetric applications. Slightly worse repeatability were obtained for 4K ultra-wide FOV mode. Calibration cameras results in the GML Camera Calibration Toolbox are less repeatability.

Future researches will concern the influence of use pre-calibrated interior orientation in receiving pre-corected images and for examining their accuracy potential in photogrammetry applications

ACKNOWLEDGMENTS

This paper has been supported by the Military University of Technology, the Faculty of Civil Engineering and Geodesy, Department of Remote Sensing, Photogrammetry and Imagery Intelligence.

AUTHOR

Damian Wierzbicki – Department of Remote Sensing, Photogrammetry and Imagery Intelligence, Institute of Geodesy, Faculty of Civil Engineering and Geodesy, Military University of Technology, Warsaw 00-908, Poland, E-mail: damian.wierzbicki@wat.edu.pl Articles

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[2] S. Mikrut, “Classical Photogrammetry and UAV – Selected Ascpects”, Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci., XLI-B1, 947–952. DOI: 10.5194/isprs-archives-XLI-B1-947-2016.

[3] M. Kedzierski, D. Wierzbicki, “Methodology of improvement of radiometric quality of images acquired from low altitudes”, Measurement, vol. 92, October 2016, 70–78. DOI: 10.1016/j. measurement.2016.06.003 [4] M. Kedzierski, A. Fryskowska, D. Wierzbicki, A. Grochala, P. Nerc, “Detection of Gross Errors in the Elements of Exterior Orientation of Low-Cost UAV Images”. In: 2016 Baltic Geodetic Congress (BGC Geomatics), Gdansk, 2016, 95–100, DOI: 10.1109/BGC.Geomatics.2016.26.

[5] H. Hastedt, T. Ekkela, T. Luhmann, „Evaluation of the Quality of Action Cameras with Wide-Angle Lenses in UAV Photogrammetry”, ISPRS-International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 2016, 851–859. DOI: 10.5194/isprsarchives-XLI-B1-851-2016.

[6] M. Kedzierski, A. Fryskowska, “Precise method of fisheye lens calibration”. In: Proceedings of the ISPRS-Congress, Beijing, China, 2008, 765–768.

[7] A. Habib, A. Pullivelli, E. Mitishita, M. Ghanma, E. M. Kim, “Stability analysis of low‐cost digital cameras for aerial mapping using different georeferencing techniques”, Photogrammetric Record, vol. 21, no. 113, 2006, 29–43.

[8] C. Balletti, F. Guerra, V. Tsioukas, P. Vernier, “Calibration of Action Cameras for Photogrammetric Purposes”, Sensors, vol. 14, no. 9, 2014, 17471–17490. DOI: 10.3390/s140917471.

[9] F. Remondino, C. Fraser, “Digital camera calibration methods: considerations and comparisons”. International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, vol. 36, no. 5, 2006, 266–272.

[10] H. Hastedt, T. Luhmann, “Investigations on the quality of the interior orientation and its impact in object space for UAV photogrammetry”, International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, vol. 40, no. 1, 2015, 321. DOI: 10.5194/isprsarchives-XL-1-W4-321-2015.

[11] A. Fryskowska, M. Kedzierski, A. Grochala, A. Braula, „Calibration of Low Cost RGB and NIR Articles

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J. G. Fryer, “Camera calibration”. In: Atkinson (Ed.), Close Range, Photogrammetry and Machine Vision, Whittles Publishing, Caithness, UK, 1996, 156–179.

T. Luhmann, S. Robson, S. Kyle, J. Boehm, Close-­ Range Photogrammetry and 3D Imaging, de Gruyter, 2014, 684.

C. S. Fraser, “Automatic camera calibration in close range photogrammetry”. Photogrammetric Engineering & Remote Sensing, vol. 79, no. 4, 2013, 381–388.

T. Luhmann, C. Fraser, H. G. Maas, “Sensor modelling and camera calibration for close-range photogrammetry”. ISPRS Journal of Photogrammetry and Remote Sensing, 115, 37–46. DOI: 10.1016/j.isprsjprs.2015.10.006.

R. Y. Tsai, “An efficient and accurate camera calibration technique for 3-D machine vision”. In: Proc. International Conference on Computer Vision and Pattern Recognition, Miami Beach, USA, 1986, 364–374.

J. Weng, P. Cohen, M. Herniou, “Camera calibration with distortion models and accuracy evaluation”, IEEE Transactions on pattern analysis and machine intelligence, vol. 14, no. 10, 1992, 965–980. DOI: 10.1109/34.159901 Z. Zhang, “A flexible new technique for camera calibration”, IEEE Trans. Pattern Anal. Mach. Intell., vol. 22, no. 11, 1330–1334, 2000.

W. S. Yin, Y. L. Luo, S. Q. Li, “Camera calibration based on OpenCV”, Computer Engineering and Design, vol. 28, no. 1, 2007, 197–199.

Y. M. Wang, Y. Li, J. B. Zheng “A camera calibration technique based on OpenCV”. In: Information Sciences and Interaction Sciences (ICIS), 3rd International Conference on, IEEE, 2010,–.

DOI: 10.1109/ICICIS.2010.5534797.

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A. De la Escalera, J. M. Armingol, “Automatic chessboard detection for intrinsic and extrinsic camera parameter calibration”, Sensors, vol. 10, no. 3, 2010, 2027–2044. DOI: 10.3390/ s100302027.

J. Kannala, S. S. Brandt, „A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses”, IEEE transactions on pattern analysis and machine intelligence, vol. 28, no. 8, 2006, 1335–1340. DOI: 10.1109/ TPAMI.2006.153.

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[24] D. Schneider, E. Schwalbe, H. G. Maas, “Validation of geometric models for fisheye lenses”, ISPRS Journal of Photogrammetry and Remote Sensing, vol. 64, no. 3, 2009, 259–266. DOI: 10.1016/j.isprsjprs.2009.01.001.

[25] T. Teo, “Video-Based Point Cloud Generation Using Multiple Action Cameras”, International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences, vol. XL-4/W5, 2015, 55–60. DOI: 10.5194/isprsarchives-XL-4-W5-55-2015.

[26] M. Ballarin, C. Balletti, F. Guerra, “Action cameras and low-cost aerial vehicles in archaeology”.

In: SPIE Optical Metrology International Society for Optics and Photonics, 2015, vol. 9528. DOI: 10.1117/12.2184692 [27] D. C. Brown, “Close-Range Camera Calibration”, Photogrammetric Engineering, vol. 37, no. 8, 1971, 855–866.

[28] GoPro 2016, Gopro Hero 4 Black user manual, http://cbcdn2.gp-static.com/uploads/product_ manual/file/490/UM_H4Black_ENG_REVA_ WEB.pdf (03.01.2017).

[29] GML Camera Calibration User's Guide http:// graphics.cs.msu.ru/en/node/910 (03.01.2017).

[30] AgiSoft PhotoScan. Available online: http:// www.agisoft.com/ (accessed on 10 July 2017).

Articles

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PD Terminal Sliding Mode Control Using Fuzzy Genetic Algorithm for Mobile Robot in Presence of Disturbances Submitted: 22nd March 2018; accepted: 18th June 2018

Walid Benaziza, Noureddine Slimane, Ali Mallem

DOI: 10.14313/JAMRIS_2-2018/11 Abstract: This paper presents a new approach in the field of trajectory tracking for nonholonomic mobile robot in presence of disturbances. The proposed control design is constructed by a kinematic controller, based on PD sliding surface using fuzzy sliding mode for the angular and linear velocities disturbances, in order to tend asymptotically the robot posture error to zero. Thereafter a dynamic controller is presented using as a sliding surface design, a fast terminal function (FTF) whose parameters are generated by a genetic algorithm in order to converge the velocity errors to zero in finite time and guarantee the asymptotic stability of the system using a Lyapunov candidate. The elaborated simulation works in the case of different trajectories confirm the robustness of the proposed approach.

Keywords: mobile robot, fast terminal function, PD sliding surface, fuzzy system, genetic algorithm, Lyapunov stability.

1. Introduction

52

The mobile robots are widely used in different fields and are known as nonholonomic system [1]. Different works are done in the field of trajectory tracking of mobile robots taking into account the model uncertainties [2]. Therefore, many researchers have done to use kinematic and dynamic model to control the motion of wheeled mobile robot [3]. The unicycle robot is generally used for the control design to track the reference trajectory in a plane [4]. The control of mobile robot has been studied in different ways for instance, point stabilization, trajectory tracking, path-following etc.

Sliding mode control has shown its robustness against the uncertainties and external disturbances [5], but conventional sliding mode control is known with the discontinuity, and the last one creates, in high oscillations, the chattering phenomenon. Many works have been proposed to resolve this kind of problem [6], [7], [8] and another suggests a fuzzy logic to reduce the effect of this phenomenon [9], [10] and [11]. However the sliding mode control displayed the problem of chattering, which is the high problem of their real implementation.

Recently, many robust controllers are proposed for the trajectory tracking and obstacle avoidance, using dynamic model [12], [13], [14] and [15].

A recent method suggests an adaptive fuzzy terminal sliding mode control for nonlinear system with non-singularity in presence of external disturbances in order to achieve a fast convergence [16], [17], [18] and [19].

A robust control using PD sliding surface for robotic system is introduced to stabilize the closed loop system and compensate the effect of the disturbances and reduce the tracking errors [20], another work used genetic sliding mode in order to achieve the optimal parameters is presented in [21] for a servo system and in [22] for a manipulator arm.

Based on the previous works using PD sliding mode control, the contribution of this paper aims to suggest a new kinematic controller for the linear and angular velocities using PD and fuzzy logic. The selected PD sliding surface is used for the angular velocity in order to accelerate the orientation error to converge to zero in short time. The added fuzzy system permits to avoid the effect of the disturbances presented in the kinematic model by selecting the appropriate value of the gain parameter. A conventional sliding mode controller used for the linear velocity aims to converge the robot position error to zeros and guarantee the asymptotic stability of the system.

A dynamic controller based on terminal sliding mode with genetic algorithm is proposed. The genetic algorithm is used in order to select the optimal values of the controller parameters, which permit to the controller to converge the dynamic velocity error to zero and guarantee the asymptotic stability by using the Lyapunov stability.

The new trajectory tracking controller is proposed, based on fuzzy logic and genetic sliding mode for mobile robot. The approach gives mainly an asymptotic stability by using Lyapunov theory.

The proposed control law can tend the tracking error and velocity error to zero in very short time with asymptotic stability by taking into account the disturbances and the system uncertainty.

The paper is organized as follow. The first section presents the kinematics and dynamic model. The second section consists to propose a kinematic controller based on PD terminal sliding mode control with fuzzy logic. The third section presents a new dynamic controller using fast terminal function (FTF) with genetic algorithm [23] into the sliding surface design by including a saturation function in order to reduce the chattering problem by taking into account the disturbances.

   e  Journal of Automation, Mobile Robotics & Intelligent Systems

Consider the differential drive which is indicated therobot differential drive whichand is the indicated inConsider Fig 1. The has two wheels speedinofFig 1.The robotis has twoaswheels and the speed of each wheel each wheel given follows: is given as follows: Y

X

Fig. 1. The robotrobot model Fig.mobile 1.The mobile model

Ra (  r   l ) Ra 2L

(1)

(1)

(2)

ω = (ϕr − ϕ l ) (2) 2L a mobile robot is given by: The Kinematic model for The Kinematic model for a mobile robot is given  x   cos  0  by:     (3) p   y    sin  x  0 cosθ 0          p 0= y 1=  sinθ 0 γ (3)

          0 1  T θ   where   v  .

T where γ = ( v ω ) .

Consider the posture error of mobile robot Consider the posture error of mobile robot T reference posture is given as: p e   x e y e T e the the reference posture is given as: pe = ( x e ye θe ) T pr p=r (x r  x ryr yθrr )T . r  The vector of the desired velocity is: γ r = (vr , �ωr )T . ω� � . as: The vector of the desired velocity is:γ � =���is � , given The position error of the mobile robot The position error of the mobile robot is given as:  x e   cosθ sinθ 0  x r − x 

pe =  ye  =  − sinθ cosθ 0  yr − y  (4) 0 1  θr − θ  θe   0

The derivative of the trajectory tracking error is expressed as:

 x e   yeω + vr cosθe − v  p e =  y e  =  − x eω + vr sinθe  (5) θe    ωr − ω

Consider the kinematic model (3) with disturbances [24], [25]:

 x   cosθ p =  y  =  sinθ  θ   0   

0  0 (γ + Ψ ) (6) 1 

Therefore, the vector Ψ presents the disturbances in the two velocities:

Where:  x   cos  0  (6) p   y    sin  v 0  (v,   )         0 1   derivative  The of model error becomes as follows:

Therefore, the cospresents θe − (v +the v ) +disturbances ye (ω + ω ) in the  x evector   vr  two velocities:  p e =  y e  =  vr sinθe − x e (ω + ω )

 (8) ~ ~ (7)   (v )θ e   

ωr − (ω + ω )

The derivative of model error becomes as follows: v ″ vmax , ω ≤ ωmax .  x e   v r cos  e  ( v  v~ )  y e (  ~ )   p e   y e    v r sin  e  x e (  ~ )  2.2. Dynamic Model  e    r  (  ~ )

2Ra

O

 

2018

Where: The proposed control law of the two velocities must take~ the tracking error to zero while t tends to ~ v   v ,    infinity by considering:

�

The linear velocity is given as: The linear velocity is given as:    l

l v  Ra ( r ) 2 v Ra r

2 The angular velocity of the mobile robot is: The angular velocity of the mobile robot is:

N° 2

Consider the kinematic model (3) with disturbances [24], [25]:

2.1.Kinematic Kinematic Model 2.1.

Model

2L

 

r 

VOLUME 12,

2. Kinematics and Dynamic Modelling 2. Kinematics and Dynamic Modelling

a

 

Ψ = (v ω ) (7)

(8)

When dynamic robot is described as The proposed controlmodel law ofofthethe two velocities must take [26,tracking 27]: error to zero while t tends to infinity by the considering: M(q)V + V (q , q )V + F (q ) + G(q) + τ d = β (q)τ + D (t ) (9)

v  v max

,

   max

.

Where: V = (vDynamic ω)T is the vector of velocities and v and ω are the 2.2.

Model linear and angular velocities respectively, [t 1 t 2 ]T is the vector of torque mobileasrobot When dynamic modelofofthe thewheels robot of is the described [26, t1 and t2 are the torques of the right and left wheel.

27]: Where ~ M (q)V V (q, q)V mF(q0) G(q) 1d 1 (q1) D(t) (9) M( q ) =  , β (q ) =  0 I  Ra  L −L  Where: 

m is(vthe mass ofvector the robot .

V= ω)T is the of velocities and v and ω are the I is the moment inertia of the robot.

T linear anddistance angularbetween velocitiesthe respectively, L is the two wheels.  1  2 is the of torque of the wheels of the mobile robot τ� Ra vector is the radius of the wheel. are the torques of the right and left wheel. and D (tτ)� represents the disturbance.

V (q , q ) is the vector of centripetal and Coriolis 1  forces. m 0 1 1 F (q ) represents the friction matrix. , Where M (q)     (q )  R  L  L  G(q) represents vector.  0theIgravitational   a  td is an unknown disturbance. m isEq. the (6) mass therobot robotis. used in order to control the of of the movement of the robot by robot. using fuzzy genetic sliding I is the moment inertia of the mode control in presence of the disturbances. Hence, Lthe is the distance between two wheels. posture error of thethe robot must tend asymptotically to zero:





lim pe = lim pr (t ) − p(t ) = 0 (10)2 t →∞

t →∞

3. Controller Design

3.1. Kinematic Controller Design 1) Angular Velocity Control Each By using PD sliding surface, the selection of switch function is given as:

s1 = ρ1e1 + ρ2e1 (11)

Where e1 e The derivative of the first sliding surface is:

s1 = ρ1e1 + ρ2e1 (12) Articles

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Journal of Automation, Mobile Robotics & Intelligent Systems

Appling the reaching control law:

s1 = −k1 sign( s1 ) (13)

In order to attenuate the chattering problem, the discontinue function is replaced by saturation function:

The control law of the angular velocity is given as:

s1 = −k1 sat ( s1 ) (14)

ω =

1 ( ρ1ωr − ρ1ω + ρ2ω r + k1 sat ( s1 )) (15) ρ2

This control law can make θe converges to zero and, r . Therefore, in order to eliminate the effect of angular disturbances, the fuzzy system is applied.

2) Fuzzy Sliding Mode Control The parameter k1 is chosen with fuzzy system in order to delete the effect of disturbances and reduces the chattering problem [28].

To select the value of k1, the relation between k1 and the sliding surface s1 must be determined. So, at this case, the inputs of fuzzy system will be s1 and s 1 and the output will be kf.

The fuzzy sets of the inputs and output can be given as: S1= {NP, NM, ZE, PS, PL} S 1 = {NP, NM, ZE, PS, PL} Kf = {NP, NM, ZE, PS, PL} The control law (15) is becoming as follows:

ω k =

1 ( ρ1ωr − ρ1ωk + ρ2ω r − k f sat ( s1 )) (16) ρ2

3) Linear Velocity Control Now, to design the control of linear velocity, the sliding mode control is applied.

When the orientation error tends to zero, the error model (7) becomes as follows: x e = ωr ye − v + v r + v (17)

y e = −ωr ye (18)

vk

vr

r

xe

r

ye

k2

s2

s2

N° 2

2018

(23)

The selected switching surfaces are given as:  s   ρ e + ρ2e1  s =  1 =  1 1  (24)  s2   x e − ye  The obtained control law is: s2   vr + ωr x e + ωr ye + k2   s2 + δ  vk    (25) = ω    1  k  ( ρ1ωr − ρ1ωk + ρ2ω r − k f sat( s1 )  ρ2 

Proof 1: to guarantee the stability of the system, the Lyapunov candidate is selected as:

1 v = s12 (26) 2

The derivative of this equation is given as:

v = s1 s1 = − s1 (k f sat ( s1 )) = − s1 k f sat ( s1 ) (27)

kf is positive and s1 sat ( s1 ) ≥ 0 Then, v ≤ 0

(28)

3.2. Dynamic Control Using Terminal Function

In this section, it is interesting to design the robot control torque in order to converge asymptotically the velocities error to zero.

Considering the dynamic model (9) and taking into account that the gravitational, centripetal, Coriolis, friction matrices and unknown disturbances are equal zero.

The dynamic model (9) becomes as follows:

M(q)V = β (q)τ + D (t ) (29)

Hence, the dynamic control is based on terminal function; therefore the design of the control law consists, firstly, in the choice of the sliding surface which is the error between the velocities of the robot:

 v −v  T eω ] =  k  (30) ωk − ω  The derivative is given as:  v − v  E v =  k  (31) ω k − ω 

By combination of the equation (13) and (14), the obtained result is:

s   v − v  S =  3 =  k  (32)  s4  ωk − ω 

By introducing (17) and (18), the control law can be obtained as:

The second sliding surface is selected as: s 2 x e ye (19)

Using a sliding mode control which is given by: s2 (20)

s2 = −k2 s2 + δ 1

s 2

x e y e

r

ye

r

xe

vr

vc

(21)

s2 = x e − y e = vr + ωr x e + ωr ye − v

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VOLUME 12,

= − k2 Articles

s2 s2 + δ 1

(22)

E v = [ev

The sliding surfaces are choosing as:

The derivative of the sliding surfaces is given as:

 s   v − v  S =  3  =  k  (33)  s4  ω k − ω 

Secondly, by using the fast terminal function (FTF), the form of the proposed sliding surface [23] is:

s + λ1 sη1 + λ2 sη2 = 0 (34)

Journal of Automation, Mobile Robotics & Intelligent Systems

VOLUME 12,

s = −λ1 sη1 − λ2 sη2 (35)

The time derivative of the first sliding surface is:

s3 = −λ1 s3η1 − λ2 s3η2 (36)

Using the sliding mode: s4 = −k3

Ra s4 ( Iω k + Ik3 + mvk + mλ1 s3η1 2 s4 + σ

+ mλ2 s3η2 )

τ2 =

Ra s4 ( −Iω k − Ik3 + mvk + mλ1 s3η1 2 s4 + σ

+ mλ2 s3η2 ) This control law can be written as follows:

A

(38)

(39)

 λ1 s3η1 + λ2 s3η2   vk    A  + B  s4  (40)  k ω  k  3 s +σ  4  

τ  R τ = 1= a τ 2  2

Where:

s4 (37) s4 + σ

The control law is obtained as:

τ1 =

I

m

m

I

,B

m m

I I

Proof 2: To ensure the stability of the system, the Lyapunov function is selected as:

2018

V2 = S T S = −λ1 s3η1 +1 − λ2 s3η2 +1 −

Where, 0 ≺ η1 ≺ 1, η2 1, η1 , η2 ∈ℜ.

The above equation can be written as:

N° 2

1 V2 = S T S (41) 2

The time derivative of the Lyapunov function is:

V2 = s3 s3 + s4 s4 (42)

− k3

s 24 ≤0 s4 + σ

(43)

So, the system is asymptotically stable.

The selection of λ1, λ2 are done by genetic algorithm, in order to achieve optimal values and accelerate the nonlinear equation (34) to converge to the origin.

1) Parameters Selection Using Genetic Algorithm A genetic algorithm (GA) is a robust search method used to find approximate solutions in research problem optimization [29].

The diagram of control system is given in Fig. 2.

The structure of genetic algorithm (GA) is given as: 1. Produce casual population of chromosomes that is given an appropriate solution.

2. Reform the fitness of any chromosome of population.

3. Produce a novel population by repeating the steps until the new population is complete.

4. Choosing two parents of chromosomes from a population depending to their fitness.

5. Crossover the parents to a novel offspring and if the condition is not satisfied, the offspring is the same of the parents.

6. Mutate the novel offspring at each position.

7. Make the novel offspring in the new population.

Finally, use the new parent populations, and if the final condition is verified, stop and return to the perfect solution of population.

The precedent operation is repeated till the condition is verified [30].

A flow chart of the general scheme of the implementation of the technique is shown in Fig. 3.

In this work, the interested parameters to optimize are (λ1, λ2). So that the controlled system can achieve a good overall performance in the slide mode control design. It is desirable to have the fast reaching velocity into the switching hyper plane during the reaching phase and the corresponding state slides to the origin.

By using GA, firstly we selected a fitness function with two variables (λ1, λ2) to achieve optimal values.

fn = fn (λ1 , λ 2 )

(44)

Fig. 2. Structure of the control system Articles

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controller are taken manually: Journal of Automation, Mobile Robotics & Intelligent Systems

k 2  25  1  20  2  0 . 1 ,  1 VOLUME  0 .9

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The parameters of the mobile robot are chosen as: The the mobileL=0.15m; robot are chosen as: m=4 kg;parameters I=3kg/m2; of Ra=0.03m; 2 ; Ra = 0.03 m; L = 0.15 m; m = 4 kg; I = 3 kg/m The kinematic disturbance is implemented in time 3 <t<4 The kinematic disturbance is implemented in time seconds into the velocity of the mobile robot which is 3 < t < 4 seconds into the velocity of the mobile robot given as follows:

~  1.5 sin(t   )u (t ) v~  1.5 sin(t   ω )u=(t1).5sin(t − π )u(t ) which is given as follows:

Where:

Where: 1

u (t )   0

v = 1.5sin(t − π ) u(t )

for 3  t  4

elsewhere 1 for 3  t  4 u(t ) =

 controller are given as: The parameters of dynamic 0 elsewhere

k 3 The  25parameters ,   0.95,  0.9, 2controller  3.5, are given as: of1 dynamic

Fig. 3. Flow chart of the genetic algorithm

1optk3 = 30 25,, σ2 opt = 0.9530 , η. 1 = 0.9, η2 = 3.5,

When the fitness function is defined, the GA prospection search will depend on the requirement of this function. The choice of the defined fitness function is an important step, which permit to the system to attain the desired performance.

The gain parameters (λ1, λ2) of dynamic controller will be optimally chosen in order to tend the function fn to zero.

The fitness function that will be used is:

fn (λ 1 , λ 2 ) = s3 + λ1 s3η1 + λ2 s3η2 (45)

Choosing two intervals of the parameters λ1 = [λ11, λ12] and λ2 = [λ21, λ22], we obtain the optimal values (λ1opt, λ2opt), that minimize the fitness function and tend rapidly fn (λ1, λ2) = 0.

By selecting optimal parameters λ1opt and λ2opt, the design control law τ is given as follows:

τ  R τ = 1= a τ 2  2

 λ1opt s3η1 + λ2opt s3η2   vk    A  + B  s4  (46)  k ω 3  k   s4 + σ  

Where:  1 p (t )   0

for 7  t  8

1 for 7  t  8 p(telsewhere )=  0 elsewhere The initial position and orientation error

� The is:����� ���initial ���� position and orientation error is: �p (2 m , 3 m , rad) 3 The membership of the inputs and output are shown in the Fig.4, and Fig. 6.

TheFig.5 membership of the inputs and output are shown in the Fig. 4, Fig. 5 and Fig. 6.

By using the kinematic and dynamic controllers of the By using the kinematic and dynamic controllers equations (25) and (46), the circular trajectory tracking is of the equations (25) and (46), the circular trajectory shown in Figure 7. tracking is shown in Fig. 7.

Figure the 9 shows the tracking errors thecontrol inputs v Fig.9 shows tracking errors and the and inputs control v and w is indicated in Fig. 8. and ω is indicated in Fig.8.

Figure 11 indicates the torques control and Fig. 12 Fig.11 indicates the torques shows the velocity error. control and Fig. 12 shows the

velocity error

Proof 3:To verify the stability of the system, new parameters are chosen, therefore the derivative of Lyapunov function is given as: V2 = S T S = −λ1opt s3η1 +1 − λ2opt s3η2 +1 − − k3

s 24 ≤0 s4 + σ

(47)

4. Results and Discussion The software MATLAB/SIMULINK is selected in vr=2,order ω� =to 1,simulate and the of the and kinematic theparameters proposed control, illuscontroller are taken manually: trate the behavior motion of the mobile robot. In this simulation, k 2  section, 25  1 we20evaluate  2  0 .through 1 ,  1  computer 0 .9 the ability of the proposed controller to stabilize the mobile robot trajectories.

Different trajectories as The parameters of the mobile robot are chosen as: 2 sinusoidal, circular and specific are taken in considm=4 kg; I=3kg/m ; Ra=0.03m; L=0.15m; eration. Indisturbance the simulation, the desiredinangular and The kinematic is implemented time 3 <t<4 linear velocities are chose as v = 2, w = 1, and r r seconds into the velocity of the mobile robot whichthe is givenparameters as follows: of the kinematic controller are taken manually: ~  1.5 sin(t k  = 25, )u (tρ) = 20, ρ = 0.1, ρ = 0.9 2 1 2 3

v~  1.5 sin(t   )u (t )

Where: 56 Articles

1

for 3  t  4

Theλdynamic disturbances are inserted in time 7<t<8 are 1opt = 30, λ2opt = 30. given as: The dynamic disturbances are inserted in time ~ 7 < t < 8 are given as: D ( t )  2 sin( t ) p ( t ) 2 sin( t ) p ( t )  D (t ) = [2sin(t )p(t ) 2sin(t )p(t )] Where:

Fig. Fig. 4. Membership ofSS1.1 4.Membershipfunction function of

· Fig. 5. Membership function of S1 Fig. 5. Membership function of

s1

3 <t<4is which hich is

Fig.10.Error orientation.

Journal of 5.

Automation, Mobile Robotics Fig. Membership function of &sIntelligent Systems

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Fig

Fig.10.Error orientation.

Fig. 5. Membership function of s1 Fig. 5. Membership function of s1

Fig

as: :

8 are

<t<8 are t<8 are

Fig.11.Torques control.

Fig. 11. Torques control

Fig. 6. Membership function of kf

Fig. 6. Membership function of kf.

Fig

Fig.11.Torques control.

Fig. 6. Membership function of kf.

Fig. 6. Membership function of kf.

Fig

errorerror error

hown wn in inin own

rs of the facking thetheis of cking ng is is

control v ontrol rol v v

Fig. 12. Velocity error

Fig.12.Velocity error.

Fig.7.Circular trajectory tracking.

Fig. Fig.7.Circular 7.Fig.7.Circular Circular trajectory tracking trajectory trajectorytracking. tracking.

p 13 shows the WithWith anFig.12.Velocity initial error������ ����, an initial error:��� , Fig. 13 (1�m , 1 mFig. , rad) error.

6 sinusoidal tracking: shows thetrajectory sinusoidal trajectory tracking:

hows the ows the

s the

�

Fig

�

Fig

With an initial error������ ��� ����, Fig. 13 shows the � sinusoidal trajectory tracking:

Fig.8.The control v and ω.

control v and ω.

Fig.9.Tracking errors.

Fig. 8.Fig.8.The The control v and w

Fig.13.Sinusoidal trajectory tracking.

Fig.13.Sinusoidal trajectory tracking.

Fig.9.Tracking errors.

Fig.8.The control v and ω.

Fig. 13. Sinusoidal trajectory tracking

Fig.9.Tracking errors.

Fig.10.Error orientation.

Fig. 9. Tracking errors Fig.10.Error orientation.

Fig.9.Tracking errors.

Fig.9.Tracking errors.

Fig.10.Error orientation.

Fig.13.Sinusoidal trajectory tracking.

6

6

Fig.13.Sinusoidal trajectory tracking.

6

trajectory tracking.

Fig.Fig.13.Sinusoidal 14.

The control vvand w Fig.14.The control and ω.

Fig.14.The control v and ω.

Fig.14.The control v and ω.

Fig.11.Torques control.

Fig.11.Torques control. orientation.

Fig. 10.Fig.10.Error Error orientation Fig.10.Error orientation.

Fig.11.Torques control.

Fig. 15. Position errors

Fig.15.Position errors.

Fig.15.Position Fig.14.The controlerrors. v and ω.

Fig.14.The control v and ω.

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57

Fig.15.Position errors.

Journal of Automation, Mobile Robotics & Intelligent Systems

VOLUME 12,

Fig.17.Torques control.

Fig.21.Position error.

Fig. 21.

Position error Fig.21.Position error.

Fig.17.Torqueserror. control.

Fig.16.Orientation

Fig.17.Torques control.

Fig.17.Torques control.

Fig.18.Velocity error.

Fig. 17.

Torques control Fig.18.Velocity error.

Fig.17.Torques control.

Fig.18.Velocity error.

Fig.17.Torques control.

Fig.18.Velocity error.

2018

Fig.21.Position error.

Fig.17.Torques control.

Fig. 16. Orientation Fig.17.Torqueserror control.

hows the

N° 2

Fig.21.Position error.

7

�

� With With an an initial initial error������ error��������� ������ rad), rad), Fig.19 Fig.19 shows shows the the � With an initial error������ ��� rad), Fig.19 shows the specific tracking: With an trajectory initial error������ specific trajectory tracking:����� rad), Fig.19 shows the specific specific trajectory trajectory tracking: tracking:

Fig.18.Velocity error.

Fig.18.Velocity error.

Fig. 18. Velocity error � Fig.18.Velocity error.

Fig.18.Velocity error.

With an initial error������ ��� � rad), Fig.19 shows the With an initial error������ ��� � rad), Fig.19 shows the � specific trajectory tracking: � p � rad), With an initial error������ the specific tracking: With an initial error:��� , 1 mFig.19 , rad)shows , Fig. 19 With antrajectory initial error������ ���(1�m rad), Fig.19 shows the 6 � specific trajectory tracking: shows the specific trajectory tracking: specific trajectory tracking: Fig.19.Specific trajectory tracking.

Fig.21.Position error.

Fig.21.Position error. error.

Fig.Fig.21.Position 22.Fig.22.Orientation Orientation Fig.22.Orientation error. error. error Fig.22.Orientation Fig.21.Position error. error.

Fig.22.Orientation error.

Fig.22.Orientation error.

Fig.22.Orientation error.

Fig. 23.Fig.23.Torques Torques error error.

Fig.23.Torqueserror. error.

Fig.22.Orientation Fig.23.Torques error.

Fig.22.Orientation Fig.23.Torques error.

Fig.19.Specific trajectory tracking.

Fig.19.Specific trajectory tracking.

Fig.19.Specific trajectory tracking.

Fig.23.Torques error.

Fig.Fig.23.Torques 24. Velocityerror. error

Fig.19.Specific trajectory tracking.

Fig. 19.

Specific trajectory tracking Fig.19.Specific trajectory tracking.

Fig.20.The control v and ω.

Fig.20.Thetrajectory control v and ω.

Fig.19.Specific tracking.

Fig.20.The control v and ω.

Fig.19.Specific trajectory tracking.

Fig.20.The control v and ω.

Fig. 20. The control v and w 58

Fig.20.The control v and ω.

Fig.20.The control v and ω.

Articles Fig.20.The control v and ω.

Fig.20.The control v and ω.

Fig.24.Velocity error.

Fig.24.Velocity Fig.23.Torques error.error.

Fig.24.Velocity error.

Fig.23.Torques error.

Fig.24.Velocity The simulationerror. results

show the effectiveness of The simulation results show the the forthe theeffectiveness dynamic andof The proposed simulationcontrol resultslaws show the effectiveness ofkinthe The simulation results show the effectiveness of the proposed control laws for the dynamic and kinematic ematic model ofresults the three (circular, sinuThe simulation show effectiveness of the proposed control laws for trajectories thethedynamic and kinematic proposed control laws for and model of the (circular, sinusoidal, proposed laws trajectories for the the dynamic dynamic and kinematic kinematic soidal, model specific). ofcontrol the three three trajectories (circular, sinusoidal, model of the three trajectories (circular, sinusoidal, specific). model of the three trajectories sinusoidal, Hence, in the Fig. 7 the mobile(circular, robot can achieve specific). specific). specific). the circular trajectory rapidly in short time among Hence, in the Fig.7 the mobile robot can achieve Hence, in the Fig.7 the mobile robot can achieve the the Hence, in the Fig.7 the mobile robot can achieve t = 4 s after inserting the kinematic disturbances circular trajectory rapidly in short time among ttand ==the 44 ss Hence, the Fig.7rapidly the mobile robot canamong achieve the circularintrajectory in short time circular trajectory rapidly in short time among tt ==time 44 ss after the and in the in theinserting time t = 9 s after inserting dynamic circular trajectory rapidly indisturbances shortthe time among after inserting the kinematic kinematic disturbances and in disturthe time after inserting the kinematic disturbances and in the time t= 9 s after inserting the dynamic disturbances, therefore bances, therefore thethe asymptotic stability ofin thethe robot after the kinematic disturbances and time t= 9 inserting s after inserting dynamic disturbances, therefore t= ss after dynamic disturbances, therefore is99assured and thethe tracking errors can converge to t= after inserting inserting the dynamic disturbances, therefore Fig.24.Velocity error. zero.

Fig.24.Velocity error.

Moreover, in the Fig. 13 and Fig. 19 the mobile ro-8 Fig.24.Velocity error.

Thebot simulation show theand effectiveness of the in8 8 can attainresults the sinusoidal specific trajectory Fig.24.Velocity error.

The simulation results show the effectiveness of kinethe 8 proposed control laws for the and kinematic the time among t = 4 s afterdynamic implementing the proposed control laws for the dynamic and kinematic The simulation results show effectiveness of the model of disturbances the three sinusoidal, matics and inthe the(circular, time t = 10 s after The simulation resultstrajectories show the effectiveness of theinmodel of control the three trajectories (circular, sinusoidal, proposed laws for the dynamic and kinematic specific). proposed control laws for the dynamic and kinematic specific). model the three trajectories (circular, sinusoidal, model of of the Fig.7 three the trajectories (circular, sinusoidal, Hence, in the mobile robot can achieve the specific).

Hence, specific).in the Fig.7 the mobile robot can achieve the

Journal of Automation, Mobile Robotics & Intelligent Systems

serting the dynamic disturbances, therefore the tracking errors and velocity errors of the robot converge asymptotically to zero.

The general control law can assure the convergence of the tracking errors to zero and guarantee the asymptotic stability of the system regarding to some kind of perturbation due to the robot wheeled slipping, however the robot can follow the reference trajectory rapidly.

5. Conclusion

In this work, a new control law is proposed for trajectory tracking of nonholonomic mobile robots. The control law is divided in two parts. Firstly, the control law for the kinematic model is proposed by using PD sliding surface and fuzzy system in order to avoid the disturbances inserted in the system. Therefore the control can bring the error state of the robot to zero rapidly in short time. However, the stability of system is guaranteed and the system is asymptotically stable. Secondly the control law for the dynamic model is proposed based on classical sliding mode and fast terminal function (FTF). This control law converge the velocity error to zero and the asymptotic stability is proved.

The general control law has the ability to maintain the robot in the reference trajectory in the presence of the disturbances, which are presented in both models, dynamic and kinematic, respectively. The simulation works show the robustness of the proposed control law regarding to the different desired trajectory using for the robot, thus the convergent time from the initial state to zero is very short.

AUTHORS

Walid Benaziza*, Noureddine Slimane, Ali Mallem – Advanced Electronic laboratory, University of Batna 2, Batna, Algeria. E-mails: w.benaziza@univ-batna2.dz, Slimane_doudi@yahoo.fr, Ali_mallem@hotmail.fr. * Corresponding author

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Journal Journal of of Automation, Automation,Mobile MobileRobotics Robotics&&Intelligent IntelligentSystems Systems

VOLUME 2018 VOLUME 13,12, N°N°2 2 2018

���������� ����� �� R��������-����� �������� ���R��� ����� �������� ��� S�������� ������� R���������� �u��i�ed: �0th April 2018; accepted: 25th June 2018

Mariusz Flasiński DOI: 10.14313/JAMRIS_2-2018/12 Abstract: Further results of research into parsable graph grammars used for s�ntac�c pa�ern recogni�on (Pattern Recognition: 21, 623-629 (1988); 23, 765-774 (1990); 24, 12-23 (1991); 26, 1-16 (1993); 43, 2249-2264 (2010), Comput. Vision Graph. Image Process. 47, 1-21 (1989), Computer-Aided Design 27, 403-433 (1995), Theoret.

Comp. Sci. 201, 189-231 (1998), Pattern Analysis Applications bf 17, 465-480 (2014)) are presented in the paper. �he genera��e power of reduc�on-based parsable ETPR(k) graph grammars is in�es�gated. �he analog� between the triad of CF - LL(k) - LR(k) string languages and the triad of NLC - ETPL(k) - ETPR(k) graph languages is discussed.

Keywords: s�ntac�c pa�ern recogni�on, graph grammar, parsing

�� ��trod�c�o� Graph grammars are the strongest descriptive/generative formalism in the theory of formal languages and automata, if compared with string or tree grammars. They are used for the synthesis of formal representations in various important areas of computer science such as: software engineering, (syntactic) pattern recognition, database design, programming languages and compiler design, computer networks, distributed and concurrent computing, logic programming, computer vision, IT systems for chemistry and biology, arti�icial intelligence (natural language processing, knowledge representation and rule-based systems) [10, 11, 45]. However, the use of graph automata/parsers as tools for the analysis of graph representations in these application areas is strongly limited because the membership problem for graph languages is PSPACE- or NP- complete.

Research into this problem has been undertaken for 50 years. The �irst graph automata were de�ined in the 1960s by Blum and Hewitt [3]. For PfaltzRosenfeld web grammars generating node-labelled graphs with embedding transformations that specify inheriting edges by pointing out proper nodes of right-hand side graphs [40], the web automata were de�ined by Rosenfeld and Milgram [43] in 1972 and in 1977 the web parser by Brayer [6]. In 1978 Franck constructed the precedence relations-based syntax analyzer, O(n2 ), n is the number of nodes, [28] for NLC-like grammars [30, 32]1 with restricted embedding transformations. (Later, the complexity is stated with respect to the number of nodes n.) In the same

year Della Vigna and Ghezzi [8] proposed the parser, O(n2 ), for grammars based on the Pratt model, in which the embedding transformation is de�ined by determining input (output) nodes of right-hand side graphs which inherit the edges of left-hand sides [41].

The precedence relations-based parser, O(n2 ), was constructed by Kaul [33] for NLC-like grammars. In the early 1980s, subclasses of graph grammars with polynomial membership problem were studied by Brandenburg [4], Slisenko [49], and Turan [51]. Subsequently, the parsing algorithm, O(n2 ), for expansive graph grammars was formulated by Fu and Shi in 1983 [48]. In 1986 the polynomial parsing algorithm for boundary NLC languages was de�ined by Rozenberg and Welzl [46]. During the �irst half of 1990s three parsing algorithms, O(n2 ), based on the analogy to LL(k) grammars [36, 44] were de�ined: for the regular ETL(1) subclass of edNLC languages [13, 14], the error-correcting parser [15], and for the context-free ETPL(k) subclass of edNLC languages [16]. The �irst (polynomial) parser for Habel-Kreowski/BauderonCourcelle hyperedge replacement grammars, HR grammars, [1, 29] was constructed by Lautemann in 1988 [35]. The succeeding parsers for this class of graph grammars were proposed by Vogler in 1991 (the Cocke-Kasami-Younger-based parser), O(n3 ), [52], by Seifert and Fischer in 2004 (the Earley-based parser), O(n3 ), [47], by Mazanek and Minas in 2008 (a method based on polynomial graph parser combinators) [37], and in 2015 by Drewes, Hoffmann and Minas for the predictively top-down parsable subclass of HR grammars, O(n2 ), [9]. Two polynomial syntax analyzers for Feder plex grammars, which generate graph-like structures (called plexes) consisting of nodes with pre-de�ined attaching points (called napes), [12] were constructed independently

by Bunke and Haller [7] and by Peng, Yamamoto and Aoki [39] in 1990. For relational grammars; in which the right-hand sides are structures de�ined with relations between labelled objects and embedding is performed in an analogous way - as with plex attaching points; parsing algorithms were proposed

by Wittenburg, Weitzman and Talley in 1991 (exponential) [54] and in 1994 by Tucci, Vitiello and Costagliola (polynomial) [50]. In 1996 Wills published a paper on exponential Earley-based parsing for attributed �low graph grammars [53], which can be treated as plex grammars with attributes generating directed acyclic graphs. The exponential parser for layered graph grammars was constructed by Rekers and Schü rr in 1997 [42]. Layered graph grammars 61

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are context-sensitive grammars with decomposing node and edge alphabets into more than two layers (i.e. terminal and nonterminal layers) and imposing a kind of lexicographical order on graphs based on layers. The polynomial syntax analyzer for reserved graph grammars, which are layered grammars with reversed productions (to make parsing ef�icient), was de�ined by �hang, �hang and Cao in 2001 [55].

The automata for Janssens-Rozenberg NCE graph languages [31] were de�ined by Brandenburg and Skodinis in 2005 [5].

Graph grammars can be divided into two large families according to the embedding mechanism: grammars with connecting embedding (the set theoretic approach, the algorithmic approach) and grammars with gluing embedding (the algebraic approach) [45].

Within each of these families two standard classes of graph grammars, which are interesting for de�ining practical parsing algorithms, are distinguished [45]. The grammars with connecting embedding are VR (vertex replacement) grammars, mainly NCE-like (Neighbourhood Controlled Embedding) grammars [31] and NLC-like (Node Label Controlled) grammars [30,32]. The grammars with gluing embedding are HR (hyperedge replacement) grammars [1, 29]. Research into de�ining the subclasses with polynomial membership problem of NLC-like grammars and their applications has been carried out for the last 30 years.

The previously mentioned parsable ETPL(k) subclass of edNLC graph grammars has been successfully used for practical applications (see below).

Moreover, the inference algorithm for ETPL(k) graph languages has been de�ined [20] and its descriptive power was characterized [19]. Nevertheless, in some cases its power limitations have been revealed. These limitations result from constraints imposed on the de�inition of ETPL(k) grammar in order to make it parsable in a top-down manner. ETPL(k) grammars have been de�ined analogously to top-down parsable (string) LL(k) grammars [36, 44]. It is also known that Knuth’s reduction-based (bottom-up) parsable (string) LR(k) grammars [34] have a greater generative power than LL(k) grammars. Therefore, the reduction-based parsable ETPR(k) subclass of edNLC graph grammars has been de�ined [23, 24]. Both classes, i.e. ETPL(k) and ETPR(k) have been applied successfully for scene analysis in robotics [13], software allocation in distributed systems [25], CAD/CAM integration [18, 22], reasoning in real-time expert systems [2, 17], mesh re�inement (�inite element method, FEM) in CAE systems [27], sign language recognition [21, 26], and computer vision [24]. However, to date the formal properties of ETPR(k) graph grammars have not been presented.

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The generative power of ETPR(k) graph grammars with polynomial membership problem is presented and the analogies between parsable subfamilies of CF string and edNLC graph languages are discussed in this paper. The de�initions pertaining to edNLC graph grammars are given in Section 2. Notions of indexed and reversely indexed edge-unambiguous graphs that Articles

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enable linear ordering on EDG graphs [32] to be introduced are presented in Section 3. The de�initions concerning edNLC graph languages with polynomial membership problem are included in Section 4.

The generative power of the reduction-based parsable ETPR(k) subclass of edNLC graph languages is investigated in Section 5. The discussion on the analogy between the triad of CF - LL(k) - LR(k) string languages and the triad of NLC - ETPL(k) - ETPR(k) graph languages is presented in Section 6 and the �inal section consists of concluding remarks.

2. Preliminaries

In this section the basic de�initions of EDG graph, edNLC graph grammar and edNLC graph language are introduced [30, 32].

ϐ ͳǤ A directed node- and edge-labelled graph, EDG graph, over Σ and Γ is a quintuple H = (V, E, Σ, Γ, ϕ),

where V is a �inite� non-empty set of nodes, Σ is a �inite� non-empty set of node labels, Γ is a �inite� non-empty set of edge labels, E is a set of edges of the form (v, λ, w), in which v, w ∈ V, λ ∈ Γ, and ϕ : V −→ Σ is a node-labelling function.

The family of the EDG graphs over Σ and Γ is denoted by EDGΣ,Γ . The components V, E, ϕ of a graph H are sometimes denoted with VH , EH , ϕH .

Let A = (VA , EA , Σ, Γ, ϕA ), B = (VB , EB , Σ, Γ,, ϕB ) and C = (VC , EC , Σ, Γ, ϕC ) be EDG graphs. An isomorphism from A onto B is a bijective function h from VA onto VB such that

ϕB ◦h = ϕA and EB = {(h(v), λ, h(w)) : (v, λ, w) ∈ EA }.

We say that A is isomorphic to B, and denote it with A∼ = B. A graph C is a (full) subgraph of B iff VC ⊆ VB , EC = {(v, λ, w) ∈ EB : v, w ∈ VC } and ϕC is the restriction to VC of ϕB .

ϐ ʹǤ An edge-labelled directed Node Label Controlled, edNLC, graph grammar is a quintuple G = (Σ, ∆, Γ, P, Z),

where Σ is a �inite� non-empty set of node labels, ∆ ⊆ Σ is a set of terminal node labels, Γ is a �inite� non-empty set of edge labels, P is a �inite set of productions of the form (l, D, C), in which l ∈ Σ\∆, D ∈ EDGΣ,Γ , C : Γ × {in, out} −→ 2Σ×Σ×Γ×{in,out} is the embedding transformation, Z ∈ EDGΣ,Γ is the starting graph called the axiom. ϐ ͵Ǥ Let G = (Σ, ∆, Γ, P, Z) be an edNLC graph grammar.

(1) Let H, H ∈ EDGΣ,Γ . Then H directly derives H in G, denoted by H =⇒ H, if there exists a node v ∈ VH G

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Fig. �. �� ��amp�� �� a ��ri�a��� ���p i� a� edNLC graph grammar. and a production (l, D, C) in P such that the following holds. (a) l = ϕH (v). (b) There exists an isomorphism from H onto the graph X in EDGΣ,Γ constructed as follows. Let D be a graph isomorphic to D such that VH ∩ VD = ∅ and let h be an isomorphism from D onto D. Then

right-hand side D of a production are shown in Fig. 1b.

Let us assume that the embedding transformation is de�ined as follows:

where

The derivation step is performed in two parts.

During the �irst stage the node labelled with B of the graph h (corresponding to the left-hand side of the production) is removed, and the graph of the right-hand side replaces the removed node. The transformed graph obtained by removing the node (cf.

VH \ {v} in De�inition 3) and its ad�acent edges (cf.

EH \ {(n, γ, m) : n = v or m = v} in De�inition 3) is called the rest graph. During the second stage, the embedding transformation is used in order to connect certain nodes of the right-hand side graph with the rest graph. The item (i) is interpreted as follows: 1) Each edge labelled with y and coming in the node corresponding to the left-hand side of the production, i.e. B, shall be replaced by

X = (VX , EX , Σ, Γ, ϕX ),

VX = (VH \ {v}) ∪ VD , { ϕH (y) if y ∈ VH \ {v} ϕX (y) = ϕ D (y) if y ∈ V D ,

EX = (EH \ {(n, γ, m) : n = v or m = v}) ∪ E D ∪{(n, γ, m) : n ∈ VD , m ∈ VX\D and there exists an edge (m, λ, v) ∈ EH such that (ϕX (n), ϕX (m), γ, out) ∈ C(λ, in)}∪ ∪{(m, γ, n) : n ∈ VD , m ∈ VX\D and there exists an edge (m, λ, v) ∈ EH such that (ϕX (n), ϕX (m), γ, in) ∈ C(λ, in)}∪ ∪{(n, γ, m) : n ∈ VD , m ∈ VX\D and there exists an edge (v, λ, m) ∈ EH such that (ϕX (n), ϕX (m), γ, out) ∈ C(λ, out)}∪ ∪{(m, γ, n) : n ∈ VD , m ∈ VX\D and there exists an edge (v, λ, m) ∈ EH such that (ϕX (n), ϕX (m), γ, in) ∈ C(λ, out)}.

* we denote the transitive and re�lexive clo(2) By =⇒ G sure of =⇒ .

G (3) The language of G, denoted L(G), is the set * H and H ∈ EDG∆,Γ }.

L(G) = {H : Z =⇒ G

An example of a derivation step of an edNLC grammar is shown in Fig. 1.

The graph h which will be transformed is shown in Fig. 1a., whereas the left-hand side l = A and the

(i) C(y, in) = {(b, a, p, out)},

(ii) C(u, out) = {(b, A, x, out), (c, A, z, in)}, (iii) C(u, in) = ∅.

2) the edge:

a) connecting the node of the graph of the righthand side of the production and labelled with b with the node of the rest graph and labelled with a, b) labelled with p, c) and going out from the node b.

Thus the item (i) of the embedding transformation generates the edge of the graph h, shown in Fig. 1c, which is labelled with p and connects the nodes labelled b and a on the basis of the edge of the graph h labelled y and connecting the nodes labelled B and a (redirection and relabelling). The item (ii) duplicates an edge, and the item (iii) deletes an edge.

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Fig. 2. An example of an IE graph (a) and an rIE graph (b).

In this paper edNLC productions are depicted according to the diagrammatical convention used in [45] (see Fig. 1d for the example production). The left-hand side is depicted with a box carrying its label in the upper left corner. The box contains the right-hand side graph. The area outside the box represents the environment of the right-hand side graph. The labelled arrows pointing to/from the box to the outside specify the domain of the embedding transformation. The labelled arrows which continue an outside arrow inside the box specify the embedding of this (outside) edge.

Thus, the outside arrow can be re-established (with possible redirection/relabelling), duplicated (if continued by more than one arrow) or deleted (if not continued).

3. edNLC Graph Languages with Polynomial Membership Problem

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As discussed in [19], there are two main reasons for the problems with constructing ef�icient parsing algorithms for graph languages (compared to the algorithms for string and tree languages) the lack of ordering of the graph structure and the complexity of the embedding transformation. Firstly, consider the ordering problem.

Note that the main concept of a reductionbased syntax analysis consists of analyzing the sentence/structure in order to identify consecutive subphrases/sub-structures (handles) that correspond to right-hand sides of the productions. Once a handle is identi�ied, it is consumed, i.e. it is reduced to the left-hand side of the appropriate production. (In a top-down parse, handles have to be identi�ied as well in order to �ind the appropriate production to be applied.) In the case of a graph structure, this means to look for a subgraph (a handle) that is isomorphic to a given graph, i.e. resolving the subgraph isomorphism problem, which is known to be NP-complete.

To resolve this problem we have introduced two subclasses of EDG graphs called indexed edgeunambiguous graphs, IE graphs [13, 15] and reverse indexed edge-unambiguous graphs, rIE graphs [23] in which a linear order on a set of nodes is de�ined. A transformation of an EDG graph into an (r)IE graph Articles

can be performed, if the former is an interpreted graph [23], i.e. it represents some real-world structure2 .

Now, let us introduce the way of indexing graph nodes, which has been used for de�ining the top-down parsable ETPL(k) graph grammars [19]. Let G = (V, E, Σ, Γ, ϕ) be an EDG graph, V = {v1 , v2 , . . . , vn }. We de�ine a set of indices Ind = {1, 2, . . . , n} for V . G is called an indexed graph if indices of Ind have been ascribed to nodes of V with a bijective function.

ϐ ͶǤ Let H be an EDG graph over Σ and Γ. An indexed edge-unambiguous graph, IE graph, over Σ and Γ de�ined on the basis of the graph H is an EDG graph G = (V, E, Σ, Γ, ϕ) which is isomorphic to H up to the direction of the edges3 , such that the following conditions are ful�illed.

1. G contains a directed spanning tree T such that nodes of T have been indexed due to the Level Order Tree Traversal (LOTT)4 .

2. Nodes of G are indexed in the same way as nodes of T. 3. Every edge in G is directed from the node having a smaller index to the node having a greater index.

The family of all the IE graphs over Σ and Γ is denoted by IEΣ,Γ .

An example of an IE graph h1 is shown in Fig. 2a.

The indices are ascribed to the graph nodes according to LOTT. The edges of the spanning tree T are thickened.

The way of indexing nodes according to LOTT is convenient if one uses a top-down parsing scheme [19]. In this paper reduction-based (bottomup) parsable ETPR(k) graph grammars are characterized. The graphs generated by these grammars should be indexed according to a scheme that allows one to apply a reduction-based parsing scheme, i.e. the parser produces the rightmost derivation in reverse.

(As it is made for Knuth’s (string) LR(k) parsers [34].)

Thus, we have to de�ine a new traversal scheme for the tree spanned on an EDG graph. Such a scheme has been introduced in [23]. It is analogous to the LOTT (BFS) scheme, however it uses a LIFO queue (i.e. a stack) instead of a FIFO queue. We call it the Reverse

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Fig. 3. An example of an ETPL(k)-ETPR(k) graph grammar Level Order Tree Traversal (RLOTT). Now reversely indexed edge-unambiguous graphs can be de�ined.

ϐ ͷǤ Let H be an EDG graph over Σ and Γ. A reversely indexed edge-unambiguous graph, rIE graph, over Σ and Γ de�ined on the basis of the graph H is an EDG graph G = (V, E, Σ, Γ, ϕ) which is isomorphic to H up to the direction of the edges, such that the following conditions are ful�illed.

1. G contains a directed spanning tree T such that nodes of T have been indexed due to the Reverse Level Order Tree Traversal (RLOTT).

2. Nodes of G are indexed in the same way as nodes of T. 3. Every edge in G is directed from the node having a smaller index to the node having a greater index.

The family of all the rIE graphs over Σ and Γ is denoted by rIEΣ,Γ .

An example of an rIE graph h2 is shown in Fig. 2b.

The indices are ascribed to the graph nodes according to RLOTT. The edges of the spanning tree T are thickened.

Let us introduce the notion of node level. We say that a node v of the IE (rIE) graph is on level n, if v is on level n of the spanning tree5 T constructed as in �e�inition 4 (�e�inition 5).

We de�ine the string-like graph representation of IE (rIE) graphs as in [19]. (This form of representation

was originally de�ined for Ω graphs in [48].)

ϐ ͸Ǥ Let vk ∈ V be the node of an IE (rIE) graph H = (V, E, Σ, Γ, ϕ). A characteristic description of vk is the quadruple ( a, r, (e1 . . . er ), (i1 . . . ir ) ), where a is the label of the node vk , i.e. ϕ(vk ) = a, r is the out-degree of vk (the out-degree of the node designates the number of edges going out from this node), (i1 . . . ir ) is the string of node indices to which edges going out from vk come (in increasing order), (e1 . . . er ) is the string of edge labels ordered in such a way that the edge having the label ex comes into the node having the index ix .

For example,

( e, 3, (p t r), (4 6 7) )

is the characteristic description of the node indexed with 3 in the graph h1 shown in Fig. 2a.

ϐ ͹Ǥ Let H = (V, E, Σ, Γ, ϕ) be an IE (rIE) graph, where V = {v1 , . . . , vk } is the set of nodes indexed such that vi is indexed with i, I(i), i = 1, . . . , k is the characteristic description of the node vi . The string I(1) . . . I(k) is called the characteristic description of the graph H. Assuming a way of indexing of the graph h1 from Fig. 2a as it has been de�ined above, we obtain the folArticles

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lowing characteristic description of this graph.

grammars, de�ined by �ozenberg and �elzl [46], nonterminal nodes cannot be adjacent (in right-hand side graphs and in the axiom). In our model [13, 16] we limit the power of the embedding transformation in the following way. Firstly, we require that all graphs in a derivation are (r)IE graphs. In fact, this requirements restricts the embedding transformation, which cannot redirect edges. Secondly, we require that a derivation process is performed according to the linear ordering imposed on IE (rIE) graphs7 . It is also assumed [13, 14, 16] that during a derivation step, the root of the right-hand side inherits the index from the replaced node (corresponding to the left-hand side) and the remaining nodes of the right-hand side get the next available indices.

a 3 (t s r) (2 3 4)

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a e 2 0 (s r) − (8 9) −

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Now, the formal properties of the ETPR(k) reduction-based parsable subclass of edNLC languages can be presented. As this subclass will be compared with the ETPL(k) top-down parsable subclass of edNLC languages6 in the next section, de�initions for both classes must be introduced.

Fortunately, most corresponding notions for both classes differ only slightly, so they may be formalized by single de�initions with modi�ications. (The modi�ications for the ETPR(k) class are written in brackets in the de�initions.)

Firstly, to reduce the computational complexity of a single step of the parsing algorithm the following constraint is imposed on the form of the right-hand side graphs of the productions.

ϐ ͺǤ Let G = (Σ, ∆, Γ, P, Z) be an edNLC graph grammar. The grammar G is called a TLP graph grammar, abbrev. from Two-Level Production, if the following conditions are ful�illed.

1. P is a �inite set of productions of the form (l, D, C), where : (a) l ∈ Σ\∆, (b) D is the IE (rIE for the ETPR(k) class) graph having the characteristic description : n1 (1) r1 E1 I1

n2 (2) r2 E2 I2

... ... ... ...

nm (m) rm Em Im

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is a characteristic description of the node i, i = 1, . . . , m, n1 ∈ ∆ (i.e. n1 is a terminal label) and i, i = 2, . . . , m are nodes on level 2, (c) C : Γ × {in, out} −→ 2Σ×Σ×Γ×{in,out} is the embedding transformation.

2. Z is an IE (rIE) graph such that its characteristic description satis�ies the condition de�ined in point 1(b).

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An example of a TLP grammar is shown in Fig. 3.

Now, we will introduce restrictions on the derivation process, i.e. on the embedding transformation.

The NLC-like embedding transformation operates at the border between the left- and right-hand sides of a production and their context. Thus, we do not have the important context freeness property stated that reordering of the derivation steps does not in�luence the result of the derivation. The lack of the order-independence property, related to the �inite Church-Rosser, fCR, property (non-overlapping steps can be done in any order), results in the intractability of the parsing. Therefore, the power of the NLC-like embedding transformation must be limited in order to obtain the fCR property and to guarantee ef�iciency of parsing. For example, in boundary NLC graph Articles

ϐ ͻǤ A TLP graph grammar G is called a closed TLP (rTLP) graph grammar G if for each derivation of this grammar Z = g0 =⇒ g1 =⇒ . . . =⇒ gn G

G

G

each graph gi , i = 0, . . . , n is an IE (rIE) graph.

ϐ ͳͲǤ Let there be given a derivation of a closed TLP (rTLP) graph grammar G: Z = g0 =⇒ g1 =⇒ . . . =⇒ gn .

G

G

G

This derivation is called a regular left-hand (righthand) side derivation, denoted =⇒ ( =⇒ ) if : rl(G) rr(G) (1) for each i = 0, . . . , n − 1 a production for a nonterminal node having the least (greatest) index in a graph gi is applied, (2) node indices do not change during a derivation.

A closed TLP (rTLP) graph grammar which rewrites graphs according to the regular left-hand (righthand) side derivation is called a closed TLPO (rTLPO) graph grammar, abbrev. from (reverse) Two-Level Production-Ordered.

In order to achieve the requirements imposed by �e�initions � and 1�, the embedding transformation C of each production (l, D, C) should ful�il the following conditions.

1. C has to re-introduce (without re-directing) the incoming edge belonging to the spanning tree T of the derived (r)IE graph (cf. �e�initions 4 and �).

2. Any other incoming edge can be re-introduced and duplicated without re-directing. It can also be deleted.

3. An outcoming edge can be: (a) deleted, (b) re-introduced without re-directing, (c) used for generating new edges coming into nodes of level 2 of the right-hand side8 .

Let us de�ine the concepts used for extracting handles in the analyzed graphs which are matched against the right-hand sides of productions during the graph parsing. These concepts will be used for the ETPL(k) class as well as for the ETPR(k). ϐ ͳͳǤ Let g be an IE (rIE) graph, l the index of some node of g de�ined by a characteristic description

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If the subgraph h of the graph g from �e�inition 11 consists of the node indexed with l, nodes having indices ia+1 , ia+2 , . . . , ir , a ≥ 0, and edges connecting the nodes indexed with: l, ia+1 , ia+2 , . . . , ir , then it is denoted h = CT L(g, l, ia+1 ).

Now, the fundamental constraint which is analogous to that used in a de�inition of string LL(k) grammars can be imposed. This constraint allows an ef�icient, non-backtracking, top-down parsing scheme for edNLC grammars to be constructed. In order to introduce the idea of this scheme in an intuitive way, an LL(k) grammar [36, 44� is de�ined.

Let G = (Σ, ∆, P, S), Σ a set of symbols, ∆ a set of terminal symbols, P a set of productions and S the starting symbol, be a context-free grammar. Let η ∈ Σ∗ , and |x| denote the length of the string x ∈ Σ∗ .

FIRSTk (η) denotes a set of all the terminal pre�ixes of strings of length k (or less than k, if a terminal string shorter than k is derived from α) that can be derived from η in the grammar G9 , i.e. * xβ ∧ |x| = FIRSTk (η) = {x ∈ ∆∗ : (η =⇒ G * x ∧ |x| < k) , β ∈ Σ∗ }. k) ∨ (η =⇒ G

* Let =⇒ denote a leftmost derivation in G, that is a l(G) derivation such that a production is always applied to the leftmost nonterminal10 , N = Σ \ ∆ be a set of nonterminal symbols.

ϐ ͳʹǤ Let G = (Σ, ∆, P, S) be a context-free grammar. The grammar G is called an LL(k) grammar if for every two leftmost derivations * * αAδ =⇒ αβδ =⇒ αx S =⇒ l(G)

l(G)

l(G)

* * αAδ =⇒ αγδ =⇒ αy, S =⇒ l(G)

l(G)

l(G)

where α, x, y ∈ ∆∗ , β, γ, δ ∈ Σ∗ , A ∈ N, the following condition holds.

If FIRSTk (x) = FIRSTk (y) then β = γ.

The LL(k) condition means that for any step during a derivation of a string w ∈ ∆ in G, we can choose a production in an unambiguous way on the basis of an analysis of some part of w which is of length at most k. We can say that an LL(k) grammar has the property of an unambiguous choice of a production given the k-length pre�ix in the leftmost derivation.

Now, by analogy, we de�ine a PL(k) graph grammar which has the property of an unambiguous choice of a production given the k − T L graph in the regular left-hand side derivation.

ϐ ͳ͵Ǥ Let G = (Σ, ∆, Γ, P, Z) be a closed TLPO graph grammar. The grammar G is called a Articles

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Journal of Journal of Automation, Automation,Mobile MobileRobotics Robotics&&Intelligent IntelligentSystems Systems

C4 p

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VOLUME N°22 2018 2018 VOLUME 13,12, N°

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PL(k), abbrev. Production-ordered k-Left nodes unambiguous, graph grammar if the following condition is ful�illed. Let * * h1 X1 AX2 =⇒ g1 =⇒ Z =⇒ rl(G)

and

rl(G)

rl(G)

* * X1 AX2 =⇒ g2 =⇒ h2 , Z =⇒ rl(G)

rl(G)

rl(G)

* is the transitive and re�lexive closure of where =⇒ rl(G) =⇒ , be two regular left-hand side derivations, such rl(G) that A is a characteristic description of a node indexed with l, and X1 and X2 are characteristic descriptions of subgraphs. Let max be a number of nodes of the graph X1 AX2 . If k − T L(h1 , l, max + 1) ∼ = k − T L(h2 , l, max + 1)

then

CT L(g1 , l, max + 1) ∼ = CT L(g2 , l, max + 1).

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For example, our graph grammar shown in Fig. 3 is PL(2). As we can see in Fig. 4 in order to identify which production has been applied to a node indexed with 3, we have to analyze 2 − T L graphs originated in this node. (1 − T L graphs for productions 2 and 3 are the same.)

For de�ining reduction-based (bottom-up) parsable graph grammars we have used the same methodology as in the case of top-down parsable grammars.

That is, we have imposed a constraint which is analogous to that used in the de�inition of �nuth�s string Articles

LR(k) grammars [34] allowing us to construct an ef�icient, non-backtracking, bottom-up parsing scheme for edNLC grammars. Therefore, we �irstly de�ine an LR(k) grammar. * denote a rightmost derivation in G, that is Let =⇒ r(G) a derivation such that a production is always applied to the rightmost nonterminal11 . A string which occurs in the rightmost derivation of some sentence is called a right-sentential form.

ϐ ͳͶǤ Let G = (Σ, ∆, P, S) be a context-free grammar. The grammar G is called an LR(k) grammar if for every two rightmost derivations * αAw =⇒ αβw S =⇒ r(G)

r(G)

* S =⇒ γBy =⇒ αβx, r(G)

r(G)

where w, x, y ∈ ∆ , α, β, γ ∈ Σ∗ , A, B ∈ N, the following condition holds. ∗

If FIRSTk (w) = FIRSTk (x) then α = γ , A = B , x = y.

The LR(k) condition means that for each rightsentential form we can identify a handle (i.e. the right-hand side of some production) and we can choose a production in an unambiguous way12 by looking at most k symbols beyond the handle. We can say that an LR(k) grammar has a property of both the identi�ication of a handle and an unambiguous choice of a production given k symbols ahead in a right-sentential form. �ow, by analogy, we de�ine a PR(k) graph grammar, which has the property of both

Journal of Journal of Automation, Automation, Mobile MobileRobotics Robotics&&Intelligent IntelligentSystems Systems

VOLUME 2018 VOLUME 13,12, N°N° 2 2 2018

1

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…s

Fig. 6. A graph h of the language L0 that cannot be generated by any ETPR(k) grammar an identi�ication of a handle and an unambiguous choice of a production given a k − T L graph beyond the handle in a regular right-hand side derivation.

ϐ ͳͷǤ Let G = (Σ, ∆, Γ, P, Z) be a closed rTLPO graph grammar. The grammar G is called a PR(k), abbrev. Production-ordered k-Right nodes unambiguous, graph grammar if the following condition is ful�illed. Let * X1 AX2 =⇒ X1 gX2 , Z =⇒ rr(G)

rr(G)

and

rr(G)

* Z =⇒ X3 BX4 =⇒ X1 gX5 , rr(G)

k − T L(X2 , 1, 2) ∼ = k − T L(X5 , 1, 2) ,

* is the transitive and re�lexive closure of where =⇒ rr(G) =⇒ , A, B are characteristic descriptions of certain rr(G)

nodes, X1 , X2 , X3 , X4 , X5 are characteristic descriptions of subgraphs, g is the right-hand side of a production: A −→ g.

Then: X1 = X3 , A = B , X4 = X5 .

The last restriction that has to be imposed concerns the embedding transformation. We have already introduced limitations for the embedding transformation which guarantee that all graphs during a derivation are IE (rIE) graphs and that node indices do not change during a derivation (�e�initions 9 and 10). Nevertheless, these conditions do not guarantee that during parsing the characteristic description of a node does not change (e.g. after its analysis by a parser). Of course, it is an unwanted effect. For example, let us modify the de�inition of production 4 of our grammar shown in Fig. 3e. A modi�ied production (4’) is shown in Fig. 5b. The results of applying productions 4 and 4’ to a graph shown in Fig. 5a are shown in �igures 5c and 5d, respectively.

One can easily notice that during parsing with the modi�ied grammar, after analyzing a node indexed with 3, its characteristic description changes, because the embedding transformation of production 4’ does not re-introduce an edge labelled with p. We will claim such edges need to be re-introduced. Let us also notice that the issue concerns only edges incoming to the root of the right-hand side, since they have already been analyzed by the parser. (If the embedding transformation for the root node v does not re-introduce

an edge outgoing from v, then the parser, analyzing the handle originated at v, ”sees” such a situation.)

ϐ ͳ͸Ǥ Let G = (Σ, ∆, Γ, P, Z) be a PL(k) (PR(k)) graph grammar. A pair (b, x), b ∈ ∆, x ∈ Γ, is called a potential previous context for a node label a ∈ Σ\∆, if there exists the IE (rIE) graph g = (V, E, Σ, Γ, ϕ) belonging to a certain regular left-hand (right-hand) side derivation in G such that : (k, x, l) ∈ E, ϕ(k) = b and ϕ(l) = a.

ϐ ͳ͹Ǥ A PL(k) (PR(k)) graph grammar G = (Σ, ∆, Γ, P, Z) is called an ETPL(k) (ETPR(k)), abbrev. from Embedding Transformation - preserving Production-ordered k-Left (k-Right) nodes unambiguous, graph grammar if for each production (l, D, C) ∈ P the following condition is ful�illed.

Let l = A, X1 , X2 , . . . , Xm , where Xi ̸= Xj , i, j = 1, . . . , m, be labels of nodes indexed with 1, 2, . . . , m of the right-hand side graph D. For each potential previous context (b, y) for A, there exists (Xi , b, z, in) ∈ C(y, in), i ∈ {1, . . . , m}. If i = 1, then z = y, i.e. (X1 , b, y, in) ∈ C(y, in).

A parsing algorithm, O(n2 ), for ETPR(k) graph grammar was de�ined in [23]. It is a slight modi�ication of the parsing scheme for ETPL(k) graph grammar [16].

�� �e�e����e ���e� �� ������� g���� ���g��� ges

In this section the generative power of ETPR(k) graph languages is characterized in an analogous way as was made for ETPL(k) graph languages in [19]. Finally, two theorems concerning both classes of languages are proved.

Let X denote a class of graph grammars. Then L(X) denotes a set of graph languages such that there exists an X grammar G and L = L(G).

Additionally, we say that a language L is ETPL(k) (ETPR(k)), if there exists an ETPL(k) (ETPR(k)) grammar G such that L = L(G).

Firstly, we will show that the class of ETPR(k) languages is a proper subclass of the class of edNLC languages. Comparing the generative power of both classes, we are interested in their intrinsic properties, which do not result from assuming the speci�ic indexing for graphs as in the case of ETPR(k) languages.

(Since, obviously, any ”ordered” version of a class of Articles

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d) (3) Fig. �. ��� a�i�m a�� ��� �r�������� �� ��� edNLCo grammar G0 .

1

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a) u

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(i)

…s

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(2m+1)

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XC

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graph languages is its ”subfamily”.) Therefore, to compare the essential generative power of both families an ordered version of ”pure” edNLC languages will be de�ined.

Let H ∈ EDGΣ,Γ . Then H[o] denotes a graph obtained from H by indexing its nodes and re-directing (if necessary) its edges in such a way that H[o] ful�ils the conditions of �e�inition �, i.e. H[o] is an rIE graph.

A grammar G = (Σ, ∆, Γ, P, Z) is called an ordered Articles

edNLC grammar corresponding to G, denoted edNLCo , if 1 =⇒ 2 =⇒ * H[o] L(G) = {H[o] : Z[o] =⇒ H[o] rr(G) rr(G) rr(G) * =⇒ H m = H and H ∈ EDG∆,Γ }. rr(G)

[o]

[o]

Theorem 1. For any k ≥ 0

L(ETPR(k)) ⊆ \ L(edNLCo ) .

Proof. PART 1: L(ETPR(k)) ⊆ L(edNLCo ).

Journal of Journal of Automation, Automation,Mobile MobileRobotics Robotics&&Intelligent IntelligentSystems Systems

Let L ∈ L(ETPR(k)), i.e. there exists an ETPR(k) grammar G such that L = L(G). We should show that L ∈ L(edNLCo ), i.e. that there exists an edNLCo grammar G such that L = L(G) = L(G).

One can easily note that setting G := G is suf�icient, because any ETPR(k) grammar is also an edNLCo grammar.

PART 2: L(ETPR(k)) ̸= L(edNLCo ).

We will de�ine a language L0 ∈ L(edNLCo ) that cannot be generated by any ETPR(k) graph grammar. Let us introduce a language which is of the complementary palindromic form. In case of strings, a complementary palindrome is a sequence of symbols which reads in reverse as the complement of the forward sequence.

It means that for each symbol its complementary symbol has to be de�ined. For example, in DNA a symbol A is complementary to T , and C is complementary to G. Thus, for example the DNA sequence GGCAT GCC is a complementary palindrome.

Let L0 consist of rIE graphs of the complementary palindromic form as the graph h shown in Fig. 6.

Let us assume that a node label c is complementary to a node label b. The graph h is ”divided” with the edge (1, u, 2). A string of node labels of a ”path” on the right-hand side of this edge (without a node indexed with 2) is a complementary palindrome of a string of node labels of the left-hand side ”path” (also without a node indexed with 2). That is, for any n-node graph h = (V, E, Σ, Γ, ϕ) ∈ L, n - an even number, the following holds. ϕ(1) = a; ϕ(2) = b or ϕ(2) = c; for an odd index k = 3, 5, . . . , (n − 1): if ϕ(k) = b then ϕ(k + 1) = c, and if ϕ(k) = c then ϕ(k + 1) = b. E = {(1, u, i), i = 2, . . . , n} ∪ {(2, y, (n − 1)}, (2, s, n)} ∪ {(k, s, (k + 2)), k = 3, 5, 7, . . . , (n − 3)} ∪ {(k, y, (k + 2)), k = 4, 6, 8, . . . , (n − 2)}. We will call L0 the complementary palindromic graph language.

Now, we de�ine an edNLCo grammar G0 = (Σ0 , ∆0 , Γ0 , P0 , Z0 ) generating the language L0 . (Without loss of generality we assume that during derivation all nodes indexed with k = 3, 4, . . . , n are generated directly as terminal nodes, i.e. not via nonterminal nodes.) Σ0 = {a, b, c, A}, ∆0 = {a, b, c}, Γ0 = {s, u, y}, P0 and Z0 are shown in Fig. 7.

Now, we will show that L0 cannot be generated by any ETPR(k) grammar. Let us assume, proving by contradiction, that there exists an ETPR(k) grammar G1 which generates L0 . Then, let us assume that we generate the n-node graph h, n ≥ 6, belonging to L0 . Let D be the following derivation of h: * * Zo =⇒ h1 =⇒ hr = h. hk =⇒ hk+1 =⇒ rr(G)

rr(G)

rr(G)

rr(G)

Let us assume that the graph hk has (2m + 2) nodes and n ≥ 2m + 6, i.e. we have still to generate at least four nodes. Let us assume also that hk+1 has more than (2m + 2) nodes, i.e. new nodes are generated during hk =⇒ hk+1 of D. rr(G)

Note that the graph hk has to be of the form shown

VOLUME 2018 VOLUME 13,12, N°N°2 2 2018

in Fig. 8a. The form of hk results from the following facts. All the nodes of hk , except for the node indexed with 2, have to be labelled with terminals. (According to De�inition 1� we apply a production to a nonterminal node having the greatest index.) The graph hk has to be of the proper, i.e. complementary palindromic, form. That is, the right-hand side ”path” has to be a complementary palindrome of the left-hand side ”path”, because we use a context-free graph grammar that does not possess the mechanisms allowing one to take into account previous derivation steps (and the terminal ”context” already derived) in a further derivation process. It means that a graph derived cannot be recti�ied later, if it does not conform to the complementary palindromic form. Ascribing indices to nodes of hk has also to be de�initive since according to De�inition 1� node indices do not change during a derivation. The edges have to be directed as in Fig. 8a according to De�inition 5. Obviously, the labels of edges connecting terminal nodes have to be de�initive.

At the step hk =⇒ hk+1 of D we have to gerr(G) nerate two new nodes simultaneously because of a palindromic-like structure of h. Let us assume that the node indexed with (2m + 3) of h is labelled with b and the node indexed with (2m + 4) of h is labelled with c. (For the opposite labelling reasoning is analogous.)

The production of G1 which is to be applied for generating the succeeding pair of complementary nodes has to be of the form shown in Fig. 8b, where - X is a nonterminal node used for generating the succeeding pairs of complementary nodes in further derivation steps, - XB is a terminal node labelled with b or XB is a nonterminal node and the production replacing XB with a terminal node labelled with b belongs to G1 , and

- XC is a terminal node labelled with c or XC is a nonterminal node and the production replacing XC with a terminal node labelled with c belongs to G1 .

Let us note that according to De�inition 8 the righthand side graph of the production (i) has to be a two-level graph. Moreover, the root of the right-hand side has to inherit the index from the replaced node.

Thus, the node X has to be the root of the right-hand side. However, it is contrary to the condition of De�inition 8 saying that the root of the right-hand side has to be a terminal node.

Q.E.D.

The parameter k in de�initions of both ETPR(k) and ETPL(k) graph grammars has shown to be very useful from a practical point of view in many applications of these grammars e.g.: robotics [13], distributed systems [25], CAD/CAM [18, 22], industrial-like control [2, 17], CAE (FEM computing) [27], sign language recognition [21, 26], computer vision [24]. In [19] we have proved that by increasing this parameter we strengthen the generative power of ETPL(k) grammars. By proving the following theorem, which establishes the hierarchy of ETPR(k) grammars, we show that the same holds for the ETPR(k) class.

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Fig. 9. The language that cannot be generated by any ETPR(m) grammar

Theorem 2. For a given k ≥ 0 L(ETPR(k)) ⊆ \ L(ETPR(k + 1)) .

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Proof. PART 1: L(ETPR(k)) ⊆ L(ETPR(k + 1)).

Let G be an ETPR(k) grammar. �ne should de�ine such an ETPR(k + 1) grammar G that L(G) = L(G).

Let us note that it is suf�icient to set : G = G. PART 2. L(ETPR(k)) ̸= L(ETPR(k + 1)).

Let us take any k = m. �e de�ine an ETPR(m + 1) language L which cannot be generated by any ETPR(m) grammar. Let L = L1 ∪ L2 . The rIE graphs belonging to both L1 and L2 are of the cascade-like form shown in Fig. 9a. Firstly, let us de�ine this cascade-like structure. Each n-node graph h = (V, E, Σ, Γ, ϕ) ∈ L, consists of: two nodes indexed with 1 and 2 such Articles

that ϕ(1) = d, ϕ(2) = e, (1, s, 2) ∈ E, and p levels, p = 1, 2, . . ., assuming that it has at least two levels.

Each level consists of (m + 1) nodes indexed as shown in Fig. 9a. If we denote the hth node of the level lev with vhlev , then its index ilev h = 2+(lev−1)·(m+1)+h.

A set of edges E contains additionally the following edges (cf. Fig. 9): - (2, t1 , i11 ), (2, t2 , i12 ), . . . , (2, t(m+1) , i1(m+1) ), - for each level l = 2, . . . , p: (l−1) (l−1) (l−1) , t1 , il1 ), (i1 , t2 , il2 ), . . . , (i1 , t(m+1) , il(m+1) ). (i1 �ow, we de�ine the way of labelling the graph nodes belonging to levels l = 1, . . . , p (cf. Figs. 9b and c.)

- If h1 ∈ L1 , then ϕ(il1 ) = a, for h = 2, . . . , m : ϕ(ilh ) = b, and ϕ(il(m+1) ) = b.

- If h2 ∈ L2 , then ϕ(il1 ) = a, for h = 2, . . . , m : ϕ(ilh ) = b, and ϕ(il(m+1) ) = c.

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Z 1

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Fig. ��. ��� a�i�m a�� ��� �r�������� �� ��� ETPR(m + 1) grammar G We will call L the (m+1)-height-step cascade graph language.

Now, let us de�ine an ETPR(m + 1) grammar G = (Σ, ∆, Γ, P, Z) which generates a language L. Σ = {a, b, c, d, e, S, A, B}, ∆ = {a, b, c, d, e}, Γ = {s, t1 , t2 , . . . , t(m+1) }, P and Z are shown in Fig. 10.

It can be easily noted that to generate a graph h1 ∈ L1 having l levels one has to apply production 1 once, production 3 l times, and production 5 once, and to generate a graph h2 ∈ L2 having l levels one has to apply production 2 once,production 4 l times, and production 6 once. The grammar G obviously does not generate any graphs not belonging to L. Thus, L = L(G).

Now, we show that G is the only grammar of the ETPR class which generates L. Firstly, let us note that graphs belonging to L are, in fact, trees.

Secondly, according to �e�inition � the right-hand side graph of any production in an ETPR grammar has to be a graph of level at most 2. A node of the righthand side graph indexed with 1 has to be terminal.

Then, a node of the right-hand side graph indexed with 2 of the productions used for developing succeeding levels13 has to be nonterminal14 .

Thirdly, let us note that a higher level of any graph belonging to L has to be generated at one derivation step, i.e. the right-hand sides of productions used for developing succeeding levels have to be two-level trees having (m + 1) children15 . To show it, let us assume, proving indirectly, that some level p can be generated in stages. It means that at the �irst stage we generate a subtree having k children, k < m + 1, indexed with: ip1 , . . . , ipk (cf. Fig. 9a). Now, on the basis of a node indexed with ipk we have to generate the Articles

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1 ̶ TL

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g

a)

b)

c)

Fig. 11. The language that cannot be generated by any ETPL(k) grammar next child, i.e. ip(k+1) with some production p. We have to use the embedding transformation of p to connect the newly-generated child with the parent, i.e. to ge(p−1) , t(k+1) , ip(k+1) ) (cf. Fig. 9). Honerate an edge (i1 wever, let us note that we will also have to destroy an edge connecting nodes ipk and ip(k+1) in a further derivation, which is forbidden by the principle of preserving a potential previous context (cf. De�inition 17).

On the other hand, G is not an ETPR(m) grammar.

During a derivation of any h ∈ L, in spite of the fact that m − T L graphs described by De�inition 15 are isomorphic, the corresponding right-hand side graph (the handle) can be reduced to various left-hand side nonterminals. For example, m − T L right-hand side graphs of productions 5 and 6 are isomorphic, however, these productions reduce to Q.E.D. different nonterminals A and B 16 .

At the end of this section we show that both classes ETPL and ETPR are incomparable.

Theorem 3. There exists

L ∈ L(ETPR(1)) such that for any k ≥ 0 L ̸∈ L(ETPL(k)).

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Proof. In [19] (cf. Theorem 4, [19]) we have de�ined a language L which cannot be generated by any ETPL(k), k ≥ 0 grammar. The language L consists of three graphs h1 , h2 , and h3 shown in Fig. 11a. If the upper path �inishes with a node labelled f , then the lower path can �inish with a node labelled either d or e. However, if the upper path �inishes with a node labelled g, then the lower path can �inish only with a node labelled d. We will call L a third-level contextual graph language, since contextual dependencies between pairs of node labels occur at the third level of a graph.

We will de�ine an ETPR(1) grammar G which generates the language L. In Figs. 11b and c we have shown the proper reductions during the bottom-up parsing. They help us to de�ine the following grammar G = (Σ, ∆, Γ, P, Z). Σ = {a, b, c, d, e, f, g, Bde , Bd , Cf , Cg , S}, ∆ = {a, b, c, d, e, f, g}, Γ = {r, s, t}, the axiom Z consists of the one-node graph labelled with S, P is shown in Fig. 12.

One can easily note that L = L(G).

Q.E.D. Theorem 4. There exists

L ∈ L(ETPL(1)) such that for any k ≥ 0 L ̸∈ L(ETPR(k)).

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S

S Cf 3

r a 1 t

Cg 3

r a 1 t

Bde 2

Bd 2

a) (1) a

t

b) (2)

Bde t 1

a b

s

d

t

Bde t

2

1

c) (3) a

t

1

s

e

2

d) (4)

Bd t

b

Cf b

s

d

2

r

r a

e) (5)

1

c

s

f

2

f) (6)

Cg r

r

a

1

c

2

s g

f) (6)

Fig. ��. ��� �r�������� �� ��� ETPR(1) grammar G Proof. Firstly, we de�ine an ETPL(1) language L. Let L = L1 ∪ L2 . Each n-node graph, n ≥ 7, h1 = (V1 , E1 , Σ1 , Γ1 , ϕ1 ) ∈ L1 is of the form shown in Fig. 13a. The graph consists of a node having the characteristic description (a(1), 2, (tr), (23)) and two paths. A sequence of nodes in each path is connected with edges labelled s. The lower path consists of: - a subsequence of nodes indexed with: 2, . . . , (2k − 2), k ≥ 2 and labelled b, - a (distinguished) node indexed with 2k, k ≥ 2 and labelled d, and - a subsequence of nodes indexed with: (2k + 2), . . . , k ≥ 2 and labelled f .

The upper path consists of: - a subsequence of nodes indexed with: 2, . . . , (2k − 1), k ≥ 2 and labelled c, - a (distinguished) node indexed with (2k + 1), k ≥ 2 and labelled e, and - a subsequence of nodes indexed with: (2k + 3), . . . , k ≥ 2 and labelled g.

The lengths of both paths (de�ined as a number of no-

des in a sequence) can be various.

Additionally, there exists a bridge = ((2k + 1), u, (2k + 2)) ∈ E1 . In other words, let dist(a, d) denote the number of nodes between the node labelled a and the node labelled d, and dist(a, e) denotes a number of nodes between the node labelled a and the node labelled e. Then, dist(a, d) = dist(a, e) and there exists a bridge ∈ E1 .

L2 consists of graphs analogous to the graphs of L1 . However, dist(a, d) ̸= dist(a, e) and, there is no edge (bridge) connecting both paths. Summing up, an edge called a bridge occurs in a graph h ∈ L iff dist(a, d) = dist(a, e). We will call L the contextuallyconditioned-bridge graph language.

Let us the de�ine an ETPL(1) grammar G = (Σ, ∆, Γ, P, Z) generating the language L. Σ = {a, b, c, d, e, f, g, B, C, D, E, F, G}, ∆ = {a, b, c, d, e, f, g}, Γ = {p, r, s, t, u, x, y}, P and Z are shown in Fig. 14.

An example of generating a bridge in case dist(a, d) = dist(a, e) is shown in Fig. 13b. One can Articles

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dist(a,e)

r h1 = a 1 t

3 (2k – 1) (2k + 1) c s … s c s e s g s …… s g u b 2

s… s s s s… s d f f b (2k – 2) (2k) (2k + 2) dist(a,d)

3

Z=

1

r a t

C p

B 2

4

6

…2

r a 1 t

(2k – 1) 3 … c s s C p 3 b 2

s… s B (2k – 2)

3 (2k – 1) (2k + 1) c s …s c s E 5 x

r a 1 t

r a 1 t

1

a)

b 2

s s …s D b (2k – 2) (2k)

3 (2k – 1) (2k + 1) … c s s c s e s G u b 2

r a 1 t

7

r a 1 t

s s s …s d b F (2k – 2) (2k) (2k + 2)

r a 1 t

3 (2k – 1) c s …s C b 2

y

s s… s D b (2k – 2) (2k)

(2k – 1) (2k + 1) 3 c s …s c s E u b 2

s s s …s d b F (2k – 2) (2k) (2k + 2)

3 (2k – 1) (2k + 1) … c s s c s e s G u 8 … s s s s …s d f F b b 2 (2k – 2) (2k) (2k + 2)

b) Fig. 13. The ETPL(1) language that cannot be generated by any ETPR(k) grammar easily see that L = L(G).

Now, we show that L cannot be generated by any ETPR(k) grammar. First of all, let us note that any rIE graph h1 ∈ L1 has to be indexed as shown in Fig. 15a.

If h1 has to belong to L1 , then dist(a, d) = dist(a, e).

Assuming indexing de�ined as in Fig. 15a (i.e. a node labelled e is indexed with k), it means that a node labelled d has to be indexed with (i + k − 3).

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According to the �e�inition 10 of ETPR(k) grammars the upper path of h1 has to be generated �irstly, as we can see in Fig. 15b. (An edge (2, r, i) is to be established in order to generate a bridge.) However, one can easily see that we do not know whether an index j of a node labelled d ful�ils the condition� j = (i + k − 3)17 . In consequence, we do not know whether to establish a bridge (in case h1 ∈ L1 ) or not (in case h1 ∈ L2 ).

Q.E.D. Articles

5. CF and NLC Languages with Polynomial Membership Problem As stated in the introduction, research into the theory of parsing for NLC graph grammars has been conducted for thirty years. The graph grammars of the edNLC class [30] were chosen as the basis for this research from the outset, which has proved to be appropriate. On the one hand, the edNLC class has been revealed as descriptively strong enough to be successfully applied for solving the previously mentioned realworld problems [2, 13, 17, 18, 21, 22, 24–27]18 . On the other hand, the edNLC class has turned out to be �lexible enough to enable us to de�ine the deterministic subclasses with polynomial membership problem, and in consequence - the ef�icient parsing algorithms. �oreover, the way of de�ining edNLC graph grammars has enabled to de�ine these deterministic subclasses,

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3

r a

1

t

C

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B

a

p

C

t

b

s

B

t

b 1

s

p s

C B 2

a

2

b

C

B t s

t

s

p

C

y

D s

b 1

s

c

c

s

s

r

2

a

1 e

s

2

G u

f

s

d s

s

s

s

f

F

2

E

b

u

y

s s

u u f 1

j) (9)

p

t

s

D

B

E d 1

x

u s

F 2

f) (5)

G

u

f 1

s

F s 2

e s

s

g s

s

h) (7) e

F d s

e

F

C

r

D

x e) (4)

u g) (6)

s

r

r

2

s

c) (2)

1 c

s

d) (3)

E

1 c

s

b) (1)

a) Z

a

s

c

g 1

s

G 2

i) (8)

G e s g s

s s

g 1

k) (10)

Fig. 14. The ETPL(1) g������ ge�e����g �he ����e������������i���e����i�ge g���h ���g��ge. using constructs and mechanisms analogous to those used in the theory of parsing of string languages. This analogy is especially noteworthy, since from the methodological/paradigmatic point of view, analogies of this kind are highly desirable [21].

First of all, let us note that the essential properties of both subclasses of edNLC graph grammars, namely top-down parsable ETPL(k) and reductionbased (bottom-up) parsable ETPR(k), expressed by �e�initions 13 and 1� are analogous to the de�initions of theirs counterparts, namely subclasses of topdown parsable LL(k) [36, 44] and bottom-up parsable LR(k) [34] CF (context-free) grammars in the parsing theory of string languages.

Secondly, these analogies have proved to be useful when studying the formal properties of ETPR(k) languages presented in a previous section. For example, the well-known fact that the (string) CF language of palindromes cannot be generated by any LR(k) gram-

mar has inspired us to construct the edNLC complementary palindromic graph language in order to show that it cannot be generated by any ETPR(k) grammar (the proof of Theorem 1). On the other hand, investigating whether ETPR(k) languages constitute a hierarchy, we have analyzed the Mickunas-LancasterSchneider strati�ication-based method used for proving that LR(k) languages do not constitute a hierarchy [38]. The study has revealed that the strati�ication trick [38] cannot be made in case of graph structures.

Knowing why this is impossible, we have been able to de�ine the ETPR (m+1)-height-step cascade graph language in order to show that ETPR(k) languages constitute a hierarchy (the proof of Theorem 2).

In our previous paper concerning the generative power of ETPL(k) languages [19] we have proved, among others, the following two theorems.

Theorem (1. in [19]) For a given k ≥ 0

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Z

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4 5 … (k – 1) k (k + 1) i 3 c s c s c s … s c s e s g s …… s g u a) s s s s s… s s s d f f b b b b … … n (i + k – 3) … (i + 1) (i + k – 2) … (i + 2) 2 5 … (k – 1) k (k + 1) 4 i 3 … … … c s c s c s s c s e s g s s g r

B 2

…

r a 1 t

4 5 … (k – 1) k (k + 1) i 3 c s c s c s … s c s e s g s …… s g u j = (i + k – 3) ? s s s s s … s d F b b b b 2 (i + 1) (i + 2) … … ( j – 1) j ( j + 1) b)

Fig. ��. �h� h���th���a� ��ri�a��n �� th� gra�h h1 ∈ L1 with an ETPR(k) grammar.

L(ETPL(k)) ⊆ \ L(ETPL(k + 1)) .

Theorem (2. in [19]) For any k ≥ 0

L(ETPL(k)) ⊆ \ L(edNLCo ) .

These two theorems together with the ones proved in a previous section allow us to establish a diagram presenting the relationships among the families of parsable edNLC languages shown in Fig. 16b. An analogous diagram for (string) CF languages is shown in Fig. 16a. The analogy between both basic classes of languages, i.e. (string) CF and (graph) edNLC, can be easily noted. However, there are also some essential differences. The �irst one consists of the lac� of a hierarchy in the case of a bottom-up parsable subclass of the edNLC class. The second difference is crucial from an application point of view. Whereas the family of LL-type languages is strictly contained in the family of LR-type languages, the classes ETPL and ETPR are not comparable. Although the insuf�icient descriptive power of ETPL-type languages for solving certain real-world application problems was the original motivation of the author for conducting research into a bottom-up parsable subclass of edNLC grammars, �inally it was shown that both parsable subclasses are needed and that they complement each other.

6. Concluding Remarks

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The following two goals were the focus of our research into Node Label Controlled (NLC) graph grammars formulated in [30].

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- To establish a theory of parsing for NLC graph languages (the theoretical-oriented research area).

- To apply this theory to various real-world problems, which re�uire ef�icient algorithmic schemes of graph (sets of graphs) processing (the application-oriented research area), for their solution.

Two generic types of parsable subclasses of languages with polynomial membership problem are considered in the theory of parsing: the top-down parsable languages and the reduction-based (bottom-up) parsable ones. These two generic subclasses have both their pros and cons. Therefore, within the theoreticaloriented area of our research two subclasses of NLC graph grammars have been developed, namely topdown parsable ETPL(k) (analogous to LL(k) grammars [36, 44]) and bottom-up parsable ETPR(k) (analogous to LR(k) grammars [34]). The generative power of the former has been presented in [19] and the latter - in this paper. Additionally, we have compared generative power of both subclasses as well. Finally, we have discussed the analogy between the triad of CF - LL(k) - LR(k) string languages and the triad of NLC ETPL(k) - ETPR(k) graph languages.

Apart from the previously discussed theoretical results, both parsable subclasses of NLC graph grammars have been successfully used in a variety of applications such as: scene analysis in robotics [13], software allocation in distributed systems [25], CAD/CAM integration [18, 22], reasoning in real-time expert systems [2, 17], mesh re�inement (�inite element met-

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edNLC

CF ⊆

⊆ LL . ..

⊆

⊆

LR . ..

⊆ LL(k) . ..

= =

LR(k)

⊆

.. .

⊆ LL(1)

⊆

⊆

= =

LR(1)

ETPL . ..

⊆

∃ L ∈ETPR: L ∉ETPL ∃ L ∈ETPL: L ∉ETPR

ETPR . ..

⊆

⊆ ETPR(k)

ETPL(k) . ..

⊆

. ..

⊆

⊆ ⊆

ETPR(1)

ETPL(1)

a)

⊆

b)

Fig. 16. The analogy between CF and edNLC� a) �ela�onshi�s a�ong �a�ilies o� �a�sable CF languages, b) �ela�onshi�s a�ong �a�ilies o� �a�sable edNLC languages hod, FEM) in CAE systems [27], sign language recognition [21, 26] and computer vision [24] within the application-oriented area of our research .

Summing up, the class of NLC graph grammars has proved to be an attractive theoretical model because of its well-balanced properties. That is, on one hand, due to its simplicity, formal elegance and strong descriptive power, and on the other hand because of its �lexibility allowing it to be used in a variety of real-world applications. In our opinion, NLC graph grammars provide an attractive reference model for the theory of parsing of graph languages and that they will play a key role in the further development of this theory.

Notes

1 NLC grammars are introduced below.

2 This condition concerning (r)IE graphs can be ful�illed easily in practice. (r)IE graphs have been used as a descriptive formalism for representing: combinations of objects of scenes analyzed by industrial robots [13], con�igurations of hardware/software components analyzed by distributed software allocators [25], structures consisting of geometrical/topological features of machine parts in CAD/CAM integration systems [18, 21], semantic networks/frames in real-time expert systems [2, 17], grids analyzed with Finite Element Analysis (FEA) methods in Computer Aided Engineering (CAE) systems [27], hand postures analyzed by sign language recognition systems [21, 26].

3 That is, (some) edges of G can be re-directed with respect to their counterparts in H. 4 Let us recall that LOTT means that for each node �irstly the node is visited, then its child nodes are put into the FIFO queue. This type of a tree traversal is also known as the Breadth First Search (BFS) scheme.

5 We assume that the root is on level 1, its children are on level 2, etc.

6 Formal properties of the ETPL(k) class have been presented in [19].

7 Analogously, as for parsable LL(k)/LR(k) string grammars a leftmost/rightmost derivation is required.

8 The formalization of these conditions is contained in the paper on inferencing ETPL(k) graph grammars [20].

9 Both notions: k−T L graph and FIRST pre�ix play an analogous k role in considered models.

10 A regular left-hand side derivation in our model is analogous to a leftmost derivation for CF grammars.

11 A regular right-hand side derivation in our model is analogous to a rightmost derivation for CF grammars.

12 That is, we can choose a proper left-hand side.

13 That is, productions 1, 2, 3, 4 in G. 14 In [19] (Lemma 1, p. 207) we have proved that the index of a replaced node is always preserved in our model.

15 As it is in productions 1, 2, 3, 4.

16 To determine a proper reduction (unambiguously) one has to analyze (m + 1) − T L graphs instead.

17 Obviously, at any derivation step we do not know how many nodes labelled with b have been generated till this step.

18 Let us note that although syntactic pattern recognition problems have been the main motivation for the application-oriented part of this research, it was not limited to this area and has included e.g. distributed systems, reasoning over ontologies in expert systems.

AUTHOR Mariusz Flasiński∗ – Information Technology Systems Department, Jagiellonian University, ul. prof. St. Łojasiewicza 4, Cracow 30-348, Poland, e-mail: Mariusz.Flasinski@uj.edu.pl, www: www.ksi.uj.edu.pl/�m�l. ∗

Corresponding author

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