Pascal Bouvry
email: pascal.bouvry@uni.lugregoire.danoy@uni.lumartin.rosalie@uni.lu

Serge Chaumette
email: serge.chaumette@labri.fr

Grégoire Danoy

Gilles Guerrini
email: gilles.guerrini@fr.thalesgroup.com

Gilles Jurquet
email: gilles.jurquet@fr.thalesgroup.com

Achim Kuwertz
email: achim.kuwertz@iosb.fraunhofer.dewilmuth.mueller@iosb.fraunhofer.dejennifer.sander@iosb.fraunhofer.de

Wilmuth Müller

Martin Rosalie

Jennifer Sander

Using Heterogeneous Multilevel Swarms of UAVs and High-Level Data Fusion to Support Situation Management in Surveillance Scenarios

published or not. The documents may come

I. INTRODUCTION

The development and usage of Unmanned Aerial Vehicles (UAVs) quickly increased in the last decades, mainly for military purposes [START_REF] Valavanis | Handbook of Unmanned Aerial Vehicles[END_REF]. This technology is also now of high interest in non-military contexts like logistics, environmental studies and different areas of civil protection, for example rescue missions support, fire fighting, police duties [START_REF] Tchouchenkov | Detection, recognition and counter measures against unwanted UAVS[END_REF]. While the technology for operating a single UAV is rather mature, additional efforts are still necessary for using UAVs in fleets (or swarms) [START_REF] Bürkle | Towards Autonomous Micro UAV Swarms[END_REF], [START_REF] Chaumette | an operational retasking application for a swarm of autonomous UAVs: First re-turn on experience[END_REF], [START_REF] Sahingoz | Mobile networking with UAVs: Opportunities and challenges[END_REF].

ASIMUT (Aid to SItuation Management based on MUltimodal, MUltiUAVs, MUltilevel acquisition Techniques), a two-year project which started in March 2015, is supported by the EDA (European Defence Agency). The project is implemented by a consortium of five partners: Thales Systèmes Aéroportés, Laboratoire Bordelais de Recherche en Informatique of the University of Bordeaux, Fraunhofer IOSB, the University of Luxembourg and the French SME Fly'n'Sense. It aims at researching and demonstrating dedicated surveillance services being essentially based on fleets of heterogeneous UAVs. The aim is to enhance the situation awareness of operators and to decrease their workload by providing a model supporting detection of threats based on multi-sensor multi-source data fusion.

The notion of heterogeneous UAVs means different types of UAVs as well as different types of sensors (electrooptical, infrared, etc.) carried by the UAVs. Collaboration between UAVs evolving at different altitudes, i.e., between large UAVs covering the considered area from a high altitude with coarse acquisition capabilities and low altitude multirotor UAVs for precise close to the field measurements, is also taken into account.

Because of the hostile environment in which the aircrafts will be deployed, the flight plans cannot be computed in an offline manner by a central entity and distributed prior to the start of the mission. Autonomous UAVs taking their own decisions concerning their movement are thus more suitable. This approach has to be addressed adequately taking selforganization processes, collective intelligence and wireless networking issues into account.

The ASIMUT system supports operators through the combination of information coming from an airborne surveillance system based on heterogeneous swarms of UAVs and additional information extracted from intelligence databases. By providing such an automated assistance to human operators (in terms of high-level data fusion), their capabilities for making efficient and effective decisions can be improved thanks to increased situation awareness and lower workload.

The ASIMUT project aims at a balanced combination of different measures, framed by the types of threats and by the operator's reaction capabilities. To reach this aim, different innovative algorithms and approaches addressing maneuverability and autonomy, reliance on extended onboard situational awareness, information sharing, and highlevel data fusion based on promising concepts from artificial intelligence and model-based reasoning, are combined profitably in the distributed surveillance system.

The remainder of this paper is organized as follows. In Sec. II, we give an overview on concepts and system design of ASIMUT: we sketch our basic approach and present the main findings with regard to the ASIMUT structure and the resulting ASIMUT system. The following sections give an overview on the main approaches and algorithms used within the ASIMUT system. Sec. III addresses FMV (Full Motion Video) analysis and focuses on moving target detection. Sec. IV presents the work done with regard to autonomy, mobility, and resilience. In Sec. V, our approaches for high-level data fusion are presented. Before we give our conclusions in Sec. VII, Sec. VI addresses our concepts and future plans with regard to system evaluation.

II. CONCEPT AND SYSTEM DESIGN

For working out concept and design of the ASIMUT system, the operational and technical needs for surveillance in the military domain have been identified and analyzed in detail. The basis has been the development of operationally relevant use cases, the definition of a basic layered model structure, and a formal functional analysis of the system as a whole. In what follows, the main findings with regard to the ASIMUT structure and the resulting ASIMUT system are presented.

A. ASIMUT Structure

Fig. 1 gives a coarse view on the ASIMUT layered structure. Different approaches are integrated in three layers, each layer serving specific purposes:

• Collaboration of UAVs

This layer enables the collaboration between the UAVs. This is done by UAV swarm flight plan optimization under consideration of the external environment.

• Detection by UAVs

This layer enables the UAVs to gather the information needed about multiple objects, events, and their relationships. To achieve this goal, each UAV carries a specific payload to perform the air-ground surveillance. The autonomy of the UAV swarm makes it possible to decide which UAVs are relevant for performing a specific analysis tasked.

• Exploitation of Data

This layer enables further exploitation of the data gathered by the UAVs and of additional information from intelligence databases and also to relate them together. It is then possible to generate data and information of higher quality and deduce additional information. Specific means like FMV analysis and higher level data fusion techniques are provided to achieve this goal.

B. ASIMUT System

The ASIMUT system is a distributed surveillance system composed of a Command and Control (C2) system and of a set of heterogeneous UAVs that constitute swarms. The swarms can operate in two different modes: area surveillance and moving object (target) tracking. As illustrated Fig. 2, there is multilevel cooperation between the UAVs, a High-Level Coordination Swarm (HLCS) commanding several Low-Level Swarms (LLSs). The LLSs have reduced onboard processing capabilities. The HLCS have more powerful processing capabilities, but still less than the C2 system. The C2 system is composed of a Ground Control Station (GCS) in charge of the control of the missions of the UAV swarms and of an Intelligence Centre (IC). The IC (for illustration purposes not depicted in Fig. 2) is responsible for correlating and aggregating the information generated by the UAVs with information available from other sources, mainly intelligence databases. The GCS controls the swarms by assigning them specific missions while the actual mission planning process is performed by the swarms themselves. The IC performs high-level data fusion for generating decision support that is presented to the commander. The intelligence personnel in the IC has access to internal and external intelligence databases, high-level data fusion tools, and to the tracks generated by the UAV swarms.

III. MOVING TARGET DETECTION IN A VIDEO

For their tactical missions, the UAVs are equipped with various on-board sensors including long-range radar and electro-optical/infrared cameras. Both of these payloads are associated with geolocation and allow effective surveillance and intelligence. Our objective is to monitor the movements of ground vehicles like trucks, cars, motorbikes and to recognize the targets of interest (according to the tasks to be fulfilled). This is done based on algorithms allowing the automatic detection of moving ground targets from a realtime video acquisition performing this detection for small targets (i.e., when the moving target in a video has a size that is lower than two pixels). The data flow is dynamic, the camera moves, and we need to make a distinction between the ground and objects whose speed and direction evolve separately. To perform automatic detection of targets in a full motion video, the process has been divided into several steps (see Fig. 3): work on optical flow, detection of points of interest, projective image transformation, compensation for camera movement, motion detection.

The optical flow allows us to determine the motion vector of a pixel between two images of the same scene, i.e., between two consecutive frames in a video flow. This technique is widely used in the field of video processing for tracking an object by determining the transition matrix between the two images. Knowing the displacement of pixels, the interest is to have landmarks in the images. With them, an analysis can be done to determine the movement of objects relative to others. A first application is the segmentation of the video into different regions according to the motion vectors. In the ASIMUT project, algorithms are developed on basis of approaches from Lucas/Kanade [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF] and Bouguet [START_REF] Bouguet | Pyramidal Implementation of the Lucas Kanade Feature Tracker, Description of the Algorithm[END_REF] to perform the detection and the tracking of moving targets in a recorded video or directly from a real-time video stream.

A point of interest is defined as a specific area of the image that has a property predefined by an image operator, for example corners. These operators are varied; the best known is the method of Harris [START_REF] Harris | A combined corner and edge detector[END_REF] for corners, corresponding to an important change in the direction of an edge in an image. As for the optical flow, we are interested in landmarks between the two images. It is possible to correlate the points of interest to obtain the displacement vector. In addition, the method of Harris allows to obtain particularly stable and favorable points for optical flow computation to perform segmentation of regions based on their motion vector and to identify those which evolve independently.

According to the use cases, different picture transformations are possible and have to be handled. The projective transformation addresses translation, rotation and scaling. This transformation allows to transform an image into the coordinate space of another, stabilizing the video acquisition. In the context of a dynamic video, the acquisition conditions constantly change. With this operator, it can be compensated for and then it is possible to have a T -1 frame in the space of the frame T and to be able to apply operators of motion detection as for a fixed camera. In order to find this transformation, we must know at least four points that are present in both images with the associated displacement vector. One can then define the same plane and find the corresponding transformation matrix. To avoid alteration of the transition matrix by outliers, the method of RANSAC [START_REF] Fischler | Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography[END_REF] is used to eliminate these values.

For compensating the camera movement, we need to know the projective transformation between each frame of a video stream. The major part of the image is the background Fig. 4.

Patterns of motion detection between two frames with the verification and the final result. and does not correspond to moving objects. Based on this reasoning, a grid of points is used on the previous image whereas the major parts of them will correspond to the ground. Then, the optical flow is calculated by applying the method of Lucas/Kanade. The output is a set of vectors that allow to find the corresponding transformation. This is done by the least squares method coupled with RANSAC to eliminate false values. The space between each point and the margin can be configured, the advantage of this is to be able to control the number of points and therefore the computation time and the quality of the result. A second solution is to calculate the points of interest obtained by the Harris corner detector which will then be the points given for optical flow calculation. With the resulting motion vectors, the transformation can be found.

For motion detection, we now regard the position of the camera to be fixed, due to compensating for the camera movement. The absolute difference of the two images is considered with the aim to determine the color distance between each pixel. In practice, the noise of the optronic sensor alters this result. Therefore, we determine a threshold below which differences are interpreted as noise. This results in a binary image according to this condition. To eliminate noise, a morphological operator, Erosion, is applied as a binary mask for each pixel.

Fig. 4 presents an example of the global algorithm to perform automatic detection of targets.

IV. AUTONOMY, MOBILITY, RESILIENCE

In the ASIMUT project, the system is fully decentralized. The GCS only sends mission requirements to the HLCS. The latter has the capability to decide who will achieve this mission: the HLCS itself or the HLCS in collaboration with one LLS. The UAVs of the considered swarms will then autonomously decide the actions they have to plan to complete this mission. If the HLCS receives too many missions, it has to stop or wait until the completion of one or more missions before assigning new ones. As a consequence, the swarms control the sharing of the load. The swarms are chosen depending on their capabilities, mainly based on the sensors they embed and on their relative position to the objective.

A. Autonomous Retasking

The LLSs need to support global retasking, i.e., retasking that involves all the multilevel swarms of the system. This need for retasking is not only because of their own behavior and internal events (e.g., loss of a UAV, refueling) but also because of the situation management requirements. In the latter case, this is the result of a HLCS decision. It might be necessary to retarget the mission and/or the role/location of each UAV depending on the sensors it embeds and on some situation management decision. The decision process of a UAV uses the data collected from the other UAVs of the swarm. This is done by building an estimated global view (Fig. 5) during the life cycle of a UAV; the latter being carried out perpetually. At any time, each UAVs receives the messages coming from other UAVs in its neighbourhood. Once per life cycle, it analyzes these messages to decide on the next action it has to perform autonomously and builds its own estimated global view.

B. Mobility Model

The UAVs mobility model is based on the ACO (Ant Colony Optimization) algorithm introduced by Dorigo [START_REF] Dorigo | Optimization, Learning and Natural Algorithms[END_REF]. The underlying idea is that ants use the environment to exchange information via pheromone deposit. The pheromone concentration then influences the behavior of UAVs (Fig. 6). This process is called stigmergy. UAVs deposit virtual repulsive pheromones in visited geographical areas so as to prevent other UAVs from revisiting the same places too often [START_REF] Kuiper | Mobility Models for UAV Group Reconnaissance Applications[END_REF]. The drawback of the method is the usage of a random process when there is no pheromone in an area: randomness prevents the GCS operators from predicting the trajectories. To overcome this issue, we use a dynamical system with chaotic behavior, i.e. a deterministic process where the solution converges to a chaotic attractor. Here we propose to transpose this technique from the robotic to the UAV domain to address the path planning problem. Recently, the synchronization of chaotic dynamical systems have been used to obtain a better method for the coverage of an area [START_REF] Fallahi | A cooperative mobile robot task assignment and coverage planning based on chaos synchronization[END_REF]. Another approach relied on a discrete dynamical map (logistic map) in chaotic regime to generate series of bits [START_REF] Ch | A chaotic path planning generator for autonomous mobile robots[END_REF]. These bits are then converted to positions points which permits to create a deterministic path planning for the robots.

In ASIMUT, another method is used to enforce a dynamical system in chaotic regime in our UAVs. More precisely, it consists of a Rössler system in a chaotic regime where the symbolic dynamic contains three symbols. Each one corresponds to a direction (left, ahead, and right) that a UAV can take when there is no pheromone to guide it. This ensures a deterministic choice that can be monitored directly by the GCS by just by having the same hardware and software as the UAVs. The main advantage of chaotic regime is that the process is deterministic while it remains unpredictable on a long term behavior. In our case, the use of the same devices guarantees to obtain the same digits. This helps the GCS operator to predict the behavior of UAVs when there is no pheromone to guide them.

C. Resilience

The main advantage of the decentralized model is that it handles the loss of UAVs. Each UAV decides on the actions to perform regularly which means that it analyzes its local environment by evaluating messages received from its neighbors. When the swarm looses a UAV, the others ones will no longer receive its messages and will accordingly decide on the appropriate actions. On the other hand, if a new UAV is tasked by the GCS to join this swarm its integration will be effective as soon as it has the capability to communicate its position and sensor data to the swarm members. This decentralized approach enables resilience and flexibility while UAVs are lost or added to reinforce the LLS actions.

V. HIGH-LEVEL DATA FUSION

Sensor data collected from the LLSs and HLCS is processed and forwarded to the IC. Here, the results have to be combined with information from internal and external intelligence databases using methods from high-level data fusion (HLDF), with the aim of supporting the commander's situational awareness. HLDF denotes the process of integrating data and information on real-world entities observed in a Fig. 7.

Overview of the Object-Oriented World Model (OOWM), a framework for information representation, integration, and exchange under consideration of uncertainties. dynamic environment into a consistent representation while, at the same time, studying the relationships between these entities [START_REF] Das | High-Level Data Fusion[END_REF]. According to the well-known JDL data fusion model [START_REF] Steinberg | Revisions to the JDL data fusions models[END_REF], the assessment and integration of object-related information like their position and attributes (level 1) forms the basis for situation assessment (level 2), where the current relationships between objects are inferred, which then allows impact assessment (level 3).

In order to integrate heterogeneous data and information provided by different sensor systems, human observers and databases, as well as to account for different data formats and levels of abstraction, a unifying approach is needed in HLDF to ensure interoperability. Thus, as a first step and prerequisite for HLDF, available data and information has to be represented in an interoperable way, which in turn enables a consistent integration as a second step. Both steps can be tackled by applying methods from world modeling, an approach aiming at consistently representing and continuously updating available information about a given environment.

The Object-Oriented World Model (OOWM) [START_REF] Kuwertz | Extending Adaptive World Modeling by Identifying and Handling Insufficient Knowledge Models[END_REF] is a system designed for modeling a given environment in an object-oriented manner. Fig. 7 depicts an overview of the OOWM. It aims at representing the current state of an observed environment based on the contained objects. Each object is described by a set of attributes, which are the results of sensor measurements on certain object features. To account for uncertainties, attributes can be represented probabilistically. In addition, background knowledge on domain objects of interest can be provided, e.g., by domain experts. It is represented as an a priori domain model, e.g., an ontology. If new information becomes available, provided for example by sensors, methods from probabilistic information processing are applied for updating the object representations. Uncertainties resulting from measuring processes can thus be accounted for. In total, the OOWM acts as information hub, integrating information provided by different sources, and the contained information can be used as a basis for displaying a common operational picture (COP) of the observed environment.

For supporting HLDF in the ASIMUT project based on the OOWM approach, information from intelligence databases, such as formal reports, and data delivered by UAV swarms have to be integrated into the OOWM. The object-based representation of real-world entities can then be used to associate different data and information with the same real-world object. For example, on one hand, information about a specific truck observed near a suspicious building could be given in an intelligence report from a database. On the other hand, current real-time sensor data about the same truck can be provided by a UAV swarm, including track data. The UAV sensor data can then be associated with the report information by the OOWM and integrated into a single object representation. This covers the data fusion on JDL level 1. For JDL level 2, relationships between represented objects have to inferred. For this purpose, background knowledge has to be employed. The OOWM allows to store domain knowledge and general facts using descriptions logics and means of formal ontology, including rules representing implications. This background knowledge can be used to interrelate observed objects and to reason about them. For example, another report in the intelligence database could state that the previously mentioned building is being used as a site for manufacturing IEDs. In addition, another report could contain information about a current raid on a fertilizer factory. Combining these facts using rules modeled in the OOWM background knowledge, a reasoning tool can conclude that the truck currently observed by the UAV swarms might be loaded with explosives and therefore is intended to be used as an IED. The HLDF component in this case would inform operators in the IC about the potential threat and present them with its findings and lines of reasoning, for them to confirm or dismiss the fusion results.

The reasoning processes in HLDF can either be deterministic, e.g., using tools for ontology reasoning in a description logic like OWL DL, or make use of probabilistic extensions to description logic like the PR-OWL [START_REF] Costa | PR-OWL: A Framework for Probabilistic Ontologies[END_REF] language based on Bayesian networks. In addition, Markov logic networks [START_REF] Domingos | Just Add Weights: Markov Logic for the Semantic Web[END_REF] can be used to achieve probabilistic reasoning on top of a rule-based domain model. In domains where no sufficient amount of training data is available, the application of data-driven machine learning techniques for probabilistic reasoning is difficult. This is also true for the domain of military surveillance. Therefore, the model-based approach to reasoning described previously is taken in ASIMUT. Currently, a proof-of-concept implementation of the reasoning processes is being realized, further results for HLDF are expected at later project stages.

VI. FUTURE WORK: EVALUATION

Evaluation is essential to support the visibility of the developments, to materialize their benefits and thus validate their relevance. In ASIMUT, evaluation will be based on the developed use case scenarios describing the missions, their relative constraints and the types of aircrafts used along with respective aeronautic properties. The scenarios will permit to use all the capabilities of the system to achieve various missions including: area surveillance, target tracking and data collection. We also work on a proof-of-concept demonstrator.

Quality metrics will be defined to assess the performance of the ASIMUT system, with respect to: coverage of the surveyed area, connectivity of the communication graph, randomness of the surveillance movements and energy con- servation. The coverage of the surveyed area, i.e. what are the characteristics of a good or efficient coverage scheme and how to quantify them. Most promising benefit targets are the maximization of the average coverage frequency, the minimization of the area that is never covered, and even the minimization of the standard deviation of the coverage frequency. The connectivity of the communication graph, i.e. how well the UAV network should remain connected through time is also an important characteristics. The intrinsic dynamicity of the system requires the adaptation of static properties and measures, such as the number of connected components or the k-vertex connectivity for more resilient structures. The randomness of the surveillance movement will also be quantified, as the mobility of the UAVs should not be predictable. The energy conservation is also an important concern as it directly influences the autonomy of the surveillance system; the energy consumption of the system will thus be modeled and measured. Furthermore, it is necessary to assess the improvement of decision quality and workload decrease for operators, which results from the automated assistance in high-level data fusion. Important characteristics for this assessment could for example be threat detection and false alarm rates as well as reaction times to such threats.

All the algorithms developed in ASIMUT will be combined in a common evaluation framework. Relying on stateof-the-art simulators, it will permit to evaluate the global chain from detection to data fusion, illustrated Fig. 8. Evaluation results shall be reported in future publications.

VII. CONCLUSION

The ASIMUT project proposes, develops and validates beyond state-of-the-art technological building blocks for the enhancement of autonomy of surveillance missions based on multilevel UAV swarms. We propose new algorithmic approaches to address the different components of the functional chain of the underlying distributed system. The swarm decisional autonomy and mobility management take profit of past expertise on swarms of UAVs and new technologies for the application to multilevel swarms. Automatic processing of the detection of (even small) moving vehicles from realtime video acquisition by an electro-optical payload looks promising to decrease the operator workload. Providing automated assistance for human operators with regard to high-level data fusion tasks, ASIMUT significantly improves the capabilities of new algorithms for making efficient and effective decisions. Solutions for smart mission management and operator support have been developed. Information coming from the swarms can now be used to display the COP and other results within the ground control station and to compare the current situation with the planned one.

The most promising approaches implemented in the different algorithms that we have designed will be combined in a single demonstration framework and evaluated via stateof-the-art simulations. The objective will be to evaluate the benefits and relevance of the developments made; a preliminary step before a future system demonstration in a real environment representative of an operational environment.

Fig. 1 .

 1 Fig. 1. ASIMUT structure.

Fig. 2 .

 2 Fig. 2. Multilevel swarms of UAVs collaborating.

Fig. 3 .

 3 Fig. 3. Overview of the different steps within the process of automated detection of targets based on motion detection in full motion video.

Fig. 5 .

 5 Fig. 5. Estimated Global View (EGV1) including the data collected from Local Views of other UAVs (LV2 and LV3) to improve decision making.

Fig. 6 .

 6 Fig. 6. LLS deposit of pheromone where stars represents UAVs and the light to dark shaded cells represents the low to high level of pheromones.

*The authors acknowledge the support of the ASIMUT project A-1341-RT-GP, which is coordinated by the European Defence Agency (EDA) and partially funded by eight contributing Members (Austria, France, Germany, Italy, Luxembourg, The Netherlands, Poland and Sweden) in the framework of the Joint Investment Programme on Innovative Concepts and Emerging Technologies 2.


Martin Rosalie
email: martin.rosalie@uni.lu

Grégoire Danoy
email: gregoire.danoy@uni.lu

Serge Chaumette
email: serge.chaumette@labri.fr

Pascal Bouvry
email: pascal.bouvry@uni.lu

From random process to chaotic behavior in swarms of UAVs

Keywords: cooperative UAVs, multilevel swarms, mobility models, ACO, chaotic systems

Unmanned Aerial Vehicles (UAVs) applications have seen an important increase in the last decade for both military and civilian applications ranging from fire and high seas rescue to military surveillance and target detection. While this technology is now mature for a single UAV, new methods are needed to operate UAVs in swarms, also referred to as fleets. This work focuses on the mobility management of one single autonomous swarm of UAVs which mission is to cover a given area in order to collect information. Several constraints are applied to the swarm to solve this problem due to the military context.

First, the UAVs mobility must be as unpredictable as possible to prevent any UAV tracking. However the Ground Control Station (GCS) operator(s) still needs to be able to forecast the UAVs paths. Finally, the UAVs are autonomous in order to guarantee the mission continuity in a hostile environment and the method must be distributed to ensure fault-tolerance of the system. To solve this problem, we introduce the Chaotic Ant Colony Optimization to Coverage (CACOC) algorithm that combines an Ant Colony Optimization approach (ACO) with a chaotic dynamical system. CACOC permits to obtain a deterministic but unpredictable system.

Its performance is compared to other state-of-the art models from the literature using several coverage-related metrics, i.e. coverage rate, recent coverage and fairness. Numerical results obtained by simulation underline the performance of our CACOC method: a deterministic method with un-. predictable UAV trajectories that still ensures a high area coverage.

INTRODUCTION

The purpose of the ASIMUT project (Aid to SItuation Management based on MUltimodal, MUltiUAVs, MUltilevel acquisition Techniques), supported by the European Defence Agency (EDA), is to improve the situation awareness of an operator through area coverage and detection of threats based on multi-sensor and multi-source data fusion (see [START_REF] Rosalie | UAV multilevel swarms for situation management[END_REF] for more details concerning this project). Information is delivered by heterogeneous swarms of autonomous Unmanned Aerial Vehicles (UAVs) flying at different altitudes. One of the objective of the ASIMUT project addressed here is area surveillance that is ensured by means of a swarm of UAVs. These UAVs collect data with their embedded sensors. It should be noted that the operation takes place in a military context.

Even if this paper is dedicated to UAVs, the problem remains the same in the case of other unmanned vehicles like ground, surface or underwater vehicles. To solve this problem two main techniques have been proposed : online and offline path planning. Offline planning consists in precomputing the flight plan of the UAVs. The main asset of this approach is that the UAVs trajectories are easily monitored from the Ground Control Station (GCS). However, these techniques are not adaptive to any change of configuration during the flight: the scheduled path can be irrelevant by the time the UAVs execute it. On the other hand, online methods compute the trajectories of the UAVs at runtime. The advantages and drawbacks of online methods are the opposite of those of to offline systems. As a consequence, in this paper we will propose a method that combines the assets of both online and offline methods.

To build such a system, we choose to use the Ant Colony Optimization method (ACO) introduced by Dorigo [START_REF] Dorigo | Optimization, learning and natural algorithms[END_REF]. We mainly rely on the work of Kuiper & Nadjm-Tehrani [START_REF] Kuiper | Mobility models for UAV group reconnaissance applications[END_REF] who adapted the ACO algorithm to the coverage problem for UAVs. The latter uses repulsive pheromones to guide the UAVs over the area they have to cover. The UAVs share a map of virtual pheromones that indicate recently visited areas when high pheromone concentrations are present. The UAVs then have a higher probability to move to the least recently visited areas. The drawback of this method is that it uses a random process which prevents any offline path planing.

The purpose of our paper is to change the random process of ACO system for a chaotic dynamical system. A chaotic dynamic is the solution to a deterministic system with the following properties. The solution is bounded and sensitive to initial conditions, and consequently, unpredictable on a long-term. We thus want to introduce these deterministic but unpredictable solutions in our UAV mobility model1 . The recent paper [START_REF] Zang | Applications of chaotic dynamics in robotics[END_REF] provides a detailed review of various applications of chaotic dynamics for mobile robots. Various systems are used to generate chaotic dynamics. In most cases, the authors use the logistic map xn+1 = αxn(1xn) to do it. A map is an iterative application that does not diverge, neither converges to a point. This logistic map converges to a dense set of points, between 0 and 1. This map generates a fully distributed chaos when α = 4 and it produces an output xn ∈ [0 : 1]. We assume that the simplicity of this equation combined with an easy to use output contributes to find this equation in many papers. For instance Iba & Shimonishi [START_REF] Iba | The origin of diversity: Thinking with chaotic walk[END_REF] use it to define a chaotic walk (in comparison to the random walk ). The shape of their process depends on the number of decimals they use to compute the next iteration. This underlines how chaotic processes are sensitive to the initial condition. Gong & Wang [START_REF] Gong | Chaos ant colony optimization and application[END_REF] also introduce a chaotic process in an ACO algorithm by testing various parameters for the logistic map to improve the results of the algorithm. This map is also successfully used with an Ant Colony Algorithm in order to find best UAV path between threats ( [START_REF] Zhou | Chaotic differential evolution approach for 3d trajectory planning of unmanned aerial vehicle[END_REF] or [START_REF] Zhang | UAV path planning based on chaos ant colony algorithm[END_REF]).

However, Arroyo et al. [START_REF] Arroyo | On the inadequacy of the logistic map for cryptographic applications[END_REF] detail some of the properties of the logistic map (including probability distributions) while the parameter α is varied. They highlight that this equation exhibits some periodic behavior between the values of the parameters used in [START_REF] Gong | Chaos ant colony optimization and application[END_REF]. The same ranges of parameters are used by Li et al. [START_REF] Li | An improved chaotic ant colony algorithm[END_REF] to solve the Traveling Salesman Problem (TSP). This highlights the fact that the logistic map works as a chaotic generator but for a small range of parameter values. Thus the logistic map is a good start but not sufficient to explore the capacities and richness of chaotic dynamical systems. Some recent works have investigated further the integration of chaotic systems in optimization algorithms. Ahmadi et al. in [START_REF] Ahmadi | Chaotic invasive weed optimization algorithm with application to parameter estimation of chaotic systems[END_REF] use Ordinary Differential Equations (ODE) systems to combine chaos and the weed invasive algorithm. Also the authors of [START_REF] Gandomi | Firefly algorithm with chaos[END_REF] use various chaotic maps (tent map, Lozi map, etc.) to explore the performance of their firefly algorithm. We choose to use the Rössler system [START_REF] Rössler | An equation for continuous chaos[END_REF] as a basic system to explore a combination of chaotic behaviour from ODE with Ant Colony Algorithm. This system is a reference in the literature as one of the first ODE systems with a simple chaotic mechanism. This work is organized as follows. We first introduce the two main topics: ACO and chaos. The second part is dedicated to the description of the problem we solve and the related work. In the third section, we present our models: one only based on the Rössler system and then CACOC. The following section contains our experimentations with the metrics and the statistics we produce. We finally give a conclusion and describe our future work.

PROBLEM DEFINITION AND RELATED WORK

Problem definition

In this paper, we consider a problem similar to the one defined by Kuiper and Nadjm-Tehrani [START_REF] Kuiper | Mobility models for UAV group reconnaissance applications[END_REF]: a squared area has to be covered by a swarm of UAVs. In addition to the non predictability constraint induced by the military context, the trajectories of the UAVs still need to be monitored from the Ground Control Station (GCS). It is indeed mandatory for users of this type of system to know and anticipate the positions of their UAVs. We remind that we also need to have an adaptive method that is resilient to UAVs failures or loss of UAV. As a consequence, the problem we address here is at the edge of the path planning for UAVs and the autonomous distributed coverage. This is not a path planning problem because of the resilience constraint.

Because of the non predictability constraint, articles giving the optimized solution of a coverage problem [START_REF] Galceran | A survey on coverage path planning for robotics[END_REF] have not been considered. Indeed, in such approaches the UAVs coverage pattern is too explicit to be used in a military context. In this section we present the algorithms from the literature that have been compared to our method.

Related work

In this paper we address the following problem: an area has to be visited regularly by UAVs in order to collect information. One of the constraints is that an observer should not be able to anticipate the reconnaissance pattern of the swarm. One solution proposed by Kuiper and Nadjm-Tehrani [START_REF] Kuiper | Mobility models for UAV group reconnaissance applications[END_REF] was to introduce a random process to prevent the UAVs trajectories to be predicted. In the same article the authors also proposed one ACO algorithm and compared it to the random appraoch. These two models are presented in detail hereinafter. We also present a third model from the literature, referred to as the chaotic walk introduced by Iba & Shimonishi [START_REF] Iba | The origin of diversity: Thinking with chaotic walk[END_REF].

Model 1: Random-based mobility model [10]

The next direction of the UAV depends on the previous one and is chosen using the probability rules given in Table 1. The initial state of each UAV is considered as "straight ahead".

Model 2:

Pheromone-based mobility model [START_REF] Kuiper | Mobility models for UAV group reconnaissance applications[END_REF] This method is a distributed pheromone repel mobility model. While UAVs fly, they also deposit virtual pheromones that indicate recently visited geographical areas and that evaporate over time. If there is pheromone in its neighbourhood, the UAV chooses its next direction using the probability indicated in Table 2. If there is no pheromone to guide the UAV, it will use the random action of Model 1 based on its previous action.

Probability of action

Left

Ahead Right

p L = total-lef t 2×total p A = total-ahead 2×total p R = total-right 2×total

Model 3: Chaotic walk [8]

The process described here was introduced by Iba & Shimonishi [START_REF] Iba | The origin of diversity: Thinking with chaotic walk[END_REF]. The authors name it chaotic walk because it only uses the logistic map to determine the next angular direction of an entity

xn+1 = 4xn(1 -xn) θn+1 = 2πxn+1 .
The logistic map makes it possible to obtain xn ∈ [0 : 1]. In our case, UAVs have constant speed and the angle is given by θn+1. Even if this method is not dedicated to coverage, we implemented it to evaluate its performance as we did for the previous models. As a consequence this simple chaotic walk serves as a basic chaotic mobility model to compare our more developed method.

UAV MOBILITY MODELS WITH CHAOS

In the previously presented methods, ACO is a good candidate for the coverage problem because the algorithm is designed in a distributed way. However, this algorithm uses probabilities to let the UAV choose its next action; this implies to have random variables for each step. On the other hand the algorithm presented by Iba & Shimonishi using the chaotic logistic map permits to make a chaotic walk. This structure does not permit an efficient coverage but it satisfies the UAVs deterministic but unpredictable constraints. This is one of the properties of chaotic dynamical systems. In this section we propose two novel chaotic mobility models for UAV swarms to tackle the coverage problem: (1) the first one is based on the Rössler system which is a different chaotic system and (2) the second one is CACOC, that combines the chaotic dynamics of the Rössler system with an ACO approach.

In methods using meta-heuristics, random processes are used to generate variability and explore the possible solutions. In order to reproduce the simulations, the authors mainly use a seed to generate pseudo-random numbers. These pseudo-random numbers satisfy several properties such as independence and non periodicity. Here the specific Models 1 and 2 require to use the previous values to decide on the next value. We do not want to use chaotic process to generate pseudo-random numbers as it as been done by Lozi [START_REF] Lozi | Emergence of randomness from chaos[END_REF]. Instead, we will build our models using the properties of chaotic dynamical systems. The fact that we can reproduce our simulation, as it is possible with seeds, is also a plus to compare the chaotic behavior versus randomness through the performance of the models. These two approaches have enough common points to be compared efficiently. In the following section we will use the specificities of chaotic systems to build a model using a chaotic process instead of a random one.

Model 4: Chaos using the Rössler system

The Rössler system [START_REF] Rössler | An equation for continuous chaos[END_REF] is a well known Ordinary Differential Equation (ODE) system

     ẋ = -y -z ẏ = x + ay ż = b + z(x -c) (1)
producing chaotic dynamics. This system is a good candidate to evaluate the impact of chaos for our systems because it has been studied several times. This system is used to validate tools and methods whose purpose is to analyse chaotic systems. As detailed in [START_REF] Rosalie | Templates and subtemplates of Rössler attractors from a bifurcation diagram[END_REF] when a parameter of the Rössler system is varied, it leads to several non equivalent dynamics. For this three-dimensional system with the parameters a = 0.1775, b = 0.215 and c = 5.995 (attractor A of [START_REF] Rosalie | Templates and subtemplates of Rössler attractors from a bifurcation diagram[END_REF]), the dynamics properties are clearly identified: the chaotic mechanism is the classical horseshoe with stretching and folding mechanisms as illustrated in Figure 1.

This system is a well documented source of various chaotic regimes. The analysis of a chaotic attractor is performed with a Poincaré section P = {(yn, zn)|xn = 0, ẋn > 0} .

(

) 2
This tool permits to discretize the continuous solution. From the Poincaré section, a first return map details the dynamical properties of the system. From the ODE system, we obtain a map (Figure 2) ρn+1 = f (ρn) with ρn ∈ [0 : 1] is the normalization of the value yn in the Poincaré section. Figure 2: First return map of the Rössler attractor (Figure 1). This map is partitioned in three parts indicating the UAV directions: L (left), A (ahead) and R (right). Orbits of period 1 and 2 illustrate patterns (AAAAA. . . ) and (ARARA. . . ).

The arrow indicates the map orientation from the center to the outside, it corresponds of the orientation from 0 to 1. The use of the Poincaré section permits to avoid considering what happens in the phase space (x, y, z). Instead we only focus on one value after the passage of the solution through the Poincaré section: ρn. This map details the dynamic properties of the Rössler system. Contrary to random processes, the independence properties imply that any value can be obtained at any iteration. Here the value is directly obtained from the previous value. To obtain the direction of the UAV from this return map, we build a partition:

• if ρn < 1/3 then direction is right;

• if 1/3 < ρn < 2/3 then direction is left;

• else the direction is ahead.

The shape of the first return map presented in Figure 2 indicates the possible transitions of the UAVs. This figure display ρn+1 depending on ρn. It indicates the next position in the Poincaré section depending on the previous one. Thus, from a previous ahead direction (A), transition to A, left (L) and right (R) are possible. From a previous L direction, transition to A is the only possible transition. Finally, from a previous R direction, only transition to L or A are possible. We intentionally remove the symmetry process (left symmetric to right) introduced in the probabilities repartition proposed for Model 1. Based on some experiments that we have conduced, we believe that this absence of symmetry is in favor of a better coverage method.

To initialize the system, we choose an initial condition for each UAV to solve the Rössler system. Giving non equal initial conditions lead to the same attractor as solution and to the same first return map. However the points constituting the return maps are not the same both in terms of values and of order of appearance.

This model satisfies the problem constraints because it uses a chaotic system to have unpredictable trajectories. The GCS uses the same hardware and runs the exact same programme as the UAVs. This ensures to obtain the same decimals as the UAVs. As a consequence, operators in the GCS can monitor the UAVs positions.

Model 5: Chaotic Ant Colony Optimi-zation for Coverage (CACOC)

To satisfy all our problem constraints, we will combine the pheromone method [START_REF] Kuiper | Mobility models for UAV group reconnaissance applications[END_REF] with the previous Rössler model. In the Model 2, random processes are used for any iteration. We propose to replace the uses of random processes with the first return map (ρn+1 = f (ρn)) as we do for Model 4.

In the first case, it happens when there is no pheromone to guide the UAV, Model 2 uses the random processes defined in Model 1. In that case, we use the Model 4 to explore a non visited area of our CACOC model (Model 5 ). If there are some pheromones in Model 2, the UAV chooses its next direction with probabilities. For CACOC, we propose to replace the random choice with the next value in the return map using the probabilities repartition of Table 2 (pL, pA and pR).

• if ρn < pR then direction is right;

• if pR < ρn < pL then direction is left;

• else the direction is ahead. Thus, the return map values are used to choose the next direction for each step of CACOC model. The use of pheromones permits to have a system resilient to UAVs failure or loss. If a UAV is out of the system, it will no more deposit pheromones.

From the point of view of the GCS operator(s), UAVs are monitored and if any failure is detected, the operator can remove the UAV in the simulator reproducing the UAVs position from the GCS. This permits to maintain the monitoring at the GCS while keeping unpredictable UAVs trajectories.

Orbits and regular structure

For our model, we use a solution of the Rössler system that is an attractor (Figure 1). Unstable periodic orbits structure attractor dynamics and topology. From the initial condition, after the transitory time, the solution converge to the attractor. It corresponds to a trajectory that evolves in the attractor. In details, the trajectory visit the orbits of the attractor. However these orbits are unstable and the trajectory reach one of them for a moment before visiting another one.

In Figure 2 we present two low period orbits: a period 1 and a period 2 orbits. We represent these two orbits in the first return map but several others orbits can be founded for this specific attractor [START_REF] Rosalie | Templates and subtemplates of Rössler attractors from a bifurcation diagram[END_REF]. We focus our attention to these orbits because it explains two simple patterns for our UAVs. First, the period 1 orbits is located in the area associated to A. This means that when the system momentarily converge close to the period 1 orbits, it corresponds to a long straight trajectory. The period 2 orbits lead to an alternation of A and R (ARARA. . . ) that corresponds to a half-turn for the UAV. These state (close to the orbits) do not persist because, these orbits are unstable. As a consequence, these regular patterns sometimes occur for our UAVs and lead them to an efficient unpredictable exploration pattern.

EXPERIMENTATIONS

In order to evaluate the performance of our proposed CA-COC mobility model, we have compared it to the other four models presented in the previous sections using simulations. The setup used for the experiments, including the different metrics, is presented in the next subsection, while the experimental results are presented and analyzed in section 4.2.

Experimental setup

The simulation area is a 100 km × 100 km square, divided in square cells of 1 km × 1 km. The UAVs have a constant speed of 1km/s and they all depart from the base station located in the middle of the bottom edge of the map, i.e. position (50,0). At each simulation step, each UAV can do one of the following three actions: (1) ahead: the UAV keeps the same direction; (2) left: the UAV turns left with a 45 • angle; (3) right: the UAV turns right with -45 • angle. To prevent collisions between our UAVs they all have non equal flight altitudes [START_REF] Kuiper | Mobility models for UAV group reconnaissance applications[END_REF].

Each of the five mobility models has been evaluated with a swarm of 10 autonomous UAVs equipped with wireless communication capabilities. We ran 30 independent simulations of 7000 steps eachs, to obtain significant results and applied statistical tests in order to compare the five models. We want to perform unpaired multiple comparisons [START_REF] Kruskal | Use of ranks in one-criterion variance analysis[END_REF].

Parameter Name

Parameter In order to evaluate the performance of the different models in terms of area coverage and network connectivity, the following four metrics from [START_REF] Danoy | Connectivity stability in autonomous multi-level UAV swarms for wide area monitoring[END_REF] have been used: coverage, recent coverage ratio, fairness and connectivity. These are detailed hereinafter.

Coverage.

The coverage is the portion of the total area visited during the whole simulation. The environment is a 100 km by 100 km square area. The coverage value variates during the whole simulation. To have a representative value of the coverage, we choose to compare the coverage value after 5000 steps for each model. This indicates how efficient the models are to visit the total area. On the other side, we want to evaluate the first steps of each model to compare their initial behaviour while there is no guidelines for UAVs (e.g. no pheromones for Models 2 and 5). We will extract the slope of a linear regression (a × x) considering the 500 first steps.

Recent coverage ratio.

This metric introduced in [START_REF] Danoy | Connectivity stability in autonomous multi-level UAV swarms for wide area monitoring[END_REF] represents the percentage of coverage during the last 100 iterations. These 100 steps correspond to the pheromones' evaporation time. We exclude the 100 first iterations of our simulation to compute the mean value of the recent coverage.

Fairness.

The fairness measures if all cells are regularly and equally scanned. This is computed by the standard deviation of their respective number of scans [START_REF] Schleich | UAV fleet area coverage with network connectivity constraint[END_REF]. To evaluate the fairness during the whole simulation, we perform a linear regression (a × x + b) using the 6500 steps. This comes to balance the coverage initial slope that only evaluates the initial UAV trajectories. Here, the slope value is considered as a measure to evaluate the fairness of the models without this initialization part.

Connectivity.

Eventually, we evaluate the network connectivity of our system by measuring the number of clusters of the swarm, i.e. the number of connected components in the induced graph. We compute the mean value of the connectivity during the whole simulation to have a representative value for the different mobility models.

Experimental results

We executed 30 runs for each model and we thus obtain a distribution of 30 values for each algorithm. We produce five statistical comparisons to evaluate our algorithms. These tests also permit to assess how our method is statistically relevant compared to the others. Models 3, 4 and 5 are deterministic. As a consequence, if we run them 30 times to compare the metrics, the solutions will be the same. We only need to introduce variability to perform statistical analysis of our models. For Models 4 and 5, we change the initial conditions of the Rössler system. We add a random value ε ∈ [0 : 0.1] to obtain the following initial conditions (x0, y0, z0) = (-0.4 + ε + 0.1 × Id, 0, 0) where Id = 1, 2, . . . , 10 for the ten UAVs. This change is enough to obtain non identical maps for each UAV due to the sensitivity to the initial condition properties of chaotic dynamical systems. We also use the ε + 0.1 × Id value as initial condition value for the logistic map (Model 3).

For the five metrics studied in this work, the 30 results do not follow normal distributions. As a consequence we performed a Kruskal-Wallis rank sum test [START_REF] Kruskal | Use of ranks in one-criterion variance analysis[END_REF] to make an unpaired multiple comparison of our model. All the results for the five metrics hereinafter have statistically significant differences with 95% confidence between any two algorithms (i.e. with a p-value < 0.05 ). We explain the good results of our Models 4 and 5 by the appearance of structured patterns when dynamics evolve close to orbit (Section 3.3).

a. Coverage after 5000 iterations. b. Slope a of the 500 first iterations.

This value evaluates the initialization phase of our system: the transition from the initial group of UAVs to a swarm. Here again Model 5 is better than Model 4, the latter being better that the three other models, as can be seen from the results in Table 5. The combination of the Rössler system with pheromone also permits an efficient coverage at the beginning of the simulation. c. Mean value of the recent coverage.

Regarding the recent coverage metrics, the 100 first iterations are not taken into account because it is the maximal time necessary to evaporate pheromones dropped in an area. Also, the maximum value of this metrics is equal to an optimal solution where, during 100 iterations, the UAVs never deposit pheromone in the same area. As a consequence the optimal recent coverage rate is (10 × 100)/(100 × = 0.1. For this metrics, we found that Model 4 is less efficient than the model using pheromones, as presented in Table 6. This underlines the need of using the ACO algorithm to be efficient for the coverage problem. These values also highlight that the combination of pheromones and chaos (Model 5) is more efficient than Model 2.

Stat

d. Fairness slope a of the 6500 last iterations.

This metrics is used as a complement to our coverage metrics: we use the last 6500 iterations to remove the initial steps. The obtained values are given in Table 7. In that case, we obtain that the Model 5 is the best model with the lower slope. Indeed, if the standard deviation grows slower during a simulation, it indicates that cells are homogeneously visited during the whole simulation.

e. Mean value of connectivity.

This last metrics helps us to evaluate how UAVs are globally organised over the area. Here again Model 5 gives the best result in terms of means but not for standard deviation, as illustrated in

CONCLUSION AND PERSPECTIVES

In this article we have tackled the problem of area coverage by a swarm UAVs in a military context. To this end, we have proposed the Chaotic Ant Colony Optimization to Coverage (CACOC) algorithm that combines an Ant Colony Optimization approach (ACO) with a chaotic dynamical system, to control the mobility of the UAVs. We have empirically demonstrated that CACOC improves the system behaviour of the Kuiper and Nadjm-Tehrani model when there is no pheromone to guide the UAVs. This has been achieved by replacing the random process with a deterministic chaotic process based on a Rössler system. Moreover, the deterministic properties of such chaotic systems permit to monitor the swarm behavior from the Ground Control Station while the UAVs movements remain unpredictable to any observer. CACOC has proven to efficiently tackle the coverage problem and satisfy its constraints.

To conclude on the impact of chaotic dynamics on our system, we insist on the richness of ODE system compared to chaotic maps as Lozi, Ikeda etc. For a given ODE system, several chaotic mechanisms can be found by modifying one parameter of the system. As a consequence, a parameter of an ODE system can be considered as a parameter of our model we can tune to optimize the solution. For future work, we plan to compare the performance of our algorithm varying a parameter of the Rössler system with the corresponding partition of the return maps.

To conclude, the keystone of this paper is that we replace transition probabilities with a chaotic return map detailing the possible transitions. In our future work, we not only propose to use a chaotic process to increase the performance of our system, but also to find the best chaotic mechanism (i.e. return maps shape) that will make our system even more efficient. CACOC is an original approach that combines assets of offline and online methods. The mobility is decided online in a deterministic and reproducible way but the trajectories of the UAVs remains unpredictable. The main advantage of CACOC is that the position of the UAVs are known on the ground without need of communication between the ground and the UAVs.

Figure 1 :

 1 Figure 1: Rössler attractor solution to (1) for parameters a = 0.1775, b = 0.215 and c = 5.995 with the Poincaré section (2).

Table 1 :

 1 Random action table for Model 1.

	Model 1	Probability of action
	Last action Left Ahead Right
	Left	0.70 0.30	0
	Ahead	0.10 0.80	0.10
	Right	0	0.70	0.30

Table 2 :

 2 Pheromone action table for Model 2. lef t is the amount of pheromone sensed at the left of the UAV, ahead is the amount of pheromone sensed in front of the UAV and right is the amount of pheromone sensed at the right of the UAV; finally total = lef t + ahead + right.

Table 3 :

 3 Main experimental parameters.

		Value
	Simulation area
	Geographical Area	100 km × 100 km
	Number of cells	100 × 100
	UAV Autopilot
	UAVs speed	1 km/min
	Possible UAV actions ahead, 45 • left, 45 • right
	Initial UAVs position middle of the bottom of the map
	Experiments
	Mobility models	[1, 2, 3, 4, 5]
	Number of UAVs	10
	Simulation steps	7000
	Independent runs	30

Table 4

 4 presents the mean coverage values after 5000 steps, together with the corresponding standard deviation, for all five mobility models. The worst is Model 3 based on the logistic map. Comparing the Kuiper and Nadjm-Tehrani methods (Models 1 and 2), the pheromo-ne based model is better. The most surprising result of this comparison is that the proposed Rössler model only using chaos without pheromone, is better than Models 1 and 2. Finally, the combination of chaos and pheromone gives not only the best result but also the smallest standard deviation.

	Stat	Model 1	Model 2	Model 3	Model 4	Model 5
	Mean	0.9579	0.9841	0.5966	0.9875	0.9917
	Std Dev	0.0171	0.0059	0.0759	0.0018	0.0015

Table 4 :

 4 Coverage after 5000 iterations.

Table 5 :

 5 Slope of the first 500 iterations.

Table 6 :

 6 Mean value of the recent coverage.

		Model 1	Model 2	Model 3	Model 4	Model 5
	Mean	0.07673	0.09047	0.05100	0.08574	0.09221
	Std Dev	3.904e-04	2.907e-04	7.238e-04	2.657e-04	2.260e-04

Table 8 .

 8 A larger value means that

	Stat	Model 1	Model 2	Model 3	Model 4	Model 5
	Mean	4.000e-04	3.081e-04	7.809e-04	3.513e-04	3.034e-04
	Std Dev	5.032e-05	4.134e-05	1.471e-04	2.618e-05	1.409e-05

Table 7 :

 7 Fairness of the coverage. the network is more disconnected which also indicates that UAVs are well distributed over the area.

	Stat	Model 1	Model 2	Model 3	Model 4	Model 5
	Mean	5.916	6.044	4.543	6.142	6.179
	Std Dev	0.203	0.156	0.538	0.101	0.122

Table 8 :

 8 Mean value of connectivity.

In the frame of the ASIMUT project, a real deployment of this model is planned on UAVs.

Acknowledgement

The authors acknowledge the support of the ASIMUT project A-1341-RT-GP, which is coordinated by the European Defence Agency (EDA) and partially funded by 8 contributing Members (Austria, France, Germany, Italy, Luxembourg, The Netherlands, Poland and Sweden) in the framework of the Joint Investment Programme on Innovative Concepts and Emerging Technologies 2. The ASIMUT project consortium is composed of Thales, Fraunhofer IOSB, Fly-n-Sense, University of Bordeaux and University of Luxembourg.

The experiments presented in this paper were carried out using the HPC facilities of the University of Luxembourg [17] (see http://hpc.uni.lu).

APPENDIX A. EXPERIMENTAL PROBABILITIES OF MODEL 4 (RÖSSLER SYSTEM)

For the reader interested in the probabilistic method instead of the deterministic one with chaos, we provide here the transition probabilities experimentally obtained from Model 4 using the Rössler system. We remind the reader that this model obtains better results than the probabilistic method (Model 1) [START_REF] Kuiper | Mobility models for UAV group reconnaissance applications[END_REF]. Even if it is not our main goal, we are able to provide a better random model with a better coverage rate (Tab. 9). In our future work, we will compare this model to Model 4 to highlight differences between the probabilistic approach and the chaotic based one.

Experimental (from