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Comparison of Eulerian and Hamiltonian circuits for evolutionary-based path planning of an autonomous surface vehicle for Monitoring Ypacarai Lake
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M. Arzamendia1, I. Espartza2, D. G. Reina3, S. L. Toral2, D. Gregor1
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Abstract — an evolutionary-based path planning is designed for an Autonomous Surface Vehicle (ASV) used in
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environmental monitoring tasks. The main objective is that the ASV covers the maximum area of a mass of water like
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the Ypacarai Lake while taking water samples for sensing pollution conditions. Such coverage problem is transformed
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into a path planning optimization problem through the placement of a set of data beacons located at the shore of the
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lake and considering the relationship between the distance traveled by the ASV and the area of the lake covered. The
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optimal set of beacons to be visited by the ASV has been modeled through two different approaches such as Hamiltonian
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and Eulerian circuits. In the case of Hamiltonian circuits, all the beacons should be visited only once. In the case of
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Eulerian circuits, the only limitation is that repeated routes cannot exist between two beacons. Both models have
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important implications in the possible trajectories of ASV throughout the lake. In this paper, we compare the
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application of both models for the optimization of the proposed evolutionary-based path planning. Due to the
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complexity of the optimization problem, a metaheuristic technique like a Genetic Algorithm (GA) is used to obtain
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quasi-optimal solutions in both models. The models have been compared by simulation and the results reveal that the
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Eulerian circuit approach can reach an improvement of 2% when comparing to the Hamiltonian circuit approach.
Sergio Luis Toral
[email protected] https://orcid.org/0000-0003-2612-0388 Mario Arzamendia
[email protected] https://orcid.org/0000-0002-6486-8402 Iratxe Esparza
[email protected] Daniel Gutierrez Reina (Corresponding Author)
[email protected] Derlis Gregor
[email protected] https://orcid.org/0000-0003-2606-5980 1
Research Department, Faculty of Engineering, National University of Asuncion, San Lorenzo, Paraguay 2
Electronic Engineering Department, University of Seville, Seville, Spain 3
Engineering Department, Loyola University Andalusia, Seville, Spain
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Index Terms— Autonomous Surface Vehicle, Coverage Path Planning, Eulerian Circuits, Hamiltonian Circuits, Genetic Algorithm.
I. INTRODUCTION
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utonomous Surface Vehicles (ASVs) have great potential for applications in aquatic environments, such as seas, rivers,
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lakes and streams [1]. ASVs do not require a crew on them, which leads to the possibility of building smaller vehicles,
A
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and consequently, reducing significantly the production and operation costs. Some examples of target applications are
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surveillance, security, monitoring, Search and Rescue operations (SAR), among others.
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The Ypacarai Lake, which is located in Paraguay, is an adequate test scenario for monitoring applications of ASVs since it
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has been recently under observation for problems related to water pollution. The monitoring of the Ypacarai Lake requires
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carrying out periodic analysis of water quality. Such tasks can be easily accomplished by equipping an ASV with the
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appropriate sensors, such as temperature, pH, dissolved oxygen (DO), turbidity, oxidation-reduction potential (ORP), etc.
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Thus, the ASV can take samples of water and analyze them as it moves throughout the lake. An area of the Ypacarai Lake is
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said to be covered if the ASV goes through such area and takes a sample of water. Therefore, the monitoring task can be
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described as a coverage problem. An adequate coverage strategy is therefore necessary to take these samples and develop an
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up-to-date map of the conditions of the lake. It is even more challenging since the water conditions can vary periodically due
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to environmental factors, such as changes of temperature and rains, and for human-made events such as massive and
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uncontrolled waste disposal.
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Furthermore, the mentioned coverage problem, which is also called Coverage Path Planning (CPP) [2], can be transformed
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into a Path Planning Problem (PPP) since the distance traveled by the ASV is highly correlated with the area of the lake sensed
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by the ASV. Therefore, the main challenge for an efficient monitoring of the Ypacarai Lake is to find an optimal path planning
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for the ASV to follow so that the coverage of the Ypacarai Lake is maximized. Path planning problem in autonomous vehicles
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is a main research topic because it allows a robot to tactically move over a specific region [3]. Generally, this region is
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represented as a map, and the path planning finds intermediate waypoints between a starting and a destination point, which in
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turn gives intermediate routes. Therefore, a visiting tour can be defined between the starting and the destination points through
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several waypoints. In general, the objective of a path planning optimization algorithm is to minimize the distance traveled by
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the robot since it implies lower operation costs. Conversely, for the proposed coverage application transformed into a path
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planning, the objective is now to maximize the distance, which implies a higher coverage of the lake as long as redundancy is
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avoided.
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The intermediate waypoints can be used to build a graph, which are described as a set of vertices and edges that connect the
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vertices between them. If the destination and starting points are the same, then it is said that the path is a cycle or circuit. In
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graph theory, a circuit that visits each vertex once is called a Hamiltonian Circuit (HC). Similarly, a circuit that visits all the
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edges of a graph once and all the vertices at least once (it can be more times), then it is called an Eulerian Circuit (EC).
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Hamiltonian Circuits are important in PPP and CPP problems because they are used to study the Travelling Salesman Problem
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(TSP), which is the problem of finding the minimum HC in a weighted graph. Both types of circuits present important
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implications in the movements and coverage of the resulting ASV path planning.
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This paper explores, by means of a comparison, the suitability of both types of circuits for CPPs. ECs allow constructing a
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path that repeats regions of a map since waypoints can be visited as many times as desired. However, HCs limit the utilization
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of the waypoints, restricting the movements of the ASV. Therefore, it is expected that the flexibility gained by applying EC
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influences positively on the distance traveled by the ASV, and consequently, on the total covered area. Furthermore by using
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an EC (non-repeated edges) it is guaranteed that paths are not repeated and in this way achieving a higher coverage of the lake.
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Regarding complexity, finding a Hamiltonian circuit for a given number of waypoints is very high (NP-complete problem)
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since it is well-studied and known as the Travelling Salesman Problem (TSP), which is a classical combinatorial optimization
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problem [4]. In contrast, finding an EC for a given number of waypoints is easier since there is only one main condition to be
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met by the graph of waypoints to be Eulerian; that is, all the vertex (waypoints) should have even number of links [5]. Notice
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that in this case the links are the routes taken by the ASV. Yet, the complexity in the Eulerian-based approach comes from
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finding the best subset of selected waypoints and their connections that form an Eulerian graph, so that at least one EC can be
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obtained from the graph. With the help of a wireless network of beacons located at the shore of the lake that act as the waypoints
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and data collections points, a complete graph can be built. Furthermore, each edge of this graph provides coverage for all the
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regions of the lake, and therefore, the optimization challenge is to find the circuit that provides a satisfactory coverage of the
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lake.
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Nowadays, there is no algorithm that solves these types of combinatorial optimization problems in polynomial time [4].
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However, there are some techniques like metaheuristics that can give a quasi-optimal solution in a reasonable time [6]. These
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techniques are for example Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), and Genetic Algorithms
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(GA), among others [7][8]. In this paper, we demonstrate the GAs are suitable search engines for finding HCs and ECs for a
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set of waypoints in a CPP for the monitoring of Ypacarai Lake.
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In previous work, the problem of the Ypacarai Lake is presented and evaluated the use of modeling the problem as the TSP
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and solved by using the GA with different objectives functions, such as the distance [9] and coverage [10]. Furthermore, in the
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last study the GA approach has been compared with random and greedy algorithms, and performed some GA parameters
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tuning to obtain the best results. In this work, we improve the previous work by applying the EC model, which we demonstrate
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that achieves better results than the HC model. Then the main contributions of this paper are twofold:
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Modeling the coverage problem of an ASV as a CPP using Hamiltonian and Eulerian circuits.
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The comparison of both models using a genetic algorithm as search engine in a real-life scenario such as the Ypacarai
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Lake.
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The structure of the paper is as follows, Section II describes some relevant related works. Section III presents the statement
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of the problem, including the application scenario and the Eulerian and Hamiltonian circuit definitions. Section IV contains
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the details of the proposed approach based on genetic algorithm. Section V contains a comparison between the performances
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of HCs and ECs in the target CPP and a comparison of the performance of GA-based approach with other metaheuristics. In
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addition, the effects of the number of beacons used is evaluated. Finally, Section VI summarizes the main conclusion of this
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paper.
II. RELATED WORK
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GA is a technique that has been used in several works related to the generation of HCs for the TSP. In [11], the authors
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present the constrained covering solid travelling salesman problems (CCSTSP), in which a salesman visits a subset of cities
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and the remained cities are covered within an imprecise predetermined distance and comes back to the initial city within a
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restricted time. They solve this problem by using a combined method of random insertion deletion with a modified genetic
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algorithm (RID-MGA). This work is different to the presented one in a way that the cities that are not visited should be covered
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by the visited ones within a certain distance. Additionally, there is a time restriction because this problem was thought for
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emergency cases, like an occurrence of an earthquake. In [12], the authors improve a GA with two local optimization strategies
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that are applied to find the shortest HC such as the four vertices and three line inequalities and reversion of local Hamiltonian
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paths with more than 2 vertices.
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Regarding the CPP, it has been mostly study in agriculture applications, and many examples with TSP modeling and the
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use of meta-heuristics can be found [13][14]. In [13], the authors propose a path planning method that generates a feasible area
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coverage plan for agricultural machines in a region that includes obstacles. They formulated the problem as a TSP and solve
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it using the ACO approach. In [14], the authors study the use of GA to find optimal path to completely cover a farm field while
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avoiding known obstacles. They analyze different aspects of the GA such as, representations of the field, obstacles, grid, path,
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etc.; and its adaptation towards the application in this area. In [15], the authors use a GA to find the best coverage solution by
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dividing the field in small squares of the size of the vehicle, and then calculate the sequence of movements of a robot that
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covers the whole field and minimizes the energy costs.
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CPP research in agriculture involves not only UGV (Unmanned Ground Vehicles), but also UAV (Unmanned Aerial
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Vehicle), which can be used to take images of a crop from high distance for evaluating its conditions, or for weed detection,
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or even for irrigation and application of fertilizers. In this line, Pham et al. [16] propose a cellular decomposition of the field
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for a given CPP with concave obstacles based on the generalization of the Boustrophedon variant, using Morse functions.
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Regarding ASV and monitoring tasks, Tokekar et al. [17] present a coverage algorithm for finding tagged fish and a localization
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algorithm to effectively finding them in a lake. Instead of covering the entire lake the algorithm searches for covering specific
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regions inside the lake. First they model the problem of visiting all the regions as a TSP with neighbors (TSPN) and then inside
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each region the ASV follows a π shape path.
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EC and HC circuit models are the base for other approaches like the Chinese Postman Problem (CPP) [18] and its extension
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Rural Postal Problem (RPP)[19], which are widely applied in logistics, where paths are chosen to deliver goods, in
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transportation services like garbage collection, in mail delivery and snow plowing, among others. The main advantage of the
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study of these problems is that billions of dollars are spent and its optimization can save large amounts of money. The CPP is
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more to the EC because it searches a solution that visits all the edges in a graph minimizing the distance traveled. The RPP,
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on the other hand, is more practical in the sense that only one sub-set is obligatory to visit and a path should be constructed
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with the rest of edges and minimizing the distance traveled as well.
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The present work is a step forward in the evaluation of HC and EC for CPP problems. The described related work shows
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that several studies have employed HCs for CPP problems using GA as search engine [11][12]. This metaheuristics engine is
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adequate to solve high complex optimization problem like CPP. However, as far as authors’ knowledge this is the first work
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that models the CPP using Eulerian circuits for Autonomous Surface Vehicles (ASV). To validate such approach, we compare
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the results of a target CPP problem like the monitoring of the Ypacarai Lake when using both HC and EC models. Furthermore,
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an evaluation of the GA as an optimization tool versus other metaheuristics is also introduced. Finally, the flexibility of using
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ECs in CPP is introduced by showing how the distance travelled, which is related to the coverage, can be adjusted according
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to the ASV autonomy requirements
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III. STATEMENT OF THE PROBLEM
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A. Coverage Path Problem transformed into Path Planning Problem
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The studied problem calculates the best path planning for an ASV to take water samples and deliver to a network of data
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beacons located at the shore of the lake. Therefore, this problem can be formulated as having a set of beacons and determining
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how the ASV can take the largest number of samples to be delivered to the beacons. These beacons can be considered as
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intermediate waypoints and the sum of the routes between the first and last waypoints leads to the total path. An image of the
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Ypacarai Lake is shown in Figure 1, which shows the set of beacons. The number and position of beacons were chosen such
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that there is an equally spaced network with wireless capabilities for collecting data with an average distance between them of
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about 1 km. Still, these can be adapted in the future for a better integration with an existing data infrastructure. This distance
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is estimated so that it is feasible to deploy links in the 2.4 GHz (Zigbee technology, IEEE 802.15.4 standard). Under these
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conditions, the total number of beacons required is n = 60.
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Figure 1. Ypacarai Lake and the data wireless beacons
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Using Figure 1 a reference map is created. The origin of this map is selected as coordinates (25˚ 22’ 21” S, 57˚ 22’ 57” W)
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and all the beacons coordinates are translated into this new reference. In addition, each beacon is given an identification number
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that will be used throughout this study, from b0 to b59. This convention is shown in Figure 2. For example, in this map beacon
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0 is located at (4,860; 13,500).
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Figure 2. Reference map
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Regarding the coverage performed by the ASV, in this work it is assumed that one sample taken at a given point represents
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the conditions of the water of an area SASV around the point. Then, the total coverage Ctot is calculated by multiplying this area
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SASV with the distance traveled by the ASV, which is denoted as PL. The distance traveled is calculated by finding the sum of
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Euclidean distances between the adjacent beacons in the sequence that forms the final path solution. Moreover, the route
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intersections are considered as duplicated sampled data. Therefore, the term 𝑛𝑟𝑜𝑢𝑡𝑒𝑖𝑛𝑡 is used to account for the number of
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intersection among routes, which should be subtracted from the area covered. The final expression is:
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2 𝐶𝑡𝑜𝑡 = 𝑆𝐴𝑆𝑉 ∗ 𝑃𝐿 − 𝑛𝑟𝑜𝑢𝑡𝑒𝑖𝑛𝑡 ∗ 𝑆𝐴𝑆𝑉 (1)
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These concepts are illustrated in Figure 3, where the green rectangle is the covered area by the ASV and the red area is the
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duplicated sensing.
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Figure 3. Illustration of ASV coverage
B. Hamiltonian Circuits vs Eulerian Circuits
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The set of beacons and their possible connections, which are determined by the routes taken by the ASV, can be modeled
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as a complete and connected graph G{V,E} where V ={v1,v2,...,vn} is the set of vertices and E={e1, e2,.., en} is the set of edges
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such that ei connects vi and vj , ∀𝑖, 𝑗 = {1,2, … , 𝑛}, where each edge has also a weight dij (dij > 0; dii = ∞). Furthermore, it is
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assumed that the path conditions between a pair of beacons is identical in either direction, then the weight is equal in both
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directions (dij = dji) and it is said that the graph is undirected. In the presented model the weight of an edge is the Euclidean
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distance between vi and vj, ∀𝑖, 𝑗 = {1,2, … , 𝑛}. For the proposed CPP the beacons are the vertices, the routes are the edges, and
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the distance between two beacons is the weight of the corresponding route. The order of a graph is |V| and it is calculated as
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the number of vertices while the size of a graph is |E| and it is calculated as the number of edges. The degree of a vertex is the
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number of edges that connect to it, where an edge that connects to the vertex at both ends (a loop) is counted twice.
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By definition, a Hamiltonian cycle is a tour in a graph that visits all the vertices and edges of a graph once and starts and
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ends at the same vertex [4]. For having this, each vertex should have only a pair of edges, one incoming and one outgoing
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edge, or what is the same, all the vertices of the graph should have a degree equal to 2. Then, a Hamiltonian cycle (HC) is
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defined for a given graph G{V,E} as:
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HC = G’{V’,E’} s. t.
V’ = V, E’ E , deg(vi) = 2 ∀i={1,2,….,n} ej is valid, ∀i={1,2,….,n}
(2)
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An edge e is valid if that edge remains entirely inside the perimeter of the lake. If not, then the edge cannot be considered
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part of the sub-graph G’. As a consequence of this definition, the order and the size of G’ are the same, i.e., |V’| = |G’|. In a
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graph with n vertices there are (n-1)!/2 possible Hamiltonian circuits. The problem of finding the Hamiltonian circuit with
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minimum length is the widely studied TSP [4].
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An Eulerian circuit (EC) is a closed tour that visits all the edges [5]. However, it can visit each vertex more than once. One
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graph has at least an EC if the degree of all the nodes is even. This condition was established by Euler in 1736 when studying
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the Koningsberg bridge problem [20]. One additional condition is to verify that it is a connected graph, meaning that there is
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a path between every pair of vertices in the graph. Similarly to the HC, the definition of EC is:
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EC = G”{V”,E”} s. t.
V” V, E” E, deg(vi) ≤ 2m ∀i={1,2,….,o} and m ≥ 1, o
