Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\n
We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\n
Throughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\n
We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
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\r\n\tThe purpose of the book vegetation index and dinamics is to present current research on vegetation science. \r\n\tThe study of populations and plant communities in their different aspects; ecological, structural, functional and dynamic, it is essential to establish a posteriori models of forest and agricultural management.
\r\n
\r\n\tFor this, the methodological approaches on the type of sampling are considered essential, since there are differences between the purely ecological and the phytosociological methods, despite the fact that both pursue the same objective. \r\n\tAlthough the ecological method for the knowledge of the vegetation is widely extended, the phytosociological one is no less so, since in the European Union it has been developed as a consequence of policies on sustainability, through which regulations have been issued, such as the habitats directive.
\r\n
\r\n\tOn the other hand, research on plant dynamics and knowledge of the landscape in an integral way, have multiplied in the last 30 years, which has favored a deep knowledge of the floristic and phytocenotic wealth, which is fundamental for agricultural management, livestock and forestry.
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He has participated in national and international congresses with about 250 papers. He has held a number of different academic positions, including Dean of the Faculty of Experimental Sciences at the University of Jaen, Spain, and founder and director of the International Seminar on Management and Conservation of Biodiversity.",coeditorOneBiosketch:"Ricardo Jorge Quinto Canas is currently an Invited Assistant Professor in the Faculty of Sciences and Technology at the University of Algarve – Portugal, and a member of the Centre of Marine Sciences (CCMAR), University of Algarve. His current research projects focus on Botany, Vegetation Science (Geobotany), Biogeography, Plant Ecology, and Biology Conservation, aiming to support Nature Conservation.",coeditorTwoBiosketch:"Ana Cano Ortiz's fundamental line of research is related to botanical bioindicators. She has worked in Spain, Italy, Portugal, and Central America. It presents more than one hundred works published in various national and international journals, as well as books and book chapters; and has presented a hundred papers to national and international congresses.",coeditorThreeBiosketch:"Carmelo Maria Musarella is a biologist, specialized in Plant Biology. He is a member of the permanent scientific committee of the International Seminar on “Biodiversity Conservation and Management” guested by several European universities. 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He\nstudied and worked in several European Universities: Messina,\nCatania, Reggio Calabria, Rome (Italy), Valencia, Jaén, Almeria\n(Spain), and Evora (Portugal). He was the Adjunct Professor\nof Plant Biology at the “Mediterranea” University of Reggio\nCalabria (Italy). His research topics are: floristic, vegetation,\nhabitat, biogeography, taxonomy, ethnobotany, endemisms, alien species, and\nbiodiversity conservation. He has authored many research articles published in\nindexed journals and books. He has been the guest editor for Plant Biosystems and a\nreferee for this same journal and others. He is a member of the permanent scientific\ncommittee of International Seminar on “Biodiversity Conservation and Management”, which includes several European universities. 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1. Introduction
Reduction of ecosystem services plays a key role in the group of phenomena that is called global ecological crisis. Population explosion has resulted in overpopulation of our planet. Energy source of this overpopulation has been fossil fuels (coal, mineral oil and natural gas) produced by the biosphere over millions of years during the history of the Earth. Exploiting and burning of these natural resources have decreased living conditions of subsequent generations and have started a global climate change at the same time. However, it is more important that urban areas and agricultural land have extended in place of natural ecosystems, causing them to decrease drastically and malfunction, which has resulted in a biodiversity crisis, mass species extinction. Besides these, the global ecological crisis includes industrial, agricultural, traffic and residential pollution, which have damaged abiotic components of habitats, that is, air, soil and water. Deteriorating ecological conditions have caused social problems directly and indirectly, such as epidemics, poverty and humanitarian crises. Besides these, pollutant and nature-destroying economic activities increase wealth and income inequality among people, which results in further social tensions (crime, terrorism, riots and wars). At the same time, problems are aggravated by favorable processes whose disadvantages are not considered at first. Increasing scientific research has led to an information explosion. Due to this, experts have been forced back to a more and more narrow intellectual space; our excellent scientific specialists are less and less able to have an overview of their own wider discipline and, especially, the whole science, and thus, they are less and less able to solve complex problems and avoid them if possible. In the database Web of Science, which is the collection of scientific articles of the highest level, there are only 224 articles with the expression “global problems” in their title (at the time of writing this text), whereas 59,957 articles can be found with Drosophila (name of a fruit fly) in their title. Thus, researchers prefer more than 250 times dealing with molecular effects of any gene of a tiny fly to the complex study of the burning problems of our time. Of course not the scientific research is the only source of our knowledge, but the global usability of local ecological knowledge also depends on scientific and social science research [1, 2]. Not even the scientific world is dealing with solving the global ecological crisis; however, it might be even graver that if scientists presented suitable solutions, there would be currently nobody to execute them. Mankind is struggling not only with overpopulation crisis, environmental crisis, biodiversity crisis, social crisis and information crisis but lacks global coordination as well, which would be essential for political guidance. Mankind does not have a central legislative and executive power necessary for saving the whole Earth, but decision-making processes are split up among 195 nation states, among which the probability of substantial consensus approaches zero even in the most important questions.
Land cover change research has an important role in understanding the intensity and dynamics of real global processes [3, 4, 5, 6, 7]. The Land-Cover and Land-Use Change (LCLUC) Program launched by the National Aeronautics and Space Administration (NASA) is studying natural and human-induced changes of the vegetation of the Earth and consequences of environment transformation processes and attempts to forecast natural disasters considering the Earth as a single complete system, with the help of satellite images, using the tools of NASA and combining them with laboratory and modeling work [8].
Survival of mankind and sustainability of the society depend on ecosystem services provided by natural ecosystems. Only a healthy biosphere is able to regulate the climate of the Earth and keep it in a range suitable for us.
1.1. Ecosystem services
Goods which mankind receives from the natural environment, from properly functioning ecosystems are called ecosystem services. These goods contribute to the survival and well-being of people directly or indirectly. Ecosystem services can be divided into four different groups [9].
Regulating services: climate regulation, water purification, river regulation, erosion prevention, pollination.
Supporting services: water and nutrient cycling, photosynthesis and primary production, soil formation (pedogenesis).
Cultural services: spiritual and religious enrichment, esthetic values, recreation and tourism.
Value of ecosystem services can be expressed in money, which indicates preferences of the users and helps to determine how much resource to expend in order to maintain or restore an ecosystem [10]. Protection of intact ecosystems helps to increase resilience against adverse effects of climate change [11]. Biodiversity, that is, diversity of life maintains and restores services. Ecosystems have certain resilience, of course; however, their ability to provide services to mankind is decreasing due to harmful human activities. This is caused by the fact that ecosystem services are less known or their importance is underestimated in political decisions [12]. They are so essential for life that people consider their existence as evident, and it is difficult to imagine that mankind can destroy these as well [13]. Ecosystem services affect each other and connect to each other in a rather complex way, and if humans use one of the services, they affect the others too [14]. For example, if the maximum yield is aimed for with intensive agriculture, this has a negative effect on the water and nutrient cycling of the area. In recent decades, the most important changes in ecosystem services have been caused by a continuous decrease in the area of intact ecosystems [15]. Between 1997 and 2011, a damage of 4.3–20.2 trillion USD was created globally due to the fact that the area of intact, properly functioning ecosystems was decreasing, and they were replaced by artificial ecosystems [16]. Urbanization is increasing, more and more people live in cities in the world. Where the soil is not covered by asphalt and concrete, plants and animals may appear and ecosystems may form. These can be alleys along the roads, parks, artificial creeks and lakes as well as gardens. Ecosystem services are present here as well, which influence people’s life positively. For example, removing dust from the air, microclimate regulation and providing more attractive environment for residents [17].
2. Evaluation methods of ecosystem services
Experts more and more often encounter the problem that the value of a certain area, ecosystem or species has to be estimated. They have to decide how to handle a certain area and what to do with plants and animals, for example, whether a forest has to be left in its natural state or has to be cultivated. In this case, the value of that forest has to be estimated. In the academic literature, there are two approaches regarding the value estimation of natural ecosystems, the anthropocentric and the biocentric one. According to the first one, anything in nature can be as valuable as it benefits mankind. However, according to the second approach, everything in nature has an inner value, independently from its benefits for mankind [18]. Supporters of the anthropocentric approach mean that since humans are the dominant species on the Earth, they have the right to determine the value of anything [18]. According to the other approach, nature has direct (use) and indirect (nonuse) values [19]. According to the Millennium Ecosystem Assessment, goods provided by nature can be divided into four categories: provisioning services (e.g., fishing, timber), regulating services (e.g., climate and flood regulation), supporting services (e.g., pollination, pest control) and cultural services (e.g., tranquility, inspiration) [20]. Since the 1960s, more and more attention is paid to ecosystem value assessment in the academic literature [21]. Since first mentioning ecosystem services in 1983, the number of articles related to these and that of their citations has been rising steeply [22]. Ecosystems provide a wide range of goods and services to mankind, which are essential for the well-being of people [23]. In order to protect ecosystems, politicians should ensure that human activities are sustainable and resources are distributed fair and efficiently [24]. Decisions of politicians and the public opinion certainly strongly influence the value and usefulness of a certain service, thus value assessment of the services is rather contradictory [25]. Some people think that it is not possible or does not make sense since economists should not give a value to incomprehensible things such as esthetics and long-term ecological benefits [26]. Thus, there can be significant differences, contradictions between economical and ecological assessments [27]. It is especially important in western countries to give a value to natural ecosystems, where great importance is attached to high productivity in economical decisions [28]. Furthermore, monetary expression of ecosystem services does not necessarily mean that these can be considered as market products or private properties [29]. For example, pollination and water regulation cannot be in private property, everybody can benefit from them; however, they cannot belong to anybody [30]. This should definitely be included in political decisions, although translation of ecosystem services assessment into suitable financial mechanisms is not completely solved yet [31]. Since it is difficult to match them with economical processes or factory goods, they have only little weight in political decisions [32]. However, economical assessment of the services and their benefits is highly important because of the control of the services [33]. Attitude toward the assessment of services is best represented by the water-diamond paradox. Water is essential for life, still little value is attached to it, diamond is not important to maintain our quality of life at all; however, it has a great monetary value [34].
While mankind is receiving beneficial services from natural ecosystems, it is changing those, thus it is extremely important to monitor changes in their status continuously since their degradation influences the quality of life of mankind as well [35]. Ecological processes are endangered by human activities, destruction and transformation of habitats and pollution result in the disappearance of natural ecosystems all over the world [36]. Despite international, national and local environmental regulations, improvement of agriculture, industry and residential areas leads to further degradation and pollution of remnant intact natural vegetation [37]. In the future, these threats will be even graver since energy and raw material demand of mankind is continuously rising [38]. Nowadays, most people live torn away from nature and often consider nature protection as a barrier of industrial development; however, ecosystem services may change the point of view, and nature protection can drive the development [39]. Assessment of ecosystem services is also a tool for decision-makers, which helps to choose from alternative management options in order to reach multiple goals [40]. It is a system that links ecology to economy, which is why economical methods should be used for assessment of components of ecological systems [41]. There are several assessment methods which help to determine the monetary value of the services, although missing data make the work more difficult [42].
2.1. Direct market valuation methods
2.1.1. Revealed preference methods
2.1.1.1. Market price method
In some cases, value of the services can be directly measured based on the market price of goods, and these goods can be directly marketed. In these cases, the value is determined by how much they are paid for during the transaction. Thus, there is no need to use complicated methods. Such goods are, for example, sawn timber, firewood, fish and other foods. The value of the goods reflects the value of the ecosystem service. The advantage of this method is that it is simple to use since it considers available price, quantity and cost information, and simple assumptions are needed. However, it has the disadvantage that several services cannot be directly marketed, and obtained information may be false and distorted; thus, the value of the service is false as well. Furthermore, it is not easy to use it in the case of large-scale changes influencing the stock and the demand on the service [43].
2.1.1.2. Production function method
This approach is used if a certain good or service is partly created by human work and partly by the contribution of an ecosystem. For example, several agricultural plants depend on pollination by insects and the value of pollination can be estimated based on the value and quality of the crops. Thus, this method has been developed to estimate indirect use values. It has the disadvantage that it is difficult to determine how tight the relationship is between ecosystem service and human contribution. Thus, this method is not often used. However, it is used to measure water quality and the change in that for example, considering lower costs of water purification, improving agricultural production data due to better pollination or improving soil quality. Thus, the quality of a marketable good has improved due to an ecosystem service. Another problem with this method can be that the researcher has to consider both human and machine contribution, which can lead to overestimation of the value of the ecosystem service. However, it has the advantage that theoretically it is rather suitable for evaluating ecosystem services since it is based on the assumption that the service and the economic advantage are strongly interconnected [44].
2.1.1.3. Cost-based methods
This method measures the value of ecosystem services so that it estimates the damage in case of loss of the service as well as it considers possible costs of substituting the ecosystem service. It is used to measure water quality and water purification costs, guard against soil erosion, storms and other natural disasters and protect natural habitats. These are not marketable goods, and the method reflects costs of creating the benefit and not the benefit itself. The method has the advantage that it supports the way the economy thinks about value and value creation. However, it has the disadvantage that in certain cases, cost of repairing the damages does not reflect the advantages obtained [45].
2.1.2. Random utility and travel cost methods
The travel cost method and the random utility method developed are based on the empiric assumption that people surely know their preferences; however, these are not always known for researchers. However, certain factors of preferences can be obtained using statistical methods. This method is mainly used to evaluate hobby fishing at lakes, rivers and seas. It measures the value of nonmarketable ecosystem services based on the money and time spent in order to get to the fishing or swimming sites. Time, money and the number of visits express the value of a site, fish and swimming [37].
2.1.3. Hedonic pricing method
This method measures the indirect value of ecosystem services, which is not marketable but can be estimated based on the observed value of a good. In order to determine the value, two goods are necessary which are the same from most points of view but differ by certain environmental conditions, for example, traffic noise or distance from a park. Difference between monetary values of the goods can be interpreted as the willingness to pay for an ecosystem service. This method is often used to estimate the benefits or costs the environmental quality has (air pollution, water pollution, noise). This means that the environmental quality can also be estimated based on the price of houses. If there are two houses which are similar almost in every respect, however, air is more polluted in the surroundings of one of them, that one may cost less. The analysis reveals if changes in the environmental conditions affect the value of a market good [46].
2.2. Stated preference methods
2.2.1. Contingent valuation
This method measures the value of ecosystem services with surveys. Filled and submitted surveys show how much people are willing to pay for certain ecosystem services. In other words, it studies how people would behave in certain situations. Since these services cannot be marketed, the questions in the surveys ask what price respondents would pay in certain situations. The survey may contain options such as a new tax, an entrance fee to a national park, annual or monthly maintenance fee or a single charge. This method is widely used to assess the value of public goods. However, respondents are often not able to determine how much they would pay for a certain service. Thus, it is rather difficult to assess what an ecosystem is worth. Several respondents highly appreciate them but cannot attach monetary value to them and the answers also depend on the income of the individuals [47].
2.2.2. Conjoint analysis
This is also a commonly used and favored method and is based on surveys. The respondent has to answer questions regarding the characteristics of a good or service. For example, he has to choose between two options which describe possible characteristics of a park (distance from the house, size, vegetation and accessibility). Statistical analysis shows the relative importance of the different features for the respondents. It reveals the distance people are willing to cover to get there. Answers can be compared with answers given regarding other recreational opportunities [48].
2.3. Biodiversity as nonmonetary evaluation approach
Individual plants or animals, which constitute the biota together, can have characteristics which directly satisfy any demand of mankind. At the same time, biota and its role in supporting the biophysical cycles in the ecosystem benefit mankind indirectly [49]. It is necessary to maintain or restore the integrity of ecosystem services so that they persist and benefit mankind in the future as well [50]. Changing biodiversity and its effect on the functioning of the ecosystem have been a rather important field of ecological research in recent decades [51]. Due to landscape transforming human activities, habitats become fragmented, isolated, and dispersion ability of species may decrease. Thus, relationship among populations and viability of species also decrease, which may lead to extension [52]. If global average temperature increases by 2–3°C by the end of the century, 20–30% of all species will be endangered by extinction [53]. Disappearance of certain species is able to change habitats physically as well, and biogeochemical cycles as well as productivity, structure and functioning of the ecosystems may also change [54]. Reduction of the number of plant species results in decreasing primary production and decomposition processes [55]. Even under stable conditions, a certain minimal number of species is necessary in order to maintain the stability of the ecosystem. Under changing conditions such as the present climate change, an even larger number of species would be necessary so that the community is able to react to changes resiliently [56].
3. Global ecological significance of ecosystem services research
Human activity is rapidly transforming the surface of the Earth, concerning biosphere, soil and water resources. This can be globally observed and changes the functioning of ecological systems. Due to this, climate changes as well because of the strong relationship between vegetation and atmosphere. Climate and vegetation mutually affect each other both locally and globally. Climate regulates the spatial distribution of vegetation types, whereas vegetation influences climate due to its physical characteristics (biogeophysical processes) and the gas exchange (biogeochemical processes) [57]. Between 1990 and 2009, 1.14 ± 0.18 Pg/year carbon was emitted to the atmosphere on average due to human activity and the disappearance of vegetation [58].
Ecological processes happen on a longtime scale, thus, damages caused by human activity will be perceptible even after decades or centuries. On a geological time scale, climatic changes were related to the changes of the Earth’s orbit around the Sun, which caused large changes in the vegetation. For example, forests disappeared in Iceland due to the so-called little ice age and the Sahara, which had rich flora and fauna previously, turned to the currently known desert 6000 years ago [59]. On a shorter time scale, extreme weather events, fires, overgrazing and human activities transformed the landscape into new ecosystems, while Pleistocene megafauna became extinct [60]. In the last 300 years, human influence became extensive and intensive globally [61]. Phenomena such as deforestation, extension and intensification of agricultural areas, desertification and urbanization can be globally observed. Significant reduction of natural vegetation results in changing climate regionally and globally, deteriorating water quality, air pollution, habitat fragmentation, decreasing biodiversity, species extinction and spreading diseases [61]. In the last 2000 years, mankind reduced plant biomass by 45% through its landscape-transforming activity, the third of which disappeared during the twentieth century [62]. Human activities change soil composition, soil-forming processes, quantity and quality of water and climate [62]. After eradication of vegetation, soil is eroded, degraded, which causes irreversible changes in ecological systems and the climate [63].
Human influence on the nature is not uniform, there are still intact areas (the Amazon Basin and the Congo Basin); however, destruction will be continued in the future and the effects of these harmful processes will be perceivable in these areas with relatively intact vegetation as well [64]. Reduction of natural vegetation results in decreasing value of the connected ecosystem services, such as biodiversity, climate regulation, carbon storage capacity and water supply [65, 66]. Change in vegetation coverage is a rather significant factor, it influences ecological systems and climate and thus human life as well [67].
The fact that rapid reduction of natural vegetation might be a serious problem and would affect the quality of human life through climate change emerged some decades ago. Disappearance of vegetation and appearance of agricultural and other artificial areas have changed the albedo of areas, that is, energy exchange between the surface and the atmosphere and the climate. Due to the continuously rising human population, demand on land is also rising. This is the most decisive cause of the further degradation of natural vegetation and loss of habitats and this poses the largest threat to biodiversity. The loss is especially large in tropical regions, where biodiversity is the highest. Between 1980 and 2000, half of the new agricultural areas was created in place of cleared, previously intact forests and 28% of them in place of secondary forests [68]. Land use certainly has economic benefits and fosters the development of the countries; however, it also has a significant negative impact on the whole planet and mankind. Agriculture supplies mankind with food, thus, this activity transforms the environment to the greatest extent and contributes to greenhouse gas emission. Loss of natural vegetation and the connected ecosystem services is a problem of the same significance as food supply of mankind and economic development, maintaining and increasing the quality of life. Thus, a compromise should be agreed regarding what has to be protected and preserved and what has to be developed considering synergetic and complementary effects which may emerge.
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Introduction",level:"1"},{id:"sec_1_2",title:"1.1. Ecosystem services",level:"2"},{id:"sec_3",title:"2. Evaluation methods of ecosystem services",level:"1"},{id:"sec_3_2",title:"2.1. Direct market valuation methods",level:"2"},{id:"sec_3_3",title:"2.1.1. Revealed preference methods",level:"3"},{id:"sec_3_4",title:"2.1.1.1. Market price method",level:"4"},{id:"sec_4_4",title:"2.1.1.2. Production function method",level:"4"},{id:"sec_5_4",title:"2.1.1.3. Cost-based methods",level:"4"},{id:"sec_7_3",title:"2.1.2. Random utility and travel cost methods",level:"3"},{id:"sec_8_3",title:"2.1.3. Hedonic pricing method",level:"3"},{id:"sec_10_2",title:"2.2. Stated preference methods",level:"2"},{id:"sec_10_3",title:"2.2.1. Contingent valuation",level:"3"},{id:"sec_11_3",title:"2.2.2. Conjoint analysis",level:"3"},{id:"sec_13_2",title:"2.3. Biodiversity as nonmonetary evaluation approach",level:"2"},{id:"sec_15",title:"3. 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Faculty of Agricultural and Environmental Science, Laboratory of Biometrics and Quantitative Ecology, Institute of Crop Production, Szent István University, Hungary
ALÖKI Applied Ecological Research and Forensic Institute Ltd, Hungary
Faculty of Agricultural and Environmental Science, Laboratory of Biometrics and Quantitative Ecology, Institute of Crop Production, Szent István University, Hungary
ALÖKI Applied Ecological Research and Forensic Institute Ltd, Hungary
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1. Introduction
The problem of area coverage is one that has been widely studied in the past decade and consists of the deployment of a sensor-equipped mobile robot team. It is usually categorized as either blanket or sweep coverage. In blanket or static coverage the goal of the robot team is a final static configuration at which an objective function is maximized [1, 2, 3]. In sweep or dynamic coverage on the other hand the mobile agents are tasked with maximizing a constantly changing objective, resulting in potentially continuous motion of the agents [4, 5, 6].
Several aspects of the area coverage problem have been studied over the years, including the effect of robot dynamics [7, 8, 9], communication constraints among agents [10, 11, 12], complex non-convex regions [13, 14, 15] or guaranteeing collision avoidance among the mobile robots [16, 17]. A wide variety of methods has also been employed for multirobot area coverage such as geometric optimization [18], optimal control [19] or event-triggered control [20]. Due to the widespread adoption of unmanned aerial vehicles (UAVs), they have become a popular platform for area coverage [21, 22, 23] since they are usually equipped with visual sensors [24, 25, 26].
In this chapter we focus on the blanket coverage problem for a convex region of interest. The techniques outlined are based on geometric optimization principles and result in distributed control schemes. In a distributed control law, each agent uses only local information from its neighboring agents in order to compute its own control input so that a common objective function is maximized. Distributed control laws are highly desirable in multiagent systems because they are easily scalable to large robot teams and because they significantly reduce the computational burden and communication requirements on the agents. Moreover, they are more robust to failures and can adapt to unexpected changes without the need to recompute a new solution as is the case with most centralized control schemes.
The chapter is organized as follows. Section 2.1 contains some mathematical preliminaries which will be relevant throughout the chapter. In Section 2.2 the problem of blanket area coverage in a convex region by a heterogeneous team of agents with omnidirectional sensors is examined. In Section 2.3 the results are extended by taking into account the uncertain positioning of the mobile robots. Section 2.4 presents a tessellation-free method for area coverage by agents with anisotropic sensing patterns. Section 2.5 contains some experimental results and it is followed by concluding remarks.
2. Area coverage using mobile agents
2.1. Mathematical preliminaries
Throughout the chapter we assume a compact, convex region Ω⊂R2 to be covered by the mobile agents and a space density function ϕ:Ω→R+. The space density function is used to encode any a priori information regarding the importance of points in Ω, for example the likelihood that an event may occur at a given point. The boundary of a set S is denoted ∂S and its interior is denoted IntS. The set 1…n is denoted In. The indicator function 1Sq for a set S and the 2×2 rotation matrix Rθ are respectively
1Sq=1ifq∈S0ifq∉S,Rθ=cosθ−sinθsinθcosθ,
while the 2×2 identity matrix is denoted I2.
2.2. Heterogeneous agents with omnidirectional sensing
One of the simplest variants of the area coverage problem is the case of a team of homogeneous agents with circular sensing footprints. This was one of the first variants to be studied and there is extensive work on the topic [27, 28]. One generalization of this problem arises by allowing each agent to have a different sensing performance, resulting in a heterogeneous team [29, 30, 31]. In this chapter we will focus in the coverage of a convex region by a team of unicycle robots equipped with heterogeneous omnidirectional sensors.
2.2.1. Problem statement
A team of n mobile ground agents is deployed inside the region of interest Ω. Each agent i∈In is approximated by a point mass located at qi∈Ω which is governed by the following kinematic model
q̇i=ui,q∈Ω,ui∈R2E1
where ui is the velocity control input of the agent.
Each agent has been equipped with an omnidirectional sensor with limited sensing radius Ri, which is allowed to differ among agents, resulting in a circular sensing pattern
SiqiRi=q∈Ω:∥q−qi∥≤Ri.E2
Since the goal of the mobile agent team is the maximization of the covered area using their sensors, while also taking into account the space density function, we define the coverage objective as
H=∫Ωmaxi∈In1Siqϕqdq.E3
The control objective is the design of a distributed control law for the mobile agents in order to guarantee monotonic increase of the coverage objective H over time.
2.2.2. Space partitioning
The first step in designing a distributed control law is finding a method to distribute the computation of the coverage objective H. Due to the heterogeneous sensing performance of the agents, the Voronoi diagram is inadequate for the task as it does not take this information into account. To that extent the power diagram will be used in order to assign a region of responsibility to each agent. In contrast to the Voronoi diagram whose generators are points, the generators of the power diagram are disks.
Given a planar region Ω and a set of disks S=S1…Sn with centers Q=q1…qn and radii R=R1…Rn, the power diagram assigns a convex cell Pi⊆Ω to each disk Si
PiΩS=q∈Ω:∥q−qi∥2−Ri2≤∥q−qj∥2−Rj2∀j∈In∖i,i∈In.
The power diagram is a complete tessellation of Ω, thus it holds that
Ω=⋃i∈InPi,IntPi∩IntPj=∅,∀i≠j.
A notable property of power diagrams is their duality with power-weighted Delaunay graphs. It has been shown that in order to compute the power cell Pi of point qi, only the power-weighted Delaunay neighbors Ni of point qi need to be considered. The power-weighted Delaunay neighbors of agent i have the property that
Ni=j∈In∖i:Pi∩Pj≠∅,E4
By using the previous definition, the power diagram can be formulated as
PiΩQ=q∈Ω:∥q−qi∥2−Ri2≤∥q−qj∥2−Rj2∀j∈Ni,i∈In.E5
Since it holds that Ni⊆In, each agent i requires information only from its power-weighted Delaunay neighbors in order to compute its own power cell Pi, thus rendering the computation of the power diagram distributed.
Remark 2.1.When the agents’ sensing radii are equal, i.e.,Ri=Rj,∀i,j, the power diagram converges to the Voronoi diagram. In that case the computation of the cell of agentirequires information only from the Delaunay neighbors of agenti. Thus the power diagram can be also utilized in the case of agents with homogeneous sensing performance.
For any two agents i and j with Si∩Sj≠∅ it holds that Si∖Pi∈Pj∩Sj and Sj∖Pj∈Pi∩Si due to the properties of the power diagram. Thus if a part of some agent i ’s sensing pattern is inside the cell of some other agent j, then that part is guaranteed to be sensed by j. Consequently, we define the r-limited power cell of agent i as PiR=Pi∩Si. Thus by utilizing the power diagram, the coverage objective H can be computed as follows
H=∑i∈In∫PiRϕqdq.E6
Since H can be written as a sum of integrals over r-limited power cells and since an agent can compute its own power cell using information just from its power-weighted Delaunay neighbors, the computation of H is distributed.
2.2.3. Control law formulation
Having found a partitioning scheme that allows distributed computation of the coverage objective H, what is left is the derivation of a distributed control law for its maximization.
Theorem 2.1.For a team of mobile ground agents with kinematics (1), sensing performance (2) and using the power diagram partitioning (5), the control law
ui=αi∫∂PiR∩∂Siniϕqdq,E7
whereαiis a positive constant andniis the outward unit normal vector on∂PiR, leading the agents to trajectories that result in monotonic increase of the coverage objective (6).
Proof. We start by evaluating the time derivative of the objective using the agent dynamics (1) we get ∂H∂t=∑i∈In∂H∂qi∂qi∂t=∑i∈In∂H∂qiq̇i=∑i∈In∂H∂qiui. By selecting the control law ui=αi∂H∂qi,αi>0, we can guarantee monotonic increase of the coverage objective.
The partial derivative ∂H∂qi is evaluated as follows
Since only the cells of power-weighted Delaunay neighbors of agent i are affected by its movement and ∂ϕq∂qi=0, by using the Leibniz integral rule [32], the previous equation becomes
∂H∂qi=∫∂PiRυiiniϕqdq+∑j∈Ni∫∂PjRυjinjϕqdq
where υji is the Jacobian matrix υji=∂q∂qi,q∈∂PjR and ni is the outward unit normal vector on ∂PiR. The boundary ∂PiR can be decomposed into three disjoint sets ∂PiR=∂PiR∩∂Si∪∂PiR∩∂Ω∪∪j∈Ni∂PiR∩∂PjR, where ∂PiR∩∂Si denotes part of the r-limited cell’s boundary that is also part of the boundary of the agent’s sensing disk, ∂PiR∩∂Ω denotes the common boundary between the r-limited cell and the region, while ∂PiR∩∂PjR denotes the common boundary with the r-limited cell of some neighboring agent j. This decomposition is presented in Figure 1 where ∂PiR∩∂Si, ∂PiR∩∂Ω and ∂PiR∩PjR are shown in solid green, red, and blue lines, respectively.
Figure 1.
Decomposition of ∂PiR into disjoint sets and corresponding normal vectors.
Since the region Ω is assumed to be static, it holds that υii=0,∀q∈∂PiR∩∂Ω. In addition, since q∈∂PiR∩∂Si are points on a circle with center qi, it holds that υii=I2,∀q∈∂PiR∩∂Si. Finally, PjR is only affected by the movement of agent i at the common boundary ∂PiR∩∂PjR, resulting in the expression
Since υii=υji and ni=−nj on the common boundary ∂PiR∩∂PjR, as shown in Figure 1, the two sums in the previous expression cancel out and by multiplying it with αi we get (7). □
2.2.4. Simulation study I
An indicative simulation is presented in this section. The region Ω is chosen as the convex polygon defined by the vertices with x and y coordinates
respectively. The space density function was ϕq=1,∀q∈Ω. A team of eight agents with heterogeneous sensing radii is deployed inside the region.
The initial and final agent configurations are shown in Figure 2a and c respectively where the agent positions are marked by black dots, the boundaries of their sensing disks are shown as dashed black lines, the boundaries of their cells are marked by solid black lines while their interiors are filled in color. The agent trajectories are shown in Figure 2b with the initial positions marked by dots and the final positions by circles. It is observed that the agents are successfully deployed inside the region, increasing the covered area in the process. In order to provide a more objective measure of the agents’ performance, two different metrics are used. The first, denoted Hr, is the value of the coverage objective H as a percentage of the objective over the whole region which in the case where ϕq=1,∀q∈Ω it is equal to the area of Ω. This metric indicates to what extent the agents cover the region Ω, with high values of Hr corresponding to high coverage over Ω. The second metric, denoted Ha, is the value of the coverage objective H as a percentage of the agents’ maximum possible covered area which is only meaningful in the case where ϕq=1,∀q∈Ω. This metric indicates to what extent the agents’ sensors are utilized, with high values of Ha indicating that the agents’ sensors are utilized close to their full capabilities. Low values of Ha simultaneously with high values of Hr indicate an overabundance of agents given the size of the current region Ω. The two metrics are more formally defined as
Hr=100H∫Ωϕqdq,Ha=100H∑i∈In∫Sidq.E8
Figure 2.
Simulation study I: (a) initial configuration, (b) agent trajectories, (c) final configuration and (d) evolution of the coverage objective over time.
Figure 2d shows Ha in solid blue and Hr in dashed red with their final values being 90.0 and 88.9% respectively, indicating that the final agent configuration is an efficient one.
2.3. Heterogeneous agents with omnidirectional sensing under positioning uncertainty
The inherent uncertainty in all localization systems’ measurements can often create unexpected problems in algorithms designed with precise localization in mind. Consequently algorithms robust to positioning errors have been sought for the area coverage problem [33, 34]. This section presents an extension to the control law presented in [35] which allows for teams of agents with heterogeneous sensing performance.
2.3.1. Agent model
The agents’ kinematic model is described by (1) and their sensing performance by (2). Due to the localization systems’ inherent inaccuracy, we assume that qi is the agent’s position as reported by the localization system, while ri is a known upper bound on the localization error. Thus we define the positioning uncertainty region Ui as follows
Uiqiri=q∈R2:∥q−qi∥≤ri,E9
which is a circular disk that contains all possible positions of agent i given its reported position qi and positioning uncertainty upper bound ri.
In order to take into account the agents’ positioning uncertainty, we define for each agent the guaranteed sensed region Sig as
SigqiriRi=⋂∀q∈UiSiqRi,E10
which is the region that is guaranteed to be sensed by agent i given all of its possible positions within its positioning uncertainty region. Given the fact that both Si and Ui are disks, the above expression can be simplified into
SigqiriRi=q∈R2:∥q−qi∥≤Rig=Ri−ri.E11
2.3.2. Space partitioning and problem statement
In order to take into account the agents’ positioning uncertainty as well as their heterogeneous sensing capabilities, the Additively Weighted Guaranteed Voronoi (AWGV) diagram is employed. The AWGV is an extension of the Guaranteed Voronoi (GV) diagram [36] that incorporates additive weights.
Given a planar region Ω, a set of uncertain regions U=U1…Un and a set of weights Rg=R1g…Rng, the AWGV diagram assigns a convex cell Gi⊆Ω to each region-weight pair UiRig as follows
Contrary to the Voronoi diagram, the AWGV diagram is not a complete tessellation of the region Ω since a part of Ω is left unassigned. This part is called the neutral region O and it holds that
Ω=O∪⋃i∈InGi.E12
A notable property of AWGV diagrams is their duality with additively weighted Delaunay graphs. It has been shown that in order to compute the AWGV cell Gi of the region-weight pair UiRig, only the additively weighted Delaunay neighbors Ni of UiRig need to be considered. By using the previous definition, the Voronoi diagram can be formulated as
Since it holds that Ni⊆In, each agent i requires information only from its additively weighted Delaunay neighbors in order to compute its own AWGV cell Gi, thus rendering the computation of the AWGV diagram distributed.
The previous definition of the AWGV can be written as Gi=⋂j∈NiHij,i∈In, where Hij=q∈Ω:∥q−qj∥−∥q−qi∥≥+ri+rj−Rig+Rjg. Thus the boundary ∂Hij is one branch of a hyperbola with foci located at qi and qi and semi-major axis aij=ri+rj−Rig+Rjg2. Since the quantity aij may be either positive or negative, ∂Hij may correspond to the ‘East’ or ‘West’ branch of the hyperbola, which results in the region Hij being convex or non-convex respectively.
We define the r-limited AWGV cell of agent i as GiR=Gi∩Sig. We now define the coverage objective as
H=∑i∈In∫GiRϕqdq,E14
which is the area of the region that is guaranteed to be closest to and at the same time sensed by each agent. Since H is a sum of integrals over r-limited AWGV cells and since an agent can compute its own AWGV cell using information just from the agents in Ni, the computation of H is distributed.
2.3.3. Control law formulation
Since the computation of the coverage objective H is distributed due to the AWGV partitioning, what is left is the derivation of a distributed control law for its maximization.
Theorem 2.2.For a team of mobile ground agents with kinematics (1), sensing performance (2), positioning uncertainty (9) and using the AWGV partitioning (13), the control scheme
whereαiis a positive constant,nithe outward unit normal vector on∂GiR, leads the agent to trajectories that result in monotonic increase of the coverage objective (14).
Proof. We start by evaluating the time derivative of the objective using the agent dynamics (1) as in the proof of Theorem 2.1 and by selecting the control law ui=αi∂H∂qi,αi>0, we can guarantee monotonic increase of the coverage objective.
The partial derivative ∂H∂qi is evaluated as follows
Since only the cells of additively weighted Delaunay neighbors of agent i are affected by its movement and ∂ϕq∂qi=0, the previous equation becomes
∂H∂qi=∫∂GiRμiiniϕqdq+∑j∈Ni∫∂GjRμjinjϕqdq
where μji is the Jacobian matrix
μji=∂q∂qi,q∈∂GjR
and ni is the outward unit normal vector on ∂GiR. The boundary ∂GiR can be decomposed into three disjoint sets as follows
∂GiR=∂GiR∩∂Sig∪∂GiR∩∂Ω∪⋃j∈Ni∂GiR∩∂Hij,E16
where ∂GiR∩∂Sig denotes part of the r-limited cell’s boundary that is also part of the boundary of the agent’s sensing disk, ∂GiR∩∂Ω denotes the common boundary between the r-limited cell and the region, while ∂GiR∩∂Hij denotes parts of the boundary that consist of hyperbolic arcs induced by some neighboring agent j. This decomposition is presented in Figure 3 where ∂GiR∩∂Sig, ∂GiR∩∂Ω, ∂GiR∩∂Hij and ∂GjR∩∂Hji are shown in solid green, red, blue and purple lines respectively.
Figure 3.
Decomposition of ∂GiR into disjoint sets and corresponding normal vectors.
Since the region Ω is assumed to be static, it holds that μii=0,∀q∈∂GiR∩∂Ω. In addition, since q∈∂GiR∩∂Sig are points on a circle with center qi, it holds that μii=I2,∀q∈∂GiR∩∂Sig. Finally, GjR is only affected by the movement of agent i at the induced hyperbolic arc ∂GjR∩∂Hji and by grouping the hyperbolic arcs in pairs and multiplying by αi we get (15). □
2.3.4. Constraining agents inside the region
When the control law (15) is used, there can be cases where the positioning uncertainty regions of some agent does not remain entirely inside Ω, i.e. it is possible that Ui∩Ω≠Ui for some agent i. This has the implication that the control law (15) may lead some agent i outside the region Ω, given the fact that an agent may reside anywhere within its positioning uncertainty region Ui.
In order to avoid such a situation, a subset Ωis⊆Ω is used instead, instead of the region Ω. This subset Ωis is in the general case different among agents due to their differing measures of positioning uncertainty ri. This subset of Ω is computed as the Minkowski difference of Ω with the disk Ui0ri=q∈Ω:∥q∥≤ri which is Ωis=q∈Ωq+Ui0⊆Ω,i∈In.
By using this subset Ωis, constraining agents inside Ω is simpler, since this is equivalent to constraining each agent’s reported position qi inside its respective subset region Ωis. This is achieved by stopping an agent if its reported position qi is located on the boundary of Ωis and simultaneously its current control input leads the agent toward the exterior of Ωis. Thus the control law being implemented is
u˜i=0ifqi∈∂Ωis∧qi+εui∉ΩisuiotherwiseE17
where ε is an infinitesimally small positive constant.
2.3.5. Simulation study II
An indicative simulation is presented in this section. This simulation is identical to the one presented in Section 2.2.4 with the only difference being the addition of positioning uncertainty to the agents.
The initial and final agent configurations are shown in Figure 4a and c respectively where the agent positioning uncertainty regions are shown as black circles, the boundaries of their sensing disks are shown as dashed black lines, the boundaries of their cells are marked by solid black lines while their interiors are filled in color. The agent trajectories are shown in Figure 4b with the initial positions marked by dots and the final positions by circles. It is observed that the agents successfully deploy inside the region, increasing the covered area in the process. In order to provide a more objective measure of the agents’ performance, the two metrics described in Section 2.2.4 are used which in the case of uncertainty positioning are more formally defined as
Hr=100H∫Ωϕqdq,Ha=100H∑i∈In∫Sigdq.
Figure 4.
Simulation study II: (a) initial configuration, (b) agent trajectories, (c) final configuration and (d) evolution of the coverage objective over time.
Figure 4d shows Ha in solid blue and Hr in dashed red with their final values being 94.0 and 70.0% respectively. In this simulation we observe that although Ha reaches a high value, this is not the case with Hr. The first reason for this result is the fact that the computation of H is based on the agents’ guaranteed sensing patterns Sig, which by definition are lower in area than their respective sensing patterns Si. Moreover, due to the definition of H being conservative, only the area of the r-limited cells GiR counts toward the value of H, thus parts of the neutral region O that are covered by the agents do not contribute to H. Consequently, in the case of the AWGV partitioning (13), coverage objective (14) and control law (15), it is expected for Hr to achieve a lower value.
2.4. Heterogeneous agents with anisotropic sensing
Although the omnidirectional sensors examined in the previous two sections can significantly simplify the problem formulation and solution, they are usually inadequate for precise modeling of real-life sensors. For this reason there have been several differing approaches to area coverage using agents with anisotropic sensing patterns [37, 38, 39, 40]. In this section we will follow the methodology presented in [41] which is a distributed optimization technique resulting in a gradient-based control law.
2.4.1. Problem statement
A team of n mobile ground agents is deployed inside the region of interest Ω. Given the anisotropic nature of the sensing patterns examined in this section, the mobile agents should be able to change their orientation as well as move around inside the region of interest. A realistic model for a mobile agent with the ability to rotate is that of the differential drive robot whose kinematic model is
where ΩiR,ΩiL are the rotational velocities of the right and left wheels, respectively, ρi is the wheel radius, and li is the length of the wheel axis. In this chapter however a simpler single integrator kinematic model is used for the agents. Each agent i∈In is approximated by a point mass located at qi∈Ω with orientation θi which is governed by the kinematic model described by
q̇i=ui,q∈Ω,ui∈R2,E18
θ̇i=ωi,θ,ωi∈R,E19
where ωi is the rotational velocity control input of the agent. This single integrator model simplifies the derivation of the control law, although the control law can be extended for differential drive robots as well.
We define the base sensing pattern Sib of agent i as the region sensed by the agent when qi=00T and θi=0. The only requirements with regards to the base sensing pattern are that qi∈IntSib and that its boundary ∂Sib can be described by a set of parametric equations. Let the radius Ri of a base sensing pattern be defined as RiSib=maxq∈∂Sib∥q∥. This is the maximum distance from the origin, which is also the base sensing pattern’s center of rotation, to its boundary.
The sensing pattern of agent i as a function of its position qi and orientation θi, can be computed by rotating around the origin and translating its base sensing pattern as follows
Siqiθi=qi+RθiSib.E20
By allowing a different base sensing pattern for each agent, teams of heterogeneous agents can be effectively utilized.
Since the goal of the mobile agent team is the maximization of the covered area using their sensors, while also taking into account the space density function, we define the coverage objective as in (3). The control objective is the design of a distributed control law for the mobile agents in order to guarantee monotonic increase of the coverage objective H over time.
2.4.2. Space partitioning
The first step in designing a distributed control law is finding a method to distribute the computation of the coverage objective H. Due to the agents’ anisotropic sensing patterns, the partitioning scheme employed in this case is highly different from Voronoi-like partitioning schemes. Instead of partitioning the whole region Ω based on the agents’ positions, only the sensed region ∪i∈InSi is partitioned based on the agents’ sensing patterns. Each agent is assigned the part of Ω that only itself is able to sense, with parts being sensed by multiple or no agents being left unassigned.
Given a planar region Ω and a set of sensing patterns S=S1…Sn, each agent i is assigned a cell Wi as follows
WiΩS=Ω∩Si∖⋃j∈In∖iSj,i∈In.
The part of Ω sensed by multiple agents is left unassigned but still contributes toward the coverage objective H. This part is called the common region and is computed as follows
WcΩS=Ω∩⋃i∈InSi∖Wi.E21
Having defined the cells and the common region, it holds that ⋃i∈InSi=⋃i∈InWi∪Wc⊆Ω .
We can define the neighbors of agent i as those agents that affect the computation of its cell. Since the computation of the cells relies entirely on the agents’ sensing patterns, the neighbors can be defined as
Ni=j∈In∖i:Si∩Sj≠∅.E22
Moreover, if the maximum base sensing radius Rmax=maxi∈InRi is known by all agents, and if each agent is able to communicate with all others within a radius
Ci=Ri+Rmax,E23
then it is guaranteed it will be able to communicate with all of its neighbors Ni. By using the neighbor definition, the proposed partitioning scheme can be computed in a distributed manner as follows
WiΩS=Ω∩Si∖⋃j∈Ni∖iSj,i∈In.E24
Remark 2.2.The partitioning scheme (24) may result in the cell of some agent being empty or consisting of multiple disjoint regions. It should be noted however that such cases are handled successfully by the control law presented in Section 2.4.3.
Thus by utilizing the aforementioned partitioning scheme, the coverage objective H can be computed as follows
ℋ=∑i∈In∫Wiϕqdq+∫Wcϕqdq.E25
Since H can be written as a sum of integrals over the assigned cells and since an agent can compute its own cell using information just from its neighbors, the computation of H is distributed.
2.4.3. Control law formulation
Having found a partitioning scheme that allows distributed computation of the coverage objective H, what is left is the derivation of a distributed control law for its maximization.
Theorem 2.3.For a team of mobile ground agents with kinematics (18, 19), sensing performance (20) and using the partitioning (24), the control scheme
ui=αi,u∫∂Wi∩∂Siniϕqdq,E26
ωi=αi,ω∫∂Wi∩∂SiniRπ2q−qiϕqdq,E27
whereαi,u,αi,ωare positive constants andniis the outward unit normal vector on∂Wi, leading the agent to trajectories that result in monotonic increase of the coverage objective (25).
Proof. We start by evaluating the time derivative of the objective using the chain rule and the agent dynamics (18, 19)
we can guarantee monotonic increase of the coverage objective.
The partial derivative ∂H∂qi is evaluated as follows
∂H∂qi=∂∂qi∫Wiϕqdq+∑j≠i∂∂qi∫Wjϕqdq+∂∂qi∫Wcϕqdq.
Due to the partitioning scheme (24) only the common region Wc is affected by the movement of agent i and since ∂ϕq∂qi=0, by using the Leibniz integral rule [32], the previous equation becomes
∂H∂qi=∫∂Wiυiiniϕqdq+∫∂Wcυcincϕqdq
where υji is the Jacobian matrix
υji=∂q∂qi,q∈∂Wj
and ni is the outward unit normal vector on ∂Wi. The boundary ∂Wi can be decomposed into three disjoint sets as follows
∂Wi=∂Wi∩∂Si∪∂Wi∩∂Ω∪∂Wi∩∂Wc,E28
where ∂Wi∩∂Si denotes part of the cell’s boundary that is also part of the boundary of the agent’s sensing disk, ∂Wi∩∂Ω denotes the common boundary between the cell and the region, while ∂Wi∩∂Wc denotes the common boundary of the cell and the common region. This decomposition is presented in Figure 5 where ∂Wi∩∂Si, ∂Wi∩∂Ω and ∂Wi∩∂Wc are shown in solid green, red and blue lines respectively.
Figure 5.
Decomposition of ∂Wi into disjoint sets and corresponding normal vectors.
Since the region Ω is assumed to be static, it holds that υii=0,∀q∈∂Wi∩∂Ω. In addition, from Eq. (20) we get that υii=I2,∀q∈∂Wi∩∂Si. Finally, on all the common boundaries ∂Wj∩∂Wc,j∈In it holds that υij=υic and nj=−nc, as shown in Figure 5, leaving only an integral over ∂Wi∩∂Si. By multiplying with αi,u we get (26). The same procedure is used for the derivation of the rotational part of the control law (27). □
2.4.4. Simulation study III
An indicative simulation is presented in this section. The region Ω, the space density function ϕq and the agent initial positions are the same as in the simulation presented in Section 2.2.4. In this simulation however the agents are equipped with heterogeneous sensors with elliptical sensing patterns.
The initial and final agent configurations are shown in Figure 6a and c respectively where the agent positions are marked by black dots, the agent orientations are marked by black arrows, the boundaries of their sensing disks are shown as dashed black lines, the boundaries of their cells are marked by solid black lines while their interiors are filled in color. The agent trajectories are shown in Figure 6b with the initial positions marked by dots and the final positions by circles. It is observed that the agents successfully deploy inside the region, increasing the covered area in the process. In order to provide a more objective measure of the agents’ performance, the two metrics defined in Eq. (8) are used. Figure 6d shows Ha in solid blue and Hr in dashed red with their final values being 91.3 and 93.5% respectively. This indicates that the final configuration results in both high coverage of Ω and efficient use of the agents sensors.
Figure 6.
Simulation study III: (a) initial configuration, (b) agent trajectories, (c) final configuration and (d) evolution of the coverage objective over time.
2.4.5. Simulation study IV
This simulation study serves to highlight the need for taking into account the agents’ anisotropic sensing patterns instead of approximating them with circular ones. To that end, Simulation Study III was repeated by approximating the agents’ elliptical sensing patterns with their maximal inscribed circles. The initial agent configuration, agent trajectories and final agent configuration are shown in Figure 7a, b and c respectively. It is observed that the agent’s performance is decreased significantly when using this underapproximation of their sensing patterns. In order to provide a more objective measure of the agents’ performance, the two metrics defined in Eq. (8) are used. Figure 7d shows Ha in solid blue and Hr in dashed red with their final values being 100% and 35.2% respectively, indicating a 62.4% decrease in the coverage of Ω compared to Simulation Study III.
Figure 7.
Simulation study IV: (a) initial configuration, (b) agent trajectories, (c) final configuration and (d) evolution of the coverage objective over time.
2.5. Experimental implementation
An experimental implementation of a simplified version of one of the previously examined control schemes is briefly presented in this section. This experimental study first appeared and is presented in greater detail in [42]. The experiment consisted of three differential-drive robots, a visual pose tracking system using fiducial markers and a computer communicating with the robots and pose tracking system via a WiFi router. The methodology presented in Section 2.3 was used in order to take into account the positioning uncertainty of the pose tracking system. The experimental results are compared with a simulation using the same initial conditions.
2.5.1. Experimental setup
The robots used in the experiment were the differential-drive AmigoBots by Omron Adept MobileRobots. The robots are 33cm×28cm×15cm in size, weigh 3.6kg and are able to carry a payload of up to 1kg. Their maximum linear and rotational velocities are vmax=1m/s and ωmax=100o/s. Although these robots are equipped with encoders measuring 39,000ticks/revolution which can be used for estimating their pose, an external pose tracking system was used instead due to the encoders’ drifting error. Since the AmigoBots lack any omnidirectional sensors, for the sake of the control law it was assumed that they were equipped with sensors with a common sensing radius of R=0.3m.
The external pose tracking system consists of a USB camera and an ODROID-XU4 computer. Pose tracking is achieved by attaching a fiducial marker on top of each robot and using the ArUco [43] library to estimate the pose of these markers. As is the case with all positioning systems, ArUco has a certain degree of uncertainty in its pose estimations. In order to get an estimate of this uncertainty, a fiducial marked was placed on the vertices and the centroid of the region Ω resulting in a maximum error of 0.032m, which was used as the measure of positioning uncertainty r for all robots.
The control scheme was implemented as a loop in the main computer with an iteration period of Ts=0.1 seconds. At each iteration, a simplified version of the control law (15) is computed for each agent, and from that, a target point qit is derived for each agent. Then a feedback controller is used in order to lead each robot to each respective target point. Once all robots are within a predefined distance dt=0.02m of their respective target points, new target points are computed from the robots’ current positions. The feedback control law used for each robot was
where qi and θi are the robot’s current position and orientation, vi and ωi the robots linear and rotational velocity control inputs respectively and dθi=∡qit−qi−θi.
2.5.2. Experimental results
The robots’ initial configuration, which is common between the experiment and simulation is shown in Figure 8a. The final configurations of the experiment and the simulation are shown in Figure 8c and d, respectively. The boundaries of the agents’ positioning uncertainty regions are shown as black circles, the boundaries of their sensing disks are shown in dashed black line and the boundaries of their cells are marked by solid black lines while their interiors are filled in color. Some photographs of the robots’ initial and final configurations are presented in Figure 9a and b respectively where the ArUco fiducial markers can be seen. In both the experiment and the simulation it is observed from the robots’ final configurations that their guaranteed sensed regions are completely contained within their respective AWGV cells, i.e. Sig⊂Gi,∀i∈In, which is a globally optimal configuration. The robots’ trajectories are depicted in Figure 8b in blue for the experiment and in red for the simulation, with the initial and final positions marked by dots and circles respectively. The simulation trajectories are smooth but the experimental trajectories have many turns due to the robots moving to target points. The robots’ final positions have an error of 9.27% the diameter of Ω between the experiment and the simulation. This large error is attributed to the difference between the implemented control laws as well as the existence of multiple global optima for this particular coverage setup. Figure shows the evolution of the metric Ha over time for the experiment in blue and the simulation in red where it is seen that it increased from 83.70 to 98.95% in the experiment. Although in the case of the experiment its increase was not monotonic, this is to be expected as the implemented control law differed from the theoretical one. The lower convergence speed is also attributed to this difference as well as the constraints on the robots’ translational and rotational velocities.
Figure 8.
Experiment: (a) initial configuration, (b) experimental (blue) and simulated (red) robot trajectories, (c) experiment final configuration and (d) simulation final configuration.
Figure 9.
Experiment: (a) initial configuration, (b) final configuration and (c) evolution of the coverage objective over time for the experiment and simulation.
3. Conclusions and future work
This chapter presented an overview of past and current work on area coverage problems. A strong theoretical background has been provided, along with indicative simulations results and an experimental implementation of one of the presented control schemes. The problem of multiagent area coverage still offers possibilities for original research. One possible extension would be the usage of more realistic sensor models, such as visual sensors. The usage of visual sensors can result in the incorporation of coverage quality metrics in the objective or in variable sensing patterns in the case of pan-tilt-zoom cameras. Another aspect of multirobot area coverage problem that has not been studied thoroughly yet is the development of communication systems and algorithms that allow the agents to exchange information in a distributed manner. Finally, implementations in actual robotic systems in order to solve practical problems are not yet common.
Conflict of interest
The authors declare no conflict of interest.
\n',keywords:"area coverage, multiagent systems, mobile robot systems, distributed control, cooperative control",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/62906.pdf",chapterXML:"https://mts.intechopen.com/source/xml/62906.xml",downloadPdfUrl:"/chapter/pdf-download/62906",previewPdfUrl:"/chapter/pdf-preview/62906",totalDownloads:457,totalViews:83,totalCrossrefCites:0,dateSubmitted:"January 26th 2018",dateReviewed:"May 18th 2018",datePrePublished:"November 5th 2018",datePublished:"March 20th 2019",dateFinished:null,readingETA:"0",abstract:"This chapter is concerned with the development of collaborative control schemes for mobile ground robots for area coverage purposes. The simplest scheme assumes point omnidirectional robots with heterogeneous circular sensing patterns. Using information from their spatial neighbors, each robot (agent) computes its cell relying on the power diagram partitioning. If there is uncertainty in inferring the locations of these robots, the Additively Weighted Guaranteed Voronoi scheme is employed resulting in a rather conservative performance. The aforementioned schemes are enhanced by using a Voronoi-free coverage scheme that relies on the knowledge of any arbitrary sensing pattern employed by the agents. Experimental results are offered to highlight the efficiency of the suggested control laws.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/62906",risUrl:"/chapter/ris/62906",signatures:"Sotiris Papatheodorou and Anthony Tzes",book:{id:"7227",title:"Applications of Mobile Robots",subtitle:null,fullTitle:"Applications of Mobile Robots",slug:"applications-of-mobile-robots",publishedDate:"March 20th 2019",bookSignature:"Efren Gorrostieta Hurtado",coverURL:"https://cdn.intechopen.com/books/images_new/7227.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"38850",title:"Dr.",name:"Efren",middleName:null,surname:"Gorrostieta Hurtado",slug:"efren-gorrostieta-hurtado",fullName:"Efren Gorrostieta Hurtado"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"242670",title:"Dr.",name:"Anthony",middleName:null,surname:"Tzes",fullName:"Anthony Tzes",slug:"anthony-tzes",email:"anthony.tzes@nyu.edu",position:null,institution:{name:"New York University Abu Dhabi",institutionURL:null,country:{name:"United Arab Emirates"}}},{id:"244840",title:"Mr.",name:"Sotiris",middleName:null,surname:"Papatheodorou",fullName:"Sotiris Papatheodorou",slug:"sotiris-papatheodorou",email:"sotiris.papatheodorou@nyu.edu",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Area coverage using mobile agents",level:"1"},{id:"sec_2_2",title:"2.1. Mathematical preliminaries",level:"2"},{id:"sec_3_2",title:"2.2. Heterogeneous agents with omnidirectional sensing",level:"2"},{id:"sec_3_3",title:"2.2.1. Problem statement",level:"3"},{id:"sec_4_3",title:"2.2.2. Space partitioning",level:"3"},{id:"sec_5_3",title:"2.2.3. Control law formulation",level:"3"},{id:"sec_6_3",title:"2.2.4. Simulation study I",level:"3"},{id:"sec_8_2",title:"2.3. Heterogeneous agents with omnidirectional sensing under positioning uncertainty",level:"2"},{id:"sec_8_3",title:"2.3.1. Agent model",level:"3"},{id:"sec_9_3",title:"2.3.2. Space partitioning and problem statement",level:"3"},{id:"sec_10_3",title:"2.3.3. Control law formulation",level:"3"},{id:"sec_11_3",title:"2.3.4. Constraining agents inside the region",level:"3"},{id:"sec_12_3",title:"2.3.5. Simulation study II",level:"3"},{id:"sec_14_2",title:"2.4. Heterogeneous agents with anisotropic sensing",level:"2"},{id:"sec_14_3",title:"2.4.1. Problem statement",level:"3"},{id:"sec_15_3",title:"2.4.2. Space partitioning",level:"3"},{id:"sec_16_3",title:"2.4.3. Control law formulation",level:"3"},{id:"sec_17_3",title:"2.4.4. Simulation study III",level:"3"},{id:"sec_18_3",title:"2.4.5. Simulation study IV",level:"3"},{id:"sec_20_2",title:"2.5. Experimental implementation",level:"2"},{id:"sec_20_3",title:"2.5.1. Experimental setup",level:"3"},{id:"sec_21_3",title:"2.5.2. Experimental results",level:"3"},{id:"sec_24",title:"3. Conclusions and future work",level:"1"},{id:"sec_28",title:"Conflict of interest",level:"1"}],chapterReferences:[{id:"B1",body:'Pierson A, Figueiredo LC, Pimenta LCA, Schwager M. Adapting to sensing and actuation variations in multi-robot coverage. The International Journal of Robotics Research. 2017;36(3):337-354'},{id:"B2",body:'Abbasi F, Mesbahi A, Velni JM. 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ISSN: 2325-5870'},{id:"B40",body:'Laventall K, Cortés J. Coverage control by multi-robot networks with limited-range anisotropic sensory. International Journal of Control. 2009;82(6):1113-1121'},{id:"B41",body:'Stergiopoulos Y, Tzes A. Cooperative positioning/orientation control of mobile heterogeneous anisotropic sensor networks for area coverage. In: Proceedings IEEE International Conference on Robotics and Automation (ICRA), IEEE. Hong Kong, China; 2014. pp. 1106-1111'},{id:"B42",body:'Papatheodorou S, Tzes A, Giannousakis K. Experimental studies on distributed control for area coverage using mobile robots. In: 25th Mediterranean Conference on Control and Automation (MED), Mediterranean Control Association (MCA). Valletta, Malta; July 2017. pp. 690-695'},{id:"B43",body:'Garrido-Jurado S, Munoz Salinas R, Madrid-Cuevas FJ, Marín-Jiménez MJ. Automatic generation and detection of highly reliable fiducial markers under occlusion. Pattern Recognition. 2014;47(6):2280-2292. 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