International audienceDans cet article, nous étudions le problème de couverture de cibles au sol par des drones. Le but est de minimiser le nombre de drones déployés et de maximiser la qualité des communications radio afin de collecter efficacement l'information des cibles au sol jusqu'à une station de base centrale. Nous modélisons ce problème à l'aide d'un programme linéaire multi-objectif dans lequel nous étudions le compromis entre le coût de déploiement et l'altitude des drones assurant une communication air-sol efficace. Nous proposons une solution équitable permettant de limiter l'ajout de drones nécessaires à l'optimisation des transmissions pour la collecte de données

Caillouet, Christelle

Razafindralambo, Tahiry

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Analyse bi-objectif pour la couverture de cibles par des drones

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HAL Id: hal-01784809https://hal.archives-ouvertes.fr/hal-01784809Submitted on 3 May 2018HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Analyse bi-objectif pour la couverture de cibles par desdronesChristelle Caillouet, Tahiry RazafindralamboTo cite this version:Christelle Caillouet, Tahiry Razafindralambo. Analyse bi-objectif pour la couverture de cibles par desdrones. Rencontres Francophones sur la Conception de Protocoles, l’Évaluation de Performance etl’Expérimentation des Réseaux de Communication, May 2018, Roscoff, France. ￿hal-01784809￿Analyse bi-objectif pour la couverture decibles par des dronesChristelle Caillouet1 † et Tahiry Razafindralambo21Université Côte d’Azur, CNRS, Inria, I3S, France2Université de la Réunion, LIM, FranceDans cet article, nous étudions le problème de couverture de cibles au sol par des drones. Le but est de minimiser lenombre de drones déployés et de maximiser la qualité des communications radio afin de collecter efficacement l’infor-mation des cibles au sol jusqu’à une station de base centrale. Nous modélisons ce problème à l’aide d’un programmelinéaire multi-objectif dans lequel nous étudions le compromis entre le coût de déploiement et l’altitude des dronesassurant une communication air-sol efficace. Nous proposons une solution équitable permettant de limiter l’ajout dedrones nécessaires à l’optimisation des transmissions pour la collecte de données.1 IntroductionUnmanned Aerial Vehicles (UAVs) with air-to-ground and air-to-air communications using directionalantennas provide potential solution to anytime and anywhere network access. Deploying UAVs to covertargets is a complex problem. Indeed, the deployment should minimize the number of UAVs to reducethe cost. UAVs should be optimally placed to cover all the targets and gather the monitored informationto a central entity for analysis. Many research focus on UAVs or robots deployment in the literature suchas in [MB06, DKK03, WXTH06]. In these works related to UAV location problem, the authors assumethat the devices evolve in a 2D space. Therefore their problem is simplified because the coverage radiusis fixed for each mobile device. In [ZDPPRG16], authors considered the same type of problem as the onewe want to tackle. However they do not ensure connectivity among the UAVs, and their primary focus isenergy consumption. Ensuring UAV connectivity with a fixed base station is very important since it ensuresuninterrupted bi-directional information exchanges between the base station and the targets. In [MSBD16],authors consider maximizing the total coverage area of the UAVs and their lifetime. But in their model, allUAVs are assumed to be placed at the same altitude and thus do not consider possible coverage quality.Moreover, they do not ensure connectivity between each UAV.In this paper, we consider the following problem : Given a set of targets deployed on the ground, the goalis to cover all the targets at minimum cost with drones flying as low as possible. Moreover, we ensure thatthe UAVs form a connected graph with a fixed base station in order to collect and analyze efficiently theinformation. We show that it is possible to develop an efficient model that computes optimal positions of theUAVs. We also show that coverage problem (minimizing the deployment cost and the UAVs altitude) andconnectivity should be optimized using Pareto optimality concept. And finally we show that the connectivityconstraint have a non negligible effect on the number of deployed UAVs.2 Problem definition and formulationLet P be the set of possible positions for the UAVs in the three dimensional space, and N be the set oftargets that has to be monitored. Each target n ∈ N is assumed to be fixed, located at position (xn,yn) onthe 2D plane. Let u = (xu,yu,hu) ∈ P be respectively the position (xu,yu) of UAV u in the 2D plane, and huits altitude. We derive the observation radius rhu of UAV u in function of hu and its directional antenna half†Travail partiellement soutenu par le programme ANR “Investments for the Future” sous la référence ANR-11-LABX-0031-01.Christelle Caillouet et Tahiry Razafindralambobeamwidth θ. The coverage area of UAV u on the 2D plane is represented by a disk of radius bounded by :rhu ≤ hu · tan( θ2 ). We say that an UAV u ∈ P covers target n ∈N if the distance between its projection on the2D plane and the target dun =√(xu− xn)2 +(yu− yn)2 is below the observation radius rhu of u. Similarly,for air-to-air communications, an UAV u can communicate with another UAV v if their distance on the 3Dplane Duv =√(xu− xv)2 +(yu− yv)2 +(hu−hv)2 ≤ Ru, where Ru is the communication range of UAV u.For efficient data collection, we enforce the deployed UAVs to be connected with each other and with afixed base station b located at coordinates (Xb,Yb,0). We also associate a cost cu with the deployment of anUAV in position u ∈ P . The goal is to deploy UAVs and choose their position and altitude in P so that :– all the mobile targets are covered by at least one UAV ;– all the UAVs are connected with each other and to the base station to efficiently collect informationthrough multi-hop communications.Let zu indicate if an UAV is deployed in position u ∈ P , and χun state if UAV u covers target n. For connec-tivity, we use variables fuv ∈ R modeling the amount of flow sent between UAVs u and v. The goal is toensure the existence of a flow between the base station and all the deployed UAVs.(i) min f 1 = ∑u∈Pcu · zu(ii) min f 2 = maxu∈Phu · zu(1)∑u∈Pχun ≥ 1,∀n ∈N (2)χun ≤ zu ·rhudun,∀u ∈ P ,n ∈N (3)∑v∈P ,v6=ufuv− ∑v∈P ,v6=ufvu ={∑v∈P zv if u = b−zu if u 6= b,∀u ∈ P ∪{b} (4)fuv ≤ zu ·RuDuv· |P |, ∀u,v ∈ P (5)fuv ≤ zv ·RuDuv· |P |, ∀u,v ∈ P (6)Constraints (2) and (3) verify that all the targets are covered by at least one deployed UAV that is locatedat distance smaller than the transmission radius of the UAV. To model the connectivity constraints, we useequations related to the existence of a flow in the graph induced by the UAVs. Since we don’t know exactlyhow many UAVs are deployed to cover the targets, the amount of flow sent by the base station to commu-nicate with the UAVs is equal to ∑u∈P zu (Constraints (4)) corresponding to the number of deployed UAVsin the solution. Also, flow can be sent between two UAVs if they are within communication range of eachother (Constraints (5) and (6)). We analyze here the trade-off between two antagonistic objectives : mini-mizing the deployment cost (Equation (1)-(i)) and the maximum UAV’s altitude (Equation (1)-(ii)). Indeed,the higher the altitude, the larger the coverage. And so we need less drones to cover all the targets. But wealso want to minimize the altitude in order to ensure good wireless communications quality. So limiting thealtitude inextricably increases the number of deployed UAVs. Since the two objectives are in conflict, it isnot relevant to combine them in an effective optimization problem. Consequently, the concept of Pareto wasintroduced to study the trade-off between the metrics (1)-(i) and (ii). The goal is to find out all the possiblenon-dominated solutions of the problem. If a solution is non-dominated, it is not possible to improve one ofthe metrics without worsening the other. Each multi-objective problem has a set of Pareto-optimal solutionsdefined as the set of non-dominated solutions. In order to generate Pareto solutions we use the ε-constraintmethod that transforms the bi-objective problem into a sequence of parametrized single objective problemssuch that the optimum of each problem corresponds to a Pareto-optimal solution [BGP09].3 ResultsThe model has been implemented in Java language and solved using IBM Cplex solver 12.7.1. Instancesare deployed in a square area of 100m×100m. We choose randomly the 2D coordinates of the targets. Foreach random set of targets, we divide the area into equal squares in which one possible point is located inAnalyse bi-objectif pour la couverture de cibles par des dronesFIGURE 1: Pareto solutions for different sizes of N and P.the center of the square. In this way, the candidate sites for placing an UAV form a regular grid. For eachpoint of coordinate (xu,yu), we set the allowed altitudes to {10m,25m,45m}. The base station is placedat coordinates (0,0,0). We generate instances of size between 5 and 50 targets, and between 50 and 300possible points for the UAVs. For each UAV, the deployment cost is set to 1, the visibility angle to 60degrees, and the communication range to 30m.We first solve twice our model with one objective each. We obtain f 1 the minimum deployment costwithout any constraint on altitude, and f 2 the smallest altitude for which the set of UAVs is connected andcovers the targets, without any constraint on the cost. Then, we iteratively solve the mono-objective problemminimizing (1)-(ii) with constraint ∑u∈P cu · zu ≤ ε, ε varying from the cost of f 1 to the one of f 2.3.1 Analysis of Pareto solutionsWe depict in Figure 1 the trade-off between altitude and cost. In the optimization problem, it can happenthat the altitude is equal for f 1 and f 2. In such cases (not represented in the figure), the optimal solution f 1minimizing the cost (1)-(i) is optimal in terms of cost and altitude. In our scenarios, it is possible to decreasethe maximum altitude by adding additional UAVs as depicted in the Pareto solutions (Figure 1). Since theobjective function (1)-(ii) minimizes the maximum altitude, we observe that the solutions quickly decreaseto f 2. It is thus possible to attain the minimum altitude with a low number of deployed UAVs. Figures 1 showthat the minimum altitude is attained with at most 6 additional UAVs, compared to the optimum of objectivefunction f 1. For topologies with a low number of ground targets (i.e. 10 and 15 targets), the maximum is3. This is a good compromise between deployment cost and communication quality as expected for ourproblem. From these observations, we derive a definition of the best trade-off solution :Definition 1. The fair optimal solution of the target covering problem with connectivity constraints amongthe UAVs corresponds to the solution (c∗,h∗max) such that :– h∗max = f 2 is the optimal solution of objective (1)-(ii) ;– c∗ is the minimum deployment cost obtained when attaining f 2.3.2 Validation of the fair optimal solutionThe fair optimal solution allows us to propose one optimal solution in terms of altitude with minimumassociated deployment cost. We first claim that the fair optimal solution is efficient in terms of altitude.The sum of all the altitudes of the fair solution, compared to the solutions f 1 and f 2, provides an UAVbackbone at lower altitude (Figure 2a). Indeed, f 1 (named as opt cost) does not optimize altitude at all,Christelle Caillouet et Tahiry Razafindralambo0 10 20 30 40 50# targets0100200300400500Sum altitudes|P| = 108opt costfair solutionopt altitude10 20 30 40 50# targets0100200300400500Sum altitudes|P| = 300opt costfair solutionopt altitude(a) Sum of altitudes of deployed UAVs5 10 15 25 30 40 50Number of Targets012345678Coverage densityopt costfair solutionopt altitude(b) Mean number of targets covered by the deployedUAVs for |P|= 108,147,300.FIGURE 2: Analysis of solutions in terms of altitude and coverage.but deploys the smallest number of UAVs (at high altitude). Solution f 2 (called opt altitude) optimizesthe altitude, but gives a solution with more UAVs with a larger value of the altitude’s sum. Therefore thefair optimal solution provides a better solution since the sum of the altitudes is lower than the two mono-objective solutions. Another metric validating the choice of the fair optimal solution is the coverage density.It corresponds to the mean number of covered targets by the deployed UAVs. We present in Figure 2b, fora given number of targets, a maximum of three bars representing the coverage density for respectively 108,147, and 300 3D locations. We remark that the coverage density of the fair optimal solution has valuebetween the density of f 2 and f 1. It cannot be lower than f 2 because this solution deploys more UAVs atlow altitude, thus balancing the target’s coverage. On the contrary, f 1 minimizes the number of deployedUAVs, each one covering more targets. But the fair optimal solution has a low coverage density, usuallyclosed to the one of f 2. This is a good result since it increases the air-to-ground communication qualitywhile ensuring the lowest altitude for the deployed UAVs.4 ConclusionIn this work, we address the problem of deploying UAVs to monitor ground targets, and forming aconnected backbone to collect efficiently information from the ground. We present an optimal bi-objectivelinear program to model the problem and obtain Pareto-optimal solutions in reasonable time for real sizeinstances. A study of the Pareto solutions help analyzing the trade-off between deployment cost and altitude.We observe that our bi-objective model allows us provide a fair optimal solution minimizing the maximumaltitude with a few number of additional UAVs than the optimum deployment cost solution. This solutionthus balances efficiently deployment cost and communication quality.Références[BGP09] J.-F. Bérubé, M. Gendreau, and J.-Y. Potvin. An exact ε-constraint method for bi-objective combinatorialoptimization problems : Application to the traveling salesman problem with profits. European Journalof Operational Research, 194(1) :39 – 50, 2009.[DKK03] K. Dasgupta, M. Kukreja, and K. Kalpakis. Topology-aware placement and role assignment for energy-efficient information gathering in sensor networks. In ICC. IEEE, 2003.[MB06] S. Martı́nez and F. Bullo. Optimal sensor placement and motion coordination for target tracking. Auto-matica, 42(4) :661–668, 2006.[MSBD16] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah. Efficient deployment of multiple unmanned aerialvehicles for optimal wireless coverage. IEEE Communications Letters, 20(8) :1647–1650, Aug 2016.[WXTH06] Q. Wang, K. Xu, G. Takahara, and H. Hassanein. Deployment for information oriented sensing coveragein wireless sensor networks. In GLOBECOM. IEEE, 2006.[ZDPPRG16] D. Zorbas, L. Di Puglia Pugliese, T. Razafindralambo, and F. Guerriero. Optimal drone placement andcost-efficient target coverage. Journal of Network and Computer Applications, 75 :16 – 31, 2016.

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Analyse bi-objectif pour la couverture de cibles par des drones

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