A Alatorre
email: aalatorr@hds.utc.fr

J Cariño
email: jcarinoe@hds.utc.fr

P Castillo
email: castillo@hds.utc.fr

R Lozano
email: rlozano@hds.utc.fr

Dynamic trajectory for landing an aerial vehicle on a mobile platform

Keywords: Autonomous landing, dynamic trajectory, rendevous, target following, UAV

In the presented work, a trajectory is designed allowing an Unmanned Aerial Vehicle (UAV) to perform landing on a mobile platform. The proposed trajectory behaves in such a way that grants a seamless integration of information about the target landing platform into its design. This allows any generic tracking controller to follow the trajectory regardless of the target's dynamics or movement, as they are indirectly applied to the trajectory's architecture. The focus of this work is for fixed-wing aircraft to land on water surface vehicles but can be adjusted for different scenarios. One of which is used for the experimental corroboration of the proposed strategy in a real-world environment.

INTRODUCTION

Researches into Unmanned Aerial Vehicles (UAV) have explored solutions in several application domains, such as the path planning for photogrammetry [START_REF] Cabreira | Energy-Aware Spiral Coverage Path Planning for UAV Photogrammetric Applications[END_REF], guidance strategies for ground reconnaissance [START_REF] Iscold | Development of a Hand-Launched Small UAV for Ground Reconnaissance[END_REF], target detection for search-rescue [START_REF] Sun | A camera-based target detection and positioning UAV system for search and rescue (SAR) purposes[END_REF], control design for tracking of ground vehicles [START_REF] Oliveira | Ground target tracking control system for unmanned aerial vehicles[END_REF], parameter estimators for payloads transportation (Wang et al, 2016), and the design of adaptive control to navigate in presence of wind disturbances [START_REF] Brezoescu | Adaptive Trajectory Following for a Fixed-Wing UAV in Presence of Crosswind[END_REF]. UAVs are commonly classified as multirotor or fixed-wing vehicles. A multirotor vehicle is comprised of more than two rotors, and is capable of vertical take-off and landing, (Castillo et al, 2006). Besides, thanks to its vertical thrust, the vehicle can remain at hover flight and perform highprecision maneuvers. However, this kind of drone presents a low performance when flying long distances, going at high speeds, and very small flight durability. Alternatively, a fixed-wing vehicle can achieve the previous tasks seamlessly, as is mentioned in [START_REF] Varga | Evaluation of control strategies for fixed-wing drones following slow-moving ground agents[END_REF]. Since the lift force is generated by the airflow through of its wings when the vehicle moves forward. This allows the airplane to glide without taxing the engine too much. Thus, it is a suitable vehicle to perform long-distance missions with low energy consumption, [START_REF] Elijah | A review on control and maneuvering of cooperative fixed-wing drones[END_REF]. Although, landing is a critical flight stage for fixed-wing drones. Often, they suffer accidents caused by the inexperience of pilots or by crosswind disturbances, [START_REF] Huh | A vision-based landing system for small unmanned aerial vehicles using an airbag[END_REF]. Therefore, some research has been carried out to achieve automatic landing to reduce the accident numbers.

The authors in [START_REF] Zhang | Autonomous Landing Control of Fixed-wing UAVs: from Theory to Field Experiment[END_REF] define the automatic landing for a fixed-wing vehicle on a runway in three flight stages: descent, flare maneuver, and taxiing for kinetic energy reduction. The challenge of an automatic landing increases when the place to land is a moving target, as any error or miscalculation could result in a failure. The goal is to develop a dynamic trajectory to facilitate the landing task on a moving target. Besides providing a safe landing solution for UAV's without landing gear such as in [START_REF] Muskardin | Landing of a fixed-wing uav on a mobile ground vehicle[END_REF]. The proposed strategy can be used along a suitable tracking controller that might already be implemented on the UAV. The literature has previously presented strategies for leading a vehicle towards a target. Among these, two remain popular to address the proposed situation: Dynamic Path Following (DPF); and Rendezvous Guidance Control. DPF is a strategy that steers a vehicle towards a trajectory, which is based on a moving target. An autonomous vehicle is often defined as the target where all its states can be known by the follower, [START_REF] Reis | Robust moving path following control for robotic vehicles: Theory and experiments[END_REF]. The rendezvous control works with the differential geometry method. This method focuses on the angular difference between the target and the follower, [START_REF] Oh | Rendezvous and standoff target tracking guidance using differential geometry[END_REF]. Comparing the strategies, rendezvous approach, using a control based on differential geometry, is a method more complex than the dynamic path following since the last method can be addressed to a trajectory tracking problem. In this work, we propose a landing dynamic trajectory to guide an aerial vehicle towards the target position at a certain rendezvous longitude, l R . The desired trajectory depends on the target position, which is the path parameter that provides the evolution of a descending slope until the target is reached.

The manuscript is organized as follows: the mathematical preliminaries are defined in Section 2. The problem statement is given in Section 3. The landing dynamic trajectory is defined in Section 4. The motion equations and control strategy for a fixed-wing drone are described in Section 5. An experimental validation is developed using a quadcopter for the trajectory evaluation. The quadcopter control strategy is described in Section 6. Landing trajectory performance is corroborated by the results of simulations and experiments in Section 7. Finally, concluding remarks and future research directions are presented in Section 8.

PRELIMINARIES

Quaternion modeling

Quaternion definitions The quaternion set is described as H. A single quaternion variable q := q 0 + -→ q ∈ H can be considered as a four tuple with one scalar part q 0 ∈ R and the sum of three imaginary parts q 1 , q 2 , q 3 ∈ R using three unit imaginary perpendicular vectors i, j, k ∈ R 3 . This last one can be summarized using the following notation:

-→ q = q 1 i + q 2 j + q 3 k as described in (Kuipers, 1999).

A quaternion can be represented using the following four tuple in vector form q = [q 0 , -→ q ] T .

(1) The main operation for a quaternion is the product, which is defined using vector operations for two quaternions q, r ∈ H as:

q ⊗ r := (q 0 r 0 -- → q • - → r ) + (q 0 - → r + r 0 - → q + - → q × - → r ) , (2)
where • and × are the vector inner and outer products. Vectors in 3D space can be considered as quaternions with null scalar parts, which are also called pure quaternions. It can be seen from equation (2) that the quaternion product is compatible with 3D vectors. A useful linear operation, akin to complex numbers, is the quaternion conjugate: q * := q 0 --→ q .

(3) The quaternion norm can be defined as : ∥q∥ := q ⊗ q * = q 2 0 + q 2 1 + q 2 2 + q 2 3 .

(4) A quaternion with unitary norm ∥q∥ = 1 is called a unitary quaternion, and these are the ones used for the description of the attitude in a rigid body. An operation that will be later used is the quaternion natural logarithm function, which is described for a unit quaternion q ∈ H, ∥q∥ = 1 as

log q :=    - → q ∥ - → q ∥ arccos q 0 , ∥ - → q ∥ ̸ = 0 0, ∥ - → q ∥ = 0 (5)
Attitude representation using quaternions The rotation operation of a 3D vector -→ v ∈ R 3 from one reference frame into another -→ v → -→ v ′ can be described using an unitary quaternion q ∈ H by means of the Euler-Rodrigues formula, [START_REF] Morais | Real quaternionic calculus handbook[END_REF]:

- → v ′ := q ⊗ - → v ⊗ q * (
6) where the quaternion is calculated as

q := cos θ 2 + - → n sin θ 2 (7)
where θ ∈ R is a rotation quantity and -→ n ∈ R 3 is a unit 3D vector that marks the axis of rotation.

In the particular case of rigid bodies, the equations ( 6) and ( 7) can be used as a means to describe the full attitude of the object.

PROBLEM STATEMENT

The objective is to design a strategy to land on a mobile target. In our case, it is assumed that a suitable tracking controller has already been implemented and is able to track a desired trajectory without issues. This also implies that the associated perturbations due to the moving platform are absorbed by the control law. The focus therefore shifts towards the design and corroboration of a suitable trajectory for the tracking control algorithm that would allow an UAV to safely land on a mobile target. The proposed UAV is a fixed-wing vehicle, but the strategy could be adapted to more agile vehicles. An advantage that the drone has is that it is assumed that the full state of the landing platform is always known.

LANDING DYNAMIC TRAJECTORY

The design of the reference trajectory is based on two techniques. The first one is the dynamic path following, that consists of steering an autonomous vehicle toward a geometric trajectory defined by the target's motion. The second strategy is named glide path, which is defined as a straight line directed to the landing point. As mentioned, the desired trajectory focuses on leading the fixed-wing drone towards the position of a moving target in the x-z plane. A ground vehicle is defined as the target. The coordinate system related to the inertial frame is denoted by {I}. The body frame of the follower vehicle is defined by {B}, and the target frame by {T }. The position vectors of the vehicles' center of mass are defined in {I}, where (x T , z T ) represents the target position, and (x, z) denotes the follower position. Figure 1 shows a diagram of the proposed trajectory. The desired trajectory is defined as:

x d = x T -(l R -x T ) cos η, (8)
z d = z T + (l R -x T ) sin η. ( 9
)
In addition, the desired velocities of the trajectory can be calculated as follows ẋd = ẋT (1 + cos η), (10

) żd = -ẋT sin η. (11)
This allows to implement different control techniques to carry out the trajectory tracking as the information of the speed of the target is available.

SIMULATION FOR A FIXED-WING DRONE

The following section describes the longitudinal model of a fixed-wing UAV and the corresponding tracking controller design that were used to validate the proposed trajectory using numerical simulations.

Longitudinal motion equations for a fixed-wing drone

A reduced-order longitudinal guidance model for a fixedwing drone can be written as in [START_REF] Beard | Small Unmanned Aircraft: Theory and Practice[END_REF], it is described as follows:

ẋ = V a cos γ (12) ż = V a sin γ (13) γ = F L mV a - g V a cos γ, (14)
where (x, z) represents the aircraft's position defined in the inertial frame. γ describes the flight path angle, and V a denotes the airspeed, which is defined as a constant value.

The terms m and g are the mass and the acceleration due to the gravity, respectively. This model comprises a combination of kinematic and dynamic equations. Moreover, it considers that the angle of attack α ≈ 0. Therefore, the F L lift force can be written of the following form:

F L = 1 2 ρV 2 a S a [C L0 + C D δe δ e ] (15)
The air density is denoted by ρ, and S a defines the surface area of the wing. Finally, the control input involved in the lift force is the elevator's deflection, which is depicted by δ e . Notice that equations (15) involves the relations with the lift coefficients, they are defined by the features of each aircraft.

Tracking controller design for a fixed-wing drone

The design of the control law to follow the desired trajectory are corroborated using Lyapunov stability analysis.

We will present the strategy to track the desired positions trajectory. Considering the aircraft dynamics, the aircraft will be guided based on a desired flight path angle, which is denoted by γ d .

From Figure 3, the desired flight angle can be calculated through

γ d = tan -1 z d -z x d -x , (16)
and its derivative is given by γd

= - (x d -x) [ żd + ż + ∆( ẋd -x)] (x d -x) 2 + (z d -z) 2 , ( 17
)
where

∆ = z d -z x d -x . ( 18
)
Fig. 3. Representation of the reference angle.

Defining the tracking error as e γ = γ -γ d . We propose a positive function V 1 = 1 2 e 2 γ , differentiating and substituting ( 14), it follows

V1 = e γ F L mV a - g V a cos γ -γd < 0. ( 19
)
Therefore, it is possible to solve the equation for the elevator control input δ e , whose result can be defined as

δ e = - V a C L δe + 2m ρV 2 a S a C L δe [g cos γ + γd -V a e γ ] . (20)
Introducing ( 20) into ( 19), then V1 = -e 2 γ , that is, γ → γ d .

EXPERIMENTAL VALIDATION USING A QUADCOPTER VEHICLE

The real-time implementation of our landing dynamic trajectory on a real-world platform will validate its performance and viability. Since the main advantage is the facility of implementation and the dependency on the movement of the target. The strategy is designed to guide one aerial vehicle towards a rendezvous point, reaching the target's position. The experimental validation is performed in an indoor environment using a quadcopter ARDrone 2.

The original firmware of this drone was replaced by the open-source software Fl-AIR (Framework Libre AIR), which takes an important roll to integrate the quadcopter with the localization system and is described in more detail in [START_REF] Sanahuja | FL-AIR framework[END_REF].

In addition, a ground vehicle, Parrot Jumping Sumo, is used as a target. It is controlled manually by a pilot. The interchange of data between target-follower is available such as the target's position and speed, which is obtained by an OptiTrack motion capture system. The control strategy used to track the trajectory in the test is described in the following section.

Quadcopter control strategy

The model and the control strategy used for the quadcopter vehicle has been discussed in detail in [START_REF] Carino | Quadrotor quaternion control[END_REF]. The model is out of the scope of this work, but the control will be briefly discussed to show how to implement the trajectory tracking. Basically, an inner position control gives a reference to an outer attitude control. The position control is defined for the tracking of the proposed trajectory as :

- → u pos := -K pos - → E -K vel - → E . ( 21
)
where -→ E represents the tracking error vector given by

- → E := [(x -x d ) , (z -z d )] T , (22)
K pos and K vel depict the gains matrices of position and velocity. The attitude control is defined as :

-→ u att := -2 K att ln q e -k ω -→ ω , (23) K att is the attitude gains, -→ ω depicts the angular velocity vector, and the quaternion attitude error q e ∈H defines as: q e := q * d ⊗ q.

(24) The attitude of the vehicles is depicted by the unit quaternion q and the desired quaternion q d is defined using the position control (21) as :

q ′ d := - → b • - → u pos + ∥ - → u pos ∥ + - → b × - → u pos q d := q ′ d ∥q ′ d ∥ (25)
where -→ b depicts a unitary vector denoting the axis in which the thrust acts in {B}. Therefore, the control input for the quadcopter platform is the attitude control torques as described in ( 23) and the scalar thrust u thrust := ∥ -→ u pos ∥ (26) The combination of both of these strategies allows for the successful tracking of the proposed trajectory in a quadcopter vehicle. This can be corroborated in practice by measuring the distance between the desired trajectory and the quadcopter's position.

RESULTS

Simulation results for the fixed-wing aerial vehicle

The landing dynamic trajectory is implemented and validated in a Simulink simulations, using the solver 0de4 (Runge-Kutta) with a sample time of 0.001 s. The aerodynamic parameters of the fixed-wing drone are obtained in [START_REF] Alatorre | Least Airspeed Reduction Strategy & Flight Recuperation of a Fixed-Wing Drone[END_REF]. As mentioned, a ground vehicle is defined as the target with an altitude of 0.5 meters and a velocity ẋT = 2 m/s. The goal is to validate the landing dynamic trajectory since it guides the aircraft towards the target position. The rendezvous position is defined in (l R , z T ) = (10, 0.5) m. Moreover, the desired trajectory angle is given by η = 12 • . The initial position of the target vehicle is defined in (x T (0), z T (0)) = (0, 0) m and the initial position of the fixed-wing drone is given by (x(0), z(0)) = (-14, 2.5) m. The airspeed of the drone is defined by 5 m/s. Figure 4 shows the simulation result of tracking the landing trajectory. The navigation path of the target is depicted by the green line, the landing dynamic trajectory is defined as the red line, and the path of the fixed-wing drone is depicted by the blue line.

Notice that the drone performs a descending flight, reaching the desired trajectory. The trajectory evolves with an η constant angle, and the distance between (x d , z d ) and (x T , z T ) reduces until reaching the target position. The target vehicle navigates in a straight line until l R where the drone converges to the target position such as is shown in the zoom of the Figure 4. Considering the elevator control input (20), the flight path angle tracks to the desired angle given in ( 16). Therefore, we can observe the behavior of γ to reach γ d in Figure 5

Experimental results

In this section, we present the experimental results of the landing dynamic trajectory using a quadcopter vehicle.

The experimental flight test is initially defined by keeping the drone at hover flight at the initial desired altitude.

Since the target has less dimension than the drone, we define z T with the height of a landing platform. It will be located in the rendezvous point, i.e. at the position of l R . The experiment stages are described as follows:

• The target is located in the origin of flight room.

• The quadcopter takes-off and keeps a hover flight on the initial position of (x d , z d ).

• The ground vehicle navigates in a straight line towards the landing platform. • The drone's controller tracks the trajectory and finally lands on the platform. The desired trajectory parameters were fixed in order to accommodate the available flight space.

Table 1. Experimental Parameters

Parameters Value Units η 50 • degrees l R 3 meters z T 0.5 meters x T (0) 0 meters (x(0), z(0)) (-2, 2.

7) meters

A video of the experimental validation of the landing dynamic trajectory can be seen at: https://youtu.be/UxMJNoK_VJ4

Figure 7 shows the performance of the desired trajectory related to the target's movement. Observe that the drone starts the mission at hover. Then, it executes the trajectory tracking algorithm where we note that the drone's path approximates to the trajectory. The target is illustrated in the zoom of Figure 7. It can be clearly seen that the drone approximates to the target position with a minimum error.

On one hand, we can visualize in Figure 8 how the displacement in the x axis of the target vehicle influences the behavior of the desired position x d . This forces the aircraft to navigates until the x T position. On the other hand, the desired altitude tends to decrease as the target vehicle moves towards the rendezvous point, see Figure 9. Therefore, the quadcopter also reduces gradually its altitude with respect to the desired altitude, which implies landing on the target. Finally, we compute the norm of the tracking error vector, which is defined in ( 22). We obtain that || -→ E || is a minimum value which approximates to zero, which is represented in logarithm scale in the Figure 10.

CONCLUSION

In this paper, a landing dynamic trajectory was presented, designed and tested to guide an aerial vehicle towards the position of a mobile target. The convergence of the position between the vehicles is achieved after the target navigates to reach a predefined distance.

The validation of the landing dynamic trajectory was made using simulations and experiments. The simulation results clearly show that a fixed-wing vehicle would be able to land on a mobile target using the proposed trajectory. The constant angle of the trajectory allows to adapt to the speed and distance towards the target just before starting the tracking. Once engaged, the proposed controller shows how the tracking error diminishes as time advances even with the movement of the target. In the experimental results, it was proven that the implemented control strategy was successful in tracking the desired trajectory in a real-world scenario. The low error implies that the quadcopter was practically always following the trajectory. The video also demonstrates how this design could be used to land an aerial vehicle on a moving target.

Future work

Even though stability for this particular scenario was guaranteed thanks to the chosen control algorithms, the fact is that a higher performance could be possible by integrating the dynamics of the ground platform into the control design.

Another interesting alternative would be to integrate a prediction model in order to have a more accurate representation of the target's dynamics.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Landing dynamic trajectory scheme.The landing dynamic trajectory develops a descending flight based on a slope directed to the target. The trajectory is a function of the target displacement, this parameter path reduces the slope's distance between the desired position and the target, keeping an η ∈ R constant angle, see Figure2. Moreover, this figure shows the association between the desired position and the target's position using reference circles. Therefore, the fixed-wing vehicle tracks the trajectory and it will reach the target's position after the target navigates a defined longitude denoted by l R ∈ R + . The l R ∈ R + path parameter represents the rendezvous point between the trajectory and the target, that is, x d = x T = l R .

Fig. 6 .

 6 Fig. 4. Trajectory tracking using a fixed-wing drone.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Experimental result of the landing dynamic trajectory executed by a quadcopter vehicle.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Behavior of the desired altitude imposing a descending flight for a quadcopter.

ACKNOWLEDGEMENTS

This paper was supported by the RPV project -UTC foundation and the Mexican National Council of Science and Technology -CONACyT.

⋆ This paper was supported by the RPV project -UTC foundation and the Mexican National Council of Science and Technology -CONACyT.


Zakaria El Ouali
email: dr.elouali.zakaria@gmail.com

Laure Gossec

Zakaria El Ouali

Challenges in interpreting sacroiliac magnetic resonance imaging for the diagnosis of axial spondyloarthritis

Keywords: spondyloarthritis, MRI, sacroiliitis, healthy subjects Word count: 1272

Axial spondylarthritis (axSpA) is a generic term referring to a spectrum of chronic inflammatory rheumatic diseases involving the axial spine, and belonging to the family of spondyloarthritis (SpA) [START_REF] Sieper | Axial spondyloarthritis[END_REF]. Magnetic resonance imaging (MRI) plays a key role in the workup of axSpA. While late forms of axSpA can be easy to diagnose, early forms are difficult to detect using plain radiographs. The lack of imaging structural damage, which can sometimes take months to years to appear [START_REF] Dougados | Sacroiliac radiographic progression in recent onset axial spondyloarthritis: the 5-year data of the DESIR cohort[END_REF], contributes to a diagnostic delay of two to eleven years [START_REF] Zhao | Diagnostic delay in axial spondyloarthritis: a systematic review and meta-analysis[END_REF]. Inflammation is however an early feature of axSpA, that can be detected on MRI. MRI has not only facilitated the diagnosis, it has also led to a significant reduction of the diagnostic delay [START_REF] Robinson | Axial spondyloarthritis: concept, construct, classification and implications for therapy[END_REF]. The Assessment of SpondyloArthritis International Society (ASAS) proposed classification criteria in 2009 including MRI features (Table 1) [START_REF] Rudwaleit | The development of Assessment of SpondyloArthritis international Society classification criteria for axial spondyloarthritis (part II): validation and final selection[END_REF]. AxSpA, as defined by ASAS, includes patients with structural damage in the sacroiliac joints (SIJ) assessable on plain radiographs (commonly referred as radiographic axSpA), but also patients without radiologic sacroiliitis, called non-radiographic axSpA (nr-axSpA) (Table 1) [START_REF] Sieper | Axial spondyloarthritis[END_REF]. Nr-axSpA often includes MRI abnormalities. Sacroiliitis on MRI was described in the initial ASAS criteria as "active (acute) inflammation of sacroiliac joints on MRI, highly suggestive of sacroiliitis associated with SpA" [START_REF] Rudwaleit | The development of Assessment of SpondyloArthritis international Society classification criteria for axial spondyloarthritis (part II): validation and final selection[END_REF]. In a more recent consensus, the ASAS/OMERACT MRI working group defined positive sacroiliac joint MRI as "either at least 2 inflammatory lesions in form of bone marrow edema (BME) in one slice, or only one BME lesion in at least 2 consecutive slices" [START_REF] Rudwaleit | Defining active sacroiliitis on magnetic resonance imaging (MRI) for classification of axial spondyloarthritis: a consensual approach by the ASAS/OMERACT MRI group[END_REF][START_REF] Lambert | Defining active sacroiliitis on MRI for classification of axial spondyloarthritis: update by the ASAS MRI working group[END_REF][START_REF] Maksymowych | MRI lesions in the sacroiliac joints of patients with spondyloarthritis: an update of definitions and validation by the ASAS MRI working group[END_REF]. However, the specificity of the ASAS definition of positive MRI is a matter of debate [START_REF] Robinson | Axial spondyloarthritis: concept, construct, classification and implications for therapy[END_REF]. Inflammation on SIJ MRI can also be found in patients without axSpA, leading to a false "positive MRI" according to the ASAS definition (4). In the present paper, we will review the frequency of these false positive MRI findings, and discuss the factors that may explain inflammation in the absence of axSpA.

Initial preliminary data from 2018 onwards indicate that sacroiliitis could be seen in healthy subjects (Table 2). In 2018, de Winter et al. compared SIJ MRI of healthy subjects to those of patients with axSpA and patients with chronic back pain [START_REF] De Winter | Magnetic Resonance Imaging of the Sacroiliac Joints Indicating Sacroiliitis According to the Assessment of SpondyloArthritis international Society Definition in Healthy Individuals, Runners, and Women With Postpartum Back Pain[END_REF]. Of the 47 healthy volunteers, 11 (23.4%) had a positive SIJ MRI versus 43/47 (91.5%) axSpA patients. Later, other small studies showed the presence of BME and fatty lesions in patients without axSpA, particularly in women with postpartum buttock/pelvic pain (3-39%, table 2) [START_REF] Seven | Magnetic Resonance Imaging of Lesions in the Sacroiliac Joints for Differentiation of Patients With Axial Spondyloarthritis From Control Subjects With or Without Pelvic or Buttock Pain: A Prospective, Cross-Sectional Study of 204 Participants[END_REF], and subjects aged over 30 (17%) [START_REF] Renson | Progressive increase in sacroiliac joint and spinal mri lesions in healthy individuals in relation to age[END_REF]. These studies raised concerns about the lack of specificity of MRI in the diagnosis of axSpA, but were considered exploratory, and the results had little impact on clinal practice. Recently, Baraliakos et al. investigated the frequency of BME and fatty lesions in a large general population-based study [START_REF] Baraliakos | Frequency of MRI changes suggestive of axial spondyloarthritis in the axial skeleton in a large population-based cohort of individuals aged <45 years[END_REF]. In all, 793 volunteers aged less than 45 years were evaluated, of whom a surprisingly high number, 136 (17.2%) had SIJ BME [START_REF] Baraliakos | Frequency of MRI changes suggestive of axial spondyloarthritis in the axial skeleton in a large population-based cohort of individuals aged <45 years[END_REF]. The investigators also found abnormalities on the spine, with a relatively high frequency of inflammatory (27%) and fatty spinal MRI lesion (81%) suggestive of axSpA [START_REF] Baraliakos | Frequency of MRI changes suggestive of axial spondyloarthritis in the axial skeleton in a large population-based cohort of individuals aged <45 years[END_REF]. Could this high rate be explained by poor interpretation? MRI sequences performed for the SIJ (i.e., Short-TI Inversion Recovery (STIR) sequence without T1-weighted sequences) were not those ideally recommended by ASAS [START_REF] Maksymowych | MRI lesions in the sacroiliac joints of patients with spondyloarthritis: an update of definitions and validation by the ASAS MRI working group[END_REF]. It is important to note that the site of the elementary lesion was not taken into account, although it has been shown that fat lesions and/or erosions located centrally or posteriorly in the SIJ are particularly significant and should be investigated further [START_REF] Seven | Anatomic Distribution of Sacroiliac Joint Lesions on Magnetic Resonance Imaging in Patients With Axial Spondyloarthritis and Control Subjects: A Prospective Cross-Sectional Study, Including Postpartum Women, Patients With Disc Herniation, Cleaning Staff, Runners, and Healthy Individuals[END_REF]. However, 2 trained readers evaluated the MR images independently to assess BME. There was good agreement in the evaluation of the images between the readers, made up of a team of rheumatologists and radiologists, guided by an experienced reader [START_REF] Baraliakos | Frequency of MRI changes suggestive of axial spondyloarthritis in the axial skeleton in a large population-based cohort of individuals aged <45 years[END_REF]. This study indicates that MRI both on SIJ and the spine, may overate inflammation. It is possible the some patients, taken from the general population, had axSpA or low back pain [START_REF] Baraliakos | Frequency of MRI changes suggestive of axial spondyloarthritis in the axial skeleton in a large population-based cohort of individuals aged <45 years[END_REF]. This study was cross-sectional without follow-up, making it impossible to assess later diagnoses. Overall, the large size of the study allowed to confidently confirm the presence of sacroiliitis in healthy subjects.

What are the factors involved in SIJ inflammation in healthy subjects (Figure 1)? First of all, age seems to have an impact. Renson et al. showed elegantly that sacroiliitis was more frequently detected in older age categories, going from 2.8% in the 20-29 category, to 17.9% in the 40-49 age group [START_REF] Renson | Progressive increase in sacroiliac joint and spinal mri lesions in healthy individuals in relation to age[END_REF]. It is possible that either mechanical stress or osteoarthritis may contribute to those findings. Moreover, physical activity is also a parameter leading to SIJ inflammation (Table 2). Of 22 elite ice hockey players and 20 recreational runners before and after running, 41% and 30-35% respectively fulfilled the ASAS definition of active sacroiliitis, with the posterior lower ilium being the most affected SI joint region, while no erosion was found [START_REF] Weber | Frequency and Anatomic Distribution of Magnetic Resonance Imaging Features in the Sacroiliac Joints of Young Athletes: Exploring "Background Noise" Toward a Data-Driven Definition of Sacroiliitis in Early Spondyloarthritis[END_REF]. Similarly, intense standardized physical training increased the rate of positive SIJ MRI in military recruits after 6 weeks of intense standardized physical training (36% positive SIJ MRI after 6 weeks versus 23% at baseline) [START_REF] Varkas | Effect of mechanical stress on magnetic resonance imaging of the sacroiliac joints: assessment of military recruits by magnetic resonance imaging study[END_REF]. However, SIJ erosions on MRI were present at the same rate both at baseline and after the 6 weeks of training (27%). These results indicate that it may be useful to ask patients to rest for a few days before undergoing MRI. However, the duration of cessation of efforts has not been explored. Another state with SIJ inflammation is postpartum. As could perhaps be expected, in 35 women within the first 10 days after giving birth, 27 (77%) had sacroiliac BME, and 60% fulfilled the ASAS definition of a positive MRI [START_REF] Renson | High prevalence of spondyloarthritis-like MRI lesions in postpartum women: a prospective analysis in relation to maternal, child and birth characteristics[END_REF]. After 6 months, 46% of the women still had BME, and 15% had a positive MRI. More surprisingly, after 12 months, 12% of the subjects still had a positive MRI [START_REF] Renson | High prevalence of spondyloarthritis-like MRI lesions in postpartum women: a prospective analysis in relation to maternal, child and birth characteristics[END_REF]. These findings were confirmed in similar studies [START_REF] Agten | Postpartum Bone Marrow Edema at the Sacroiliac Joints May Mimic Sacroiliitis of Axial Spondyloarthritis on MRI[END_REF][START_REF] Hoballah | MRI of sacroiliac joints for the diagnosis of axial SpA: prevalence of inflammatory and structural lesions in nulliparous, early postpartum and late postpartum women[END_REF]. It appears that postpartum has long standing effects.

False positive MRIs in daily practice could lead to overdiagnosis of axSpA, and potentially overtreatment [START_REF] Ortolan | The Risk of overdiagnosis and overtreatment in spondyloarthritis[END_REF][START_REF] Wendling | Tight control and treat-totarget in axial spondyloarthritis. Where are we?[END_REF]. We propose some elements to avoid overdiagnosis. It is important to perform the proper MRI sequences to analyze the SIJ in accordance with the ASAS/OMERACT recommendations, which are the T1-weighted and the STIR sequences [START_REF] Maksymowych | MRI lesions in the sacroiliac joints of patients with spondyloarthritis: an update of definitions and validation by the ASAS MRI working group[END_REF]. Moreover, the development of expertise in reading SIJ MRI is fundamental. The aim is to learn how to distinguish inflammatory lesions that may signal axSpA, from inflammatory lesions caused by other conditions, and to improve interobserver agreement [START_REF] Rueda | Interobserver Agreement in Magnetic Resonance of the Sacroiliac Joints in Patients with Spondyloarthritis[END_REF]. The interpretation should take into account the location and the extent of the SIJ lesions, the existence of erosion or fatty metaplasia, and anterior involvement of the lumbar spine [START_REF] Herrada | Diagnostic performance of sacroiliac and spinal MRI for the diagnosis of non-radiographic axial spondyloarthritis in patients with inflammatory back pain[END_REF]. Of note, the interpretation of MRI should take into account both inflammatory and structural lesions, since taken separately, none of them have perfect specificity. More attention could be paid to structural damage, since the development of new sequences like the volumetric interpolated breath-hold examination (VIBE) have improved the detection of erosions in the SIJ compared with a standard T1-weighted sequence [START_REF] Diekhoff | Improved detection of erosions in the sacroiliac joints on MRI with volumetric interpolated breathhold examination (VIBE): results from the SIMACT study[END_REF]. It is possible that the current definition of SIJ MRI abnormalities used for the classification of axSpA is too wide and should be reconsidered. On a global level, clinical findings remain key elements in the investigation of SpA, particularly extra-articular manifestations such as uveitis, dactylitis, psoriasis, or signs of inflammatory bowel disease (4). In the absence of such clinical symptoms, caution is needed when interpreting a "positive MRI".

In conclusion, inflammatory SIJ MRI lesions suggestive of axSpA can be present in healthy subjects. SIJ MRI should be interpreted with caution and clinical expertise must remain at the forefront of the diagnosis of axSpA.

  spondyloarthritis ; F: female ; M: male ; n: number of subjects ; MRI: magnetic resonance imaging ; % positive MRI: percentage of the patients with positive MRI according to the ASAS definition. The frequency of positive MRI in patients with axSpA ranged 56-92% in studies[START_REF] De Winter | Magnetic Resonance Imaging of the Sacroiliac Joints Indicating Sacroiliitis According to the Assessment of SpondyloArthritis international Society Definition in Healthy Individuals, Runners, and Women With Postpartum Back Pain[END_REF][START_REF] Seven | Magnetic Resonance Imaging of Lesions in the Sacroiliac Joints for Differentiation of Patients With Axial Spondyloarthritis From Control Subjects With or Without Pelvic or Buttock Pain: A Prospective, Cross-Sectional Study of 204 Participants[END_REF].

FiguresFigure 1 .

 1 Figures

  Figure 1. Factors of positive sacroiliac MRI findings outside of axSpA (9-12, 14-18, 24). The size of the diamonds reflects the number of patients. % IRM +: percentage of the patients with positive MRI according to the ASAS definition.

Table 1 .

 1 Role of imaging in the ASAS classification criteria for axSpA (derived from 5) Entry

criterion: patients with ≥3 months back pain and age at onset <45 years Arm 1 : Radiographic axSpA Arm 2 : Non-radiographic axSpA SpA features

	Sacroiliitis	on	plain	Sacroiliitis on MRI according to the	• Inflammatory back pain
	radiograph according to the	ASAS consensus definition	• Arthritis
	modified New York criteria	+ ≥ 1 feature of SpA	• Enthesitis (heel)
	+ ≥ 1 feature of SpA			• Uveitis
				OR	• Dactylitis
					• Psoriasis
				HLA-B27	• Crohn's/colitis
				+ ≥ 2 other features of SpA	• Good response to
					NSAIDs
					• Family history for SpA
					• HLA-B27
					• Elevated CRP
	axSpA: axial spondyloarthritis ; ASAS: Assessment of SpondyloArthritis International Society.
	CRP: C-reactive protein. NSAIDs: non-steroidal anti-inflammatory drugs. SpA :
	spondyloarthritis.

Table 2 .

 2 Frequency of positive sacroiliac MRI findings outside of axSpA(9-12, 14-18, 24).

	Study population	Ref	n	Gender	Age, years %	positive
						MRI
	Healthy subjects
		(11)	36	M + F	20-29	3%
		(11)	31	M + F	30-39	16%
		(11)	28	M + F	40-49	18%
		(12)	793	M + F	37.3 ± 6.3	17%
		(10)	29	M	30.9 ± 6.4	0%
		(9)	47	M + F	30.9 ± 6.5	23%
	Physical activity / athletes
	Long-distance runners	(10)	23	M + F	32.7 ± 6.2	4%
	Runners	(9)	24	M + F	34.3 ± 7.7	13%
	Military recruits after 6 weeks of training	(15)	11	M	25 ± 0.8	36%
	Ice hockey players	(14)	22	M	25.9 ± 4.6	41%
	Recreational runners	(14)	20	M + F	27.2 ± 5.4	30-35%
	Postpartum
	Within 10 days of delivery	(16)	35	F	29.7 ±2.62	77%
	Early postpartum period	(17)	30	F	34.0	63%
	Given birth within 12 months	(18)	60	F	31.2 ±4.5	33%
	Given birth more than 24 months	(18)	144	F	38.3 ±6.2	21%
	Without buttock and/or pelvic pain	(10)	14	F	33.1 ± 4.1	21%
	With buttock and/or pelvic pain	(10)	46	F	32.6 ± 3.3	41%
	With back pain	(9)	7	F	30.1 ± 7.2	57%
	Low back pain
	Lumbar disc herniation
	Chronic back pain
	Low back pain


Vidolov
email: bvidolov@hds.utc.fr

C Souza

P Castillo
email: castillo@hds.utc.fr

B Vidolov

Local interaction and navigation guidance for hunters drones: A chase behavior approach with real-time tests

Keywords: hunter drones, autonomous navigation, tracking, pursuit, real-time tests

A behavioral-based strategy for cooperative hunting using drones is proposed in this paper. In this decentralized scheme, each drone acts as an individual agent computing its guidance strategy toward the target based on the relative position of its neighbors without the use of direct communication. The algorithm is based on the Deviated Pure Pursuit methodology, and the emerged behavior mimics a natural hunting formation. Simulations and real-time experiments with varying conditions were carried out to validate the effectiveness of the proposed hunting scheme. Videos of the system in action can be seen on: https://youtu.be/g2dODbd6ZLA.

Introduction

The study and analysis of pursuit applied to mobile robots has been gaining the attention of the research community, motivated by the increasing popularity of small robots, especially drones. The need for surveillance of civil areas has motivated the application of classical war-like pursuit techniques, see refs. [START_REF] Lefebvre | Conceptual design study of an anti-drone drone[END_REF][START_REF] Allen | Quadrotor intercept trajectory planning and simulation[END_REF][START_REF] Solinger | Creating a dogfight agent[END_REF][START_REF] Hert | Autonomous Predictive Interception of a Flying Target by an Unmanned Aerial Vehicle[END_REF][START_REF] Arneberg | Guidance Laws for Partially-Observable UAV Interception Based on Linear Covariance Analysis[END_REF]. Nevertheless, using a single agent for pursuit has several limitations, such as when the target has a velocity equal or superior to the pursuer, or a chaotic behavior. A recent alternative to overcome these limitations is the use of multiple pursuers; see refs. [START_REF] Yulan | Study on multi-robot cooperation stalking using finite state machine[END_REF][START_REF] Yamaguchi | A cooperative hunting behavior by multiple nonholonomic mobile robots[END_REF][START_REF] Wu | A multi-robot cooperative hunting approach based on dynamic prediction of target motion[END_REF][START_REF] Weitzenfeld | A biologically-inspired wolf pack multiple robot hunting model[END_REF], where it is possible to increase the spatial distribution of the agents to allow for a more fault-tolerant system. However, with an increase in the number of pursuers, the task increases in complexity -as the problem now includes issues such as collision-free navigation, formation control, task, and roles allocation. Many researchers approach this complex task in an approach base on collective hunters in nature refs. [START_REF] Yulan | Study on multi-robot cooperation stalking using finite state machine[END_REF][START_REF] Yamaguchi | A cooperative hunting behavior by multiple nonholonomic mobile robots[END_REF][START_REF] Muro | Wolf-pack (canis lupus) hunting strategies emerge from simple rules in computational simulations[END_REF][START_REF] Shishika | Robust lyapunov control design for bioinspired pursuit with autonomous hovercraft[END_REF][START_REF] Angelani | Collective predation and escape strategies[END_REF][START_REF] Lin | Abaid Collective behavior and predation success in a predator-prey model inspired by hunting bats[END_REF].

The algorithms found in the literature can be distributed in four main categories: pursuit, game theory, swarming, and bio-inspired group chasing.

Pursuit

The pursuit-evasion phenomena have been observed in almost all biological relations and have been traditionally analyzed mathematically and applied to modern humans. Since the classical and traditional analyses of Pierre Bourger, in Les courbes de Poursuite (1732), where he describes the interception of merchant's vessels by pirates ship, many pursuit techniques have been formalized and implemented. Classical guidances laws, such as the Deviated Pursuit Guidance (DPP), the Proportional Navigation Guidance (PNG), and its variations, have been used notably in missile guidance; see refs. [START_REF] Shneydor | Missile guidance and pursuit: kinematics, dynamics and control[END_REF][START_REF] Nahin | Chases and Escapes The Mathematic[END_REF]. These guidance laws have been extensively explored for ground robots, see refs. [START_REF] Belkhouche | A method for robot navigation toward a moving goal with unknown maneuvers[END_REF][START_REF] Belkhouche | Parallel navigation for reaching a moving goal by a mobile robot[END_REF][START_REF] Teimoori | A biologically inspired method for robot navigation in a cluttered environment[END_REF]. In ref. [START_REF] Belkhouche | A method for robot navigation toward a moving goal with unknown maneuvers[END_REF], the authors explored the use of geometric rules to navigate a single agent towards a target with unknown movements and in the presence of obstacles. They applied two navigation approaches, the velocity pursuit guidance (a special case of PNG) and the DPP methodology. In ref. [START_REF] Belkhouche | Parallel navigation for reaching a moving goal by a mobile robot[END_REF], the authors implemented a controller that keeps a constant line-of-sight from pursuers to the target, using the principle of parallel navigation to ensure, under some conditions, the rendezvous course. Similarly, in ref. [START_REF] Teimoori | A biologically inspired method for robot navigation in a cluttered environment[END_REF], the authors applied a sliding-mode-based control to keep a constant line-of-sight.

In ref. [START_REF] Allen | Quadrotor intercept trajectory planning and simulation[END_REF], the author applied PP and PNG guidance algorithms into quadcopters for target interception. They compared those techniques against an optimal control trajectory planner, analyzing the time and energy required to achieve the task. However, results are only presented in simulations. In ref. [START_REF] Arneberg | Guidance Laws for Partially-Observable UAV Interception Based on Linear Covariance Analysis[END_REF], the author analyses three missile guidance laws in the case of a partially observable target. They formulate a pursuitevasion game to determine the guarantees of capture under evasive maneuvers, deriving an optimal controller. However, during the approaching phase, the PP algorithm is used.

The classic guidance navigation approaches have been useful in more applications than the pursuit problem; for example, in ref. [START_REF] Tan | Proportional navigation (pn) based tracking of ground targets by quadrotor uavs[END_REF] the authors used the PNG scheme in a drone landing on a moving platform. Their final solution mixed PNG with proportional derivative (PD) algorithms, effectively completing the task with less oscillation and smaller errors than traditional tracking PD. The proportional navigation approach has also been used as guidance law in autonomous racing of drones in ref. [START_REF] Jung | Perception, guidance, and navigation for indoor autonomous drone racing using deep learning[END_REF]. In that case, the authors adapted the PNG algorithm to handle the decoupled dynamics of the quadrotor.

Differential Games

Recently, the research community has been interested in a more complex version of the pursuit problem, increasing the number of pursuers and evaders. This can be modelled using a differential game, which uses game theory to model conflict in the context of a dynamic system.

In ref. [START_REF] Huang | Guaranteed decentralized pursuitevasion in the plane with multiple pursuers[END_REF], the authors used a differential game formulation for multi-pursuers chasing a single evader. In their approach, each agent minimizes the action area of the evader, which is characterized by the Voronoi cells. Similarly, in ref. [START_REF] Li | Coordinated multi-robot target hunting based on extended cooperative game[END_REF], a cost function is proposed and computed based on the relative angle and distance between agents and target. These methodologies guarantee the optimality input for pursuit. However, the large amount of assumptions and precise knowledge needed about the whole system makes it infeasible for application in the real-world.

Multi-agents

The benefits of the group work against the individual alone can be observed in many social behaviors in nature. For example, in ref. [START_REF] Zafeiris | Why We Live in Hierarchies?: A Quantitative Treatise[END_REF], the authors completed quantitative analysis of the probability to survive from predators, feeding, reproducing, and living in a group; similarly, they have discussed how hierarchical systems emerge.

In ref. [START_REF] Stander | Cooperative hunting in lions: the role of the individual[END_REF], the authors analyzed the cooperative chase with lionesses. Although lions commonly chase alone, they can cooperate -which increases hunting success. The formation of the lions, with center and wings lions, has inspired our approach.

The attempt to use a group of robots to stalk a moving target started early, basing on the extensive multi-agents formation theory, and using a feedback controller to haunt tasks and keep formation; see refs. [START_REF] Yamaguchi | A cooperative hunting behavior by multiple nonholonomic mobile robots[END_REF][START_REF] Gong | A ga based combinatorial auction algorithm for multi-robot cooperative hunting[END_REF][START_REF] Wang | A rapid hunting algorithm for multi mobile robots system[END_REF].

Although the hunting task can be performed in argued to be completed in all above cases, all of them rely on fixed encirclement points around the target and keeping formation, making the system not so flexible to a dynamical and real-world application. Some improvements to these strategies have been made in refs. [START_REF] Wu | A multi-robot cooperative hunting approach based on dynamic prediction of target motion[END_REF][START_REF] Duan | Multi-robot dynamic virtual potential point hunting strategy based on fis[END_REF], although these propositions do not rely on keeping formation, they require fixed encirclement points around the target. These points are virtual and global information, which requires, for instance, the centralization of information. Nevertheless, the complexity of these algorithms, which require optimization for path planning, is not scalable (in both cases, only four agents were used) and would be unfeasible to implement in a bigger group.

Bio-inspired group chase

Another relevant field to group pursuit is artificial swarming and flocking. This approach considers each agent as an autonomous entity, following similar rules based on local interaction, resulting in emergent collective behaviors. This methodology is based on theoretical work as in refs. [START_REF] Reynolds | Flocks, herds and schools: A distributed behavioral model[END_REF][START_REF] Vicsek | Collective motion[END_REF], where the authors proposed complex collectives behaviors, for flocking and herding, starting from simple local rules.

Several works consider the agent-based model to represent multi-agent pursuit; in ref. [START_REF] Angelani | Collective predation and escape strategies[END_REF], the author designed an algorithm based on the Vicsek particle model. In ref. [START_REF] Lin | Abaid Collective behavior and predation success in a predator-prey model inspired by hunting bats[END_REF] the authors modeled a predation system using agents inspired by swarms of bats. In ref. [START_REF] Saito | Group chase and escape model with chasers' interaction[END_REF], the authors studied the emergent behavior of agents with an interaction between pursuers in a discrete environment (two-dimensional square lattice). The simplicity of the model allowed simulations with large numbers of players (e.g. 10e 4 ). In ref. [START_REF] Janosov | Group chasing tactics: how to catch a faster prey[END_REF], the hunting behavior of agents is studied in a more realistic environment using a framework to simulate a fleet of UAVs (see ref. [START_REF] Vásárhelyi | Outdoor flocking and formation flight with autonomous aerial robots[END_REF]). In this methodology, the interaction of each agent is represented by physical forces in bi-and tri-dimensional space with boundary conditions. The action of each agent is calculated by the sum of different interaction factors: chase, inter-agent repulsion, alignment and, short-range collision. The 'chase' factor assumes access to information about the position and velocity vector of the evader, this allows the agent to predict of the future position of the target. Although the work showed a powerful algorithm to analyze realistic pursuit with robots, they have strong assumptions, and neglected the limited perception of the robot (often using a camera) and the coupled dynamics between perception and motion (when using drones). Furthermore, results are only validated in simulations.

In ref. [START_REF] Weitzenfeld | A biologically-inspired wolf pack multiple robot hunting model[END_REF], the author proposed a model inspired by wolf pack hunting to design a multi-robot application. This involves a hierarchical system containing at least one alpha and one beta agent and a finite state machine that determines the robot's motion. Nevertheless, the system relies on a fixed formation and fixed roles for the agents reducing the flexibility in a more reactive application. In ref. [START_REF] Muro | Wolf-pack (canis lupus) hunting strategies emerge from simple rules in computational simulations[END_REF], they are again inspired by strategies observed in a wolf pack. They observed interaction between agents in a wolf pack which allowed them to encircle the prey and start to attack it. However, the algorithm's simplicity does not fully capture the complexity of a coordinated pursuit of a faster prey, which requires more sophisticated techniques to track and ambush the target.

Authors' contribution

Our approach is a cycling hierarchical agent-based methodology to implement behavior for a group of chasing robots. The bio-inspiration lies in allocating dynamically changing roles, and the pursuit itself. The pursuit behavior is given by the Deviated Pure Pursuit (DPP) technique. Furthermore, our approach introduces an alternative to conventional approaches based on attraction-repulsion forces to reduce the recurrent problem of oscillations. In addition, the technique is flexible to the number of pursuers and robust to abrupt changes in the formation. Moreover, the approach has no explicit communication between agents and is closer to individual pursuer agents Our approach avoids two common problems in multi-robot cooperation: oscillations and the observation of neighbors. The first is produced by the spring effect caused by the repulsive/attractive forces. This problem is accentuated in real-world applications where delays, loss of data, or long sample times are presented. To account for this, the behavior of our system is not controlled by repulsive/attractive forces. Secondly, we model the observation of neighbors just using bearing-only information, which can be easily obtained from a single camera. This provides a more realistic observation model simplifying transfer to real-world applications.

Finally, we provide a numerical validation with different pursuers and experimental results for three different scenarios.

The paper's outline is presented as follows. Section 2 introduces the methodology and the dynamics of the agents. In Section 3, the structure of the pursuit control strategy is described. Numerical validations are done describing the performance of the proposed algorithms are shown in Section 4. Flight tests are carried out to validate the pursuit strategy. Section 5 illustrates the behavior when a drone fleet is pursuing an intruder drone, before concluding in Section 6.

Setup and Problem Statement

The objective of pursuit is that at least one pursuer intercepts the target. We adopt the terminology target to describe the target of the pursuit. This is not a fixed target, but has different behaviors described in Section 3. This problem can be stated as follows.

Our multi-agents pursuit-evasion problem involves N pursuers and a single target moving in a horizontal plane. The pursuit takes place inside a bounded work-space W in IR 2 . We consider different bounded workspaces in this paper (square and a R#2Q8 circle). The final goal for the pursuer team is r iT < R cap , i.e. that at least one of the pursuers has a relative distance (r iT ) towards the target less than a threshold R cap . The target trajectories are described by P T (t) = [x T (t), y T (t)], where (x T , y T ) is its Cartesian coordinates in W . Similarly, the pursuer paths are described as

P i (t) = [x i (t), y i (t)], where i ∈ 1, 2, ...N .
We aim to validate the approach experimentally in real time, hence, for the safety of our prototypes, the capture radius in final capture of the intruder (target) drone will not be included in this work, i.e., the interception of the target will occur just in simulations, in the experimental results, the goal will be to corral the target.

Agent

The term agent, in this work, is equivalent to the self-propelled particles proposed by Vicsek, [START_REF] Janosov | Group chasing tactics: how to catch a faster prey[END_REF], in the sense that each individual is autonomous, has a limited range of actuation, and they are driven by simplistic rules. We extend this definition to include the definition of an agent by Muro in ref. [START_REF] Muro | Wolf-pack (canis lupus) hunting strategies emerge from simple rules in computational simulations[END_REF]. Therefore, we consider that the agents are homogeneous (neither physically differentiation or a static hierarchy), do not use explicit communication and no explicit cooperation. All agents have a common goal, and they can differentiate between neighbors and target, however, not between individuals.

The aerial drones used in this work are quadcopter vehicles as shown in Figure 1-left. From the Newton-Euler approach, the mathematical equations for this vehicle can be written as: The bold letters represent vectors. F denotes the thrusts generated by the motors, F g is the gravity force. η represents the vector of Euler angles, Ω represents the angular velocity in the body frame. m indicates the mass of the drone, and v = [u v w] T defines the velocity vector of the aerial robot in the body system, R describes the rotation matrix generated in the order yaw-pitch-roll. B(η) represents the matrix that relates the angular velocity and the derivative of the Euler angles. J is the inertia matrix of the drone.

m v = F + R T F g η = B(η)Ω J Ω = τ -[Ω] × JΩ (1)
[Ω] × represents the skew-symmetric matrix of angular velocity and τ defines the torques applied to the vehicle.

As previously mentioned, we focus on pursuing an intruder drone, assuming it is flying at a constant altitude. We use a nonlinear controller to stabilize the orientation in the axes x, y and the altitude of the aerial vehicles. The desired altitude will be given by the z position of the intruder drone. The remaining states for the agents are ψ, x, and y as illustrated in Figure 2. The goal of this work is not to propose a control regulation for the desired trajectory; but the collective tracking of a mobile target with unknown movements in the x -y plane. From Figure 2, the kinematics of the vehicle are as follows ẋi =v i cos ψ i , (2a)

ẏi =v i sin ψ i (2b)
where x i , y i defines the position of the agent in the plane x, y, v i denotes the magnitude of the vector velocity v i = [v xi , v yi ] T in the body frame, and ψ i represents the yaw angle of the agent i to the horizontal axis x. The linear and angular velocities define the control inputs such that

v i =f i , ( 3a
) ψi =f ψi ( 3b
)
where f i = [f ui , f vi ] signifies the control inputs for the longitudinal and lateral velocities, respectively and f ψi introduces the control input for the yaw rate. This motion constraint has an interesting property while considering the frontal camera as a navigation sensor: the drone will displace only in the direction of the field of vision, tightly coupling the perception to the motion.

For the same reason, this kind of drone motion is commonly observed in first-personview piloting, where the only information for the pilot is the frontal camera image. In this case, the remote-pilot cancels the drone's lateral drift by combining a roll movement while turning the drones heading (yaw).

Target

The ultimate advantage of using multi-agents in pursuit is the possibility of cooperative ambushing for intercepting a more agile target. The term "agile" can mean having superior velocity or having more degrees of freedom. We assume both of these hypotheses in our work.

We consider the target will have a velocity up to twice as large as the pursuers. Furthermore, the intruder will not be constrained to non-holonomic equations (2) but by a simple particle model, which gives locomotion advantages. This configuration is a well-studied problem in differential games, commonly called "the homicidal chauffeur", see ref. [START_REF] Rufus | Differential Games[END_REF].

We consider two strategies regarding the target behavior: (1) predefined trajectories and (2) a reactive target. First, the trajectories can be defined off-line; for example, by collecting real-flight data, or using fixed paths, e.g, circular or square trajectories. A repulsion model defines the reactive behavior of the target. Each pursuer and the arena border exert a repulsive force towards the target. Details about this repulsive force implementation is given in the subsection 3.5.

Relative Engagement

In this work, the pursuer behavior is given by a modified version of a classical guidance law, DPP. The relative kinematic engagement is a useful way to analyze convergence conditions in a pursuit-evasion problem. The kinematics equation can be deduced from the geometrical engagement of the pursuer-target, as represented in Figure 2.

From Figure 2, the Line-of-Sight (LOS) connecting the pursuer and target is denoted by r iT . The bearing angle, defined between the LOS and the inertial axis-x, is denoted by λ iT . The vector v i is the velocity in the body frame, and ψ i represents the heading (yaw) of the pursuer i.

As shown in ref. [START_REF] Shneydor | Missile guidance and pursuit: kinematics, dynamics and control[END_REF][START_REF] Belkhouche | Parallel navigation for reaching a moving goal by a mobile robot[END_REF], from the engagement a relative kinematic model can be obtained. Considering the relative velocity between, ṙiT = ṖT -Ṗi . The relative velocity can be decomposed along (v ) and perpendicular (v ⊥ ) to the LOS :

v = ṙiT = v T cos(α T ) -v i cos(α i ), v ⊥ = r iT λiT = v T sin(α T ) -v i sin(α i ). ( 4
)
where α T = ψ i -λ iT and α i = ψ T -λ iT . This equation represents the velocities of target seen by the pursuer along the the LOS.

Pursuit strategy

In the following, we introduce our group pursuit strategy. Firstly, we will describe the Deviated Pure Pursuit (DPP) guidance law, the base method for our proposition. Then, we discuss the hierarchy and the roles in our pursuit methodology, relating them with the different offsets in a DPP. Finally, our group pursuit will be explored, and details of the implementation, such as collision avoidance, will be given.

Deviated Pure Pursuit -DPP

As stated in ref. [START_REF] Shneydor | Missile guidance and pursuit: kinematics, dynamics and control[END_REF], the DPP can be seen as a variation of the classical Pure Pursuit (PP); it can be a result of manufacturing flaws or can be intentionally designed in order to point the agent towards a position ahead of the target, causing a "lead pursuit." However, if the offset drives the agent towards a position behind the target, it causes a "lag pursuit."

The basic principle relies in driving the heading error to a constant offset angle (φ iλ iT ) → α 0 . Considering this principle, the relative kinematics become: R#2Q3 ṙiT = v T cos(α T ) -v i cos(α 0 ),

r iT λiT = v T sin(α T ) -v i sin(α 0 ). (5)
The above equations establishes one necessary condition about the offset angle: |α 0 | < π/2. Nevertheless, under this strategy the capture cannot be guaranteed for all v T > v i . R#2Q2 R#2Q4 Our work explores the idea that each agent can assume different offset angles, therefore, each agent will have different degrees of "lead", and "lag" pursuit. This will cause spreading of the pursuers around the target, as shown in Figure 3. In ref. [START_REF] Shneydor | Missile guidance and pursuit: kinematics, dynamics and control[END_REF], an intuitive law to apply the DPP is given as R#2Q3

f ψi = -K dpp (ψ i -λ iT -α 0 ) ( 6
)
where K dpp defines a positive constant gain.

Hierarchy

Our system has a mixed hierarchical architecture. The homogeneity of the agents means that all agents are physical equals, and no identification is required. We define a hierarchy based on the advantage of information, i.e., one agent with better information will lead the group, this makes a kind of cycling hierarchy. On the other hand, they play a role in differentiation, where each agent can assume different behavior depending only on the current relative position of the pursuer to its neighbors and target. As shown by Stander, see ref. [START_REF] Stander | Cooperative hunting in lions: the role of the individual[END_REF], in natural systems, the central position is commonly related to dominance. In our system, the central agents have priority in moving and capturing since they have a more direct trajectory towards the target. We differentiate roles based purely on agent positioning and not due to social or biological factors (compared to natural systems). Here, each agent's trajectory is determined by computing the DPP angle assumed in the pursuit.

In Figure 4, each agent has a different angle offset, hence, each agent has a different role. The agent i (blue) and agent k (green) have consecutively positive and negative offset, therefore, they do not go in a straight line to target, rather, they ambush it from left and right, respectively. In ref. [START_REF] Stander | Cooperative hunting in lions: the role of the individual[END_REF], the author identified this pattern in a group of hunting lionesses and called the agents at the extremities as "wings," alluding to football players' roles.

The differentiation in roles has a significant effect on the performance of the pursuit of the target since the "wings"-agents do not follow a straight line to the target but enclose it depending on the offset angle. The roles are defined such that the condition for an agent to be considered the 'center' is based on the proximity to the intruder. This allows that an agent very close to the target can execute a terminal attack independently of the position of its neighbors. R#2Q4 Fig. 4. Three agents pursuit an intruder drone playing different roles.

.

Pursuit control algorithm

The objective of the control scheme is not to control the position of each agent as it is likely to produce oscillations, due to the random and fast movements of the intruder. The control scheme's goal is to follow a deviate pursuit towards the target, with a given angle offset which will avoid collisions with the neighbors.

The heading of agent i will be controlled by its angular rate related to the pursuit conditions as shown in Figure 4. Each agent has different offset angles (α), causing different trajectories for each agent, aiming to increase the possibility of capturing the intruder using an ambush strategy. Therefore, rewriting (6) for the whole system yields R#2Q5

f ψi = -K p   λ iT + α j =i N δ i,j (ξ i , ξ j )   , (7)
δ i,j = -1, if j is in the left side 1, if j is in right the side δ i,j can be obtained from visual information, for this work we computed it by

δ i,j = r ij × r iT r ij × r iT (8)
where

r ij = [x j -x i , y j -y i , 0] T and r iT = [x T -x i , y T -y i , 0]
T , correspond to the planar vector of the positions for neighbor j and to target T , respectively, in the body coordinates of the pursuer i.

From Figure 4 and equation (3a), we can deduce that the longitudinal control input f ui varies with respect to the magnitude of the desired vector velocity v i of the pursuer and also is related with the target and neighbors distances. R#2Q6 Thus, we propose

f ui = u max σ a d iT -R cap R cap [C rep , C wall ] min (9)
where u max is the maximal velocity imposed to the agents, d iT defines the distance between agent i and the target T and R cap denotes the radius to capture the intruder. 0 ≤ [C rep , C wall ] min ≤ 1 is a safety coefficient that regulates the linear velocity of the agent i when approaching to agent j (collision avoidance) or when the agents are close to the limits of the workspace. Similarly, the term σ a (...), with a = 1, is set to saturate the normalized distance error ( d iT -Rcap

Rcap

). R#2Q7 The saturation function, σ a (.), can be defined as follows.

σ a (x) =      a, if x > a x, if a ≥ x ≥ -a -a, if x < -a (10)

R#2Q9

Please note that controlling the velocity of the pursuers become relevant mainly when physical integrity of the materials are considered, such as in our experimental validations. However, in the further numerical simulation, where the collision is not a concern, the velocity of the pursuer is set constant.

Collision Avoidance

Most works in artificial flocking use the principle of attraction-repulsion to keep the cohesion of the group. However, those forces might cause oscillatory responses that could allow for an over-damped response of the fleet. As a consequence, this will reduce the velocity response of the agents.

We do not use attraction-repulsion principle between pursuers, but use an attraction factor towards the target, which acts as the point of convergence for all pursuers. For practical safety, a short-range repulsion was implemented, however, this is not responsible for the hunting formation.

There are two terms representing the collision avoidance (C rep and φ rep i ). The first one acts as a brake coefficient, limiting the velocity input (f ui ) to zero. The second imposes a boundary in the lateral velocity, and it will be explained in Section 3.6.

For the longitudinal part this term can be denoted by

C rep = C repi,j , C rep i,k , ... min (11)
C rep defines a term acting in a small rage and only aims to avoid agents collision. When an agent i detects agent j inside its repulsion area (delimited by the field of vision and a radius r rep ) it decreases its velocity. The coefficient of repulsion takes the minimum value of the individual repulsion from agent i to all neighbors inside the repulsion area, and it is represented as

C repi,j = 1, if d ij < R col + a 0 dij -a0 R col , otherwise (12)
where R col means the radius of collision (see Figure 4) while a 0 a constant offset. When the agents are moving, the distance d ij between agents i and j could be smaller than the radius of collision, then the term a 0 will impose a distance where the velocity of the agents will be completely null. This term can be defined as the shortest distance possible between two agents.

Virtual wall

Consider the scenario when the drones' fleet is approaching the workspace boundary (e.g, surveillance area). We implement a virtual repulsion force when an agent is getting closer to the border of its workspace or even a forbidden area. Along this paper, we consider two shapes for the workspace, square and circular arena.

R#2Q8

For designing the virtual forces, we consider a projection of the agent in the border of the workspace, see Figure 5. Since we have one virtual agent for each agent, the break coefficient C wall is computed in the same way of the agent's collision avoidance.

Lateral control

We design a controller to eliminate the lateral drift caused by the inertia of the drone. The drift occurs when an agent changes in its heading while moves forward. In addition, this controller also implements a lateral short-range force repulsion between two drones or between the drone and a virtual wall.

This controller is proposed as:

f vi = 0, if d ij < R col δ ij R col -dij R col , otherwise (13)
where

φ rep i = R col -dij R col .
The lateral velocity increases linearly when the neighbors invade the collision radius (d ij < R col ). The function δ ij was defined in (equation ( 8)), and gives the direction of the velocity in the opposite side of the collision.

Simulations

In order to analyze our proposition, we implemented the framework in MATLAB. The agents were modeled as a particle with constant velocity and moving in a 2D plane. We use a bounded circular workspace for these simulations. The sample period (T e) in simulations is 0.01s. For simulation purposes, the crosses (red) represent the path of the target, while the path of the pursuers is illustrated by the diamond (blue), the asterisk (black), the circle (green), and the square (yellow). The black arrows denote the final heading (yaw) of each pursuer. The stop condition in the simulation is that the target is in a distance less than the capture radius R cap = 0.3m of at least one agent. Three scenarios are considered: r Fixed target and three pursuers, to illustrate the formation pattern in a simple case; r Escape trajectories, varying number of agents, to show the flexibility of the strategy; r Swarm of pursuers, to show our approach of these properties to deal with scalability issues, involving large amount of agents.

Finally, we analyse the effect of the angle offset in the capture rate and on the effect on collisions between pursuer.

Fixed target

The first simulation is where the target remains in a fixed position. Figure 6 shows that the patterns of the pursuers drones is similar to the trajectories executed by a group of hunter animals. Fig. 6. Trajectories generated for three agents and a fixed target. The pursuers start in the bottom of the square and they pursuit the target with constant velocity. The deviation offset (α 0 ) for this pursuit is 30 degrees.

Escape trajectories

Due to this strategy's flexibility and decentralized nature, it can be easily transferred between environments with a different amount of agents. Several simulations were carried out with different numbers of pursuers (1, 2, 3 and 4), see Figure 7. For these simulations, the intruder velocity was set to 10m/s and for the pursuers 5m/s (half of the target speed). For visualization's sake, the maximum time for simulation was limited to 4s. In addition, in this scenario, we propose reactive escape behavior for the target. The behavior is described in Section 2.2, all the pursuers and the borders exert a repulsive force towards the target. Fig. 7. Snapshots for pursuits with different numbers of agents (1, 2, 3 and 4). The target in all cases does an escape trajectory based in repulsion forces, with velocity of 2 times bigger than the pursuers, i.e., v T = 2v i . The end of the target trajectories is signalized by the red circles.

From Figure 7, in the two firsts cases (A and B), the target was not caught until the end of simulations. However, in the two lasts cases (C and D), the target was successfully intercepted before the limit of time, this shows the importance of the increasing number of pursuers in the success of the chase. In addition, in Figure 7 the velocity of the agents is implicitly indicated in the diagram. The positions of agents are shown at a constant time stamp, therefore, the sparser a trajectory of an agent, the faster it is. In Figure 7-A, for example, we can see that the target (red cross) has superior to the single pursuer (blue diamond).

Swarm

Although collective hunting is usually employed to small groups of pursuers; our algorithm can easily be applied to larger groups of agents. In the following simulation (Figure 8), we illustrate the pursuit behavior applied to a group of 15 pursuers. The agents demonstrate an emergent flocking pattern, where each agent computes independent trajectories following the target with no information of the neighbors distance.

Effect of the offset angle on the capture

This quantitative evaluation aims to assess the effect of the angle offset (α 0 ) on the total number of successful captures. To this end, we subject our strategy to a group pursuit benchmark, previously proposed in ref. [START_REF] Souza | Decentralized Multi-Agent Pursuit Using Deep Reinforcement Learning[END_REF]. This benchmark consists of a circular bounded arena. We tested different angle offset values, incrementally from π/32, measuring the number of captures, as indicated in Figure 9.

For each benchmark evaluation, 700 pursuit episodes are completed, with a variable target relative velocity, v T /v i = 0.8, 1.0, ..., 2. More details about the benchmark implementation is given in ref. [START_REF] Souza | Decentralized Multi-Agent Pursuit Using Deep Reinforcement Learning[END_REF].

Figure 9-left shows the number of captures as function of the amount of pursuers. Figure 9-right, represents the average time taken to capture.

The first value (α 0 = 0) is equivalent to the classic Pure Pursuit, where all agents point to the target's current position. However, applying this configuration, the cooperative behavior does not emerge, therefore, the total catches remain practically the same during all the numbers of agents.

Therefore, with the smallest increment of offset (α 0 = π/32), the size of the group has an affect on the total amount of captures. The number of captures tends to increase, culminating in a cumulative total of 4233 (76%) catches for an offset of π/8. For an angles offset of α 0 = 3π/32, our approached completed all captures during the benchmark for 5 pursuers onwards, therefore, this is as an effective strategy for chasing a faster target.

For angle offsets larger than π/8 the capture's performance tends to decay. This is due to the fact that large offset values result in very open trajectories towards the target, allowing the target to escape between two pursues. Fig. 9. Effects of the offset value in the pursuit.

Effect of the offset angle on the collision

We use the same benchmark, to analyse the amount of collisions between two pursuers, i.e., r ij < R col , where the collision radius R col is equal to 0.15m. The graph on Figure 10 clearly illustrates the effect of the offset on collision avoidance. For α 0 = 0, collisions tend to grow exponentially, reaching values up to 395 times higher than using α 0 = 3π/32. Fig. 10. Effect of the offset value (α 0 ) on the collision between pursuers.

Experimental Validation

The proposed pursuit algorithm was also validated in a real-world experiments. We use quadcopter Parrots -AR Drone 2 for aerial robots. We used Framework libre Air1 an open source framework which provides a low-level connection to the drones. Opti-Track was used for motion capture to locate each agent. For experimental validation and security reasons, the final capture does not occur here; instead, the pursuers must stop at a security distance of d = 1m. The drones operate in a 6m × 6m square workspace.

Drone control

In our application, the control of the quadcopters is divided into two levels, as illustrated in Figure 11. The lowest level is the control for stabilizing the drone in orientation and altitude. The higher level is responsible for controlling the pursuers (f ψi , f ui , f vi ) in desired angles (θ i d , φ i d and ψ i d ). The pursuit algorithm (the green box) is responsible for the "flocking intelligence" for each individual. Each agent can perceive the environment (neighbors, obstacles and target) and gives to the the lower levels the velocities references (f ψi , f ui , f vi ). Therefore, from (1) and using the PD controller in the altitude and the orientation2 , it follows that, w i → 0, z Bi → z Bi d , ηi → 0 and η i → η i d , where z Bi d defines the desired altitude of the aerial vehicle i and η i d represents its desired orientation. Choosing η i d = 0 the aerial vehicle is stabilized at hover. The goal is the capture of the target drone, thus from (1) and without loss of generality for x and y (in the body frame) it follows that ui ≈ -g sin θ i vi ≈ g cos θ i sin φ i

The prototype has an embedded controller for the attitude, thus η is small enough such that cos Ξ i ≈ 1 and sin Ξ i ≈ Ξ i , being Ξ i a generic angle representing θ i and φ i .

u i = ẋi B v i = ẏi B ui ≈ -gθ i vi ≈ gφ i
where x i B and y i B are the position coordinates in the body frame. From (3a) and ( 13), it follows that the following stabilize the lateral dynamics vi = -k vi v i -f vi [START_REF] Shneydor | Missile guidance and pursuit: kinematics, dynamics and control[END_REF] where k vi denotes a positive gain. Notice from (13) that f vi defines a position lateral feedback for avoiding the obstacles. Thus, it is easy to deduce the desired roll angle as

φ i d = -k ẏi v -k yi f vi (15)
where k ẏi = kv i g and k yi = 1 g . Similarly, from (3a) and ( 9), it follows that ui = -k ui (u i -f ui ) [START_REF] Belkhouche | A method for robot navigation toward a moving goal with unknown maneuvers[END_REF] with k ui denotes a positive gain. Notice here that f ui defines the desired velocity of the agent i that will vary with respect the position of the target T . Therefore,

θ di = k ẋi e ui (17)
where k ẋi = ku i

g and e ui = u i -f ui . From (3b) and ( 7), it follows that ψi = -K p e ψi [START_REF] Teimoori | A biologically inspired method for robot navigation in a cluttered environment[END_REF] where e ψi = ψ i -ψ i d .

All control laws and pursuit algorithms are computed on-board each agent, the ground station only analyzes the states and re-takes the control in manual mode in emergency cases. For improving experimental validation, a Kalman Filter was implemented to estimate states in cases where the agent does not receive the data, see ref. [START_REF] Souza | Enhanced uav pose estimation using a kf: experimental validation[END_REF].

Lateral control takes great importance in real-world experiments, as lateral drifts appear when the agents avoid obstacles which does not happen in simulation.

Experimental Evaluation

Two scenarios are proposed to validate the pursuit algorithms.

1. The intruder drone remains in a fixed position, and three pursuers drones track it. 2. The target, moving with random movements given by a user, is tracked for three pursuers aerial drones.

Three pursuers and one static target

The target is placed in the ground at a distance of 6m from the pursuers. The pursuers are placed in an initial triangular formation with a distance of approximately 2m from their closest neighbors. This experiment aims to reproduce the pattern presented in simulation (see Figure 6). The parameters used in this experiment are presented in Table I Figure 12 shows the real-world result for this scenario. The agents take on a similar strategy compared to the hunting pattern described in Figures 1 and6. However, the irregularities observed in the pursuer's trajectories are caused by the airflow produced by the proximity of the drones and the absence of position control.

Three pursuers and one faster intruder

An operator manually controls the target, trying to get away from the three pursuers' drones. The goal for the pursuers is to track the target, limiting their distances to the target to ensure no collisions. In addition, to further ensure the safety of our drones, the target flies at a different altitude. We choose z d = 0.8m for the intruder and z di = 1.7m to the pursuers. The parameters for this flight are also shown in Table I. In Figure 13 some snapshots are shown which illustrate the real-world performance of the drones pursuing the faster target. The figure is composed of 9 snapshots taken at an interval of 2.75s. For each snapshot, the position of each agent was plotted ten times, with a sample time of 0.275s. The larger velocity of the target (2m/s) is highlighted by the distance traveled by it in each snapshot.

The collision avoidance property was implemented in the pursuers and can be seen in Figure 14, where the inter-distances are plotted as a function of time. The minimum distance between two pursuers throughout the episode was approximately 1m. However, since the target was piloted manually, there were no guarantees of a collision-free trajectory Observe from these figures that the cooperative hunting pattern emerged during these real-world experiments. Even facing a complex pursuit with a faster reactive target, the pursuer's drones could track and intercept the futures movements of the target. No formation is intended to be kept, instead, the drones re-arrange between themselves in other to corral the intruder. The roles of positioning (center, left-, and right-wings) are constantly changing as a function of the relative position of the fleet to the target.

Moreover, taking into account the size of the virtual arena (6 × 6m) and the presence of 4 drones flying inside, this approach has demonstrated a robust, practical performance with respects to the disturbances produced by the airflow of the aerial vehicles.

The flight tests can be seen on: https://youtu.be/g2dODbd6ZLA.

Conclusion

A decentralized strategy for cooperative group pursuit against a single intruder has been proposed and validated in this work. Our approach, inspired by lionesses' hunting behavior, applies the geometrical rules of Deviated Pure Pursuit to generate different trajectories for each agent. This strategy avoids using attractive and repulsive forces for pattern formation, which occasionally results in the system's oscillations. Numerical simulations were carried out to demonstrate the efficiency of the proposed chase strategy. The effect of the angle offset was analyzed quantitatively through a pursuit benchmark. We observed the emergence of the behavior of collective pursuit through two main metrics, the increase in the number of target's capture and the decrease in the amount of collision between pursuers. Furthermore, a proof-of-concept with real drones is shown, were we could verify the pursuit behavior in a real-world experiment with three pursuers and one target.

This preliminary work focused mainly on the formalization of our algorithm and the qualitative description of the emerging behavior. However, our proposal also proved to be effective in capturing a faster target, as shown by the group pursuit benchmark described in section 4. Furthermore, our proposal requires very minimal assumptions; the pursuit algorithm, disregarding the safety layer of collision avoidance, requires only relative bearing angle information in relation to the target and neighbors. This minimalism of implementation puts our algorithm in a very advantageous position compared to the previous ones, such as in ref [START_REF] Muro | Wolf-pack (canis lupus) hunting strategies emerge from simple rules in computational simulations[END_REF][START_REF] Angelani | Collective predation and escape strategies[END_REF][START_REF] Lin | Abaid Collective behavior and predation success in a predator-prey model inspired by hunting bats[END_REF], which requires more information from the target and neighbors, such as position or speed in global coordinates.

We believe that a more in-depth theoretical analysis of the collision-free phenomenon observed in this work is also interesting and reaffirmed from simulations in the pursuit benchmark.

Although these first experimental results indicate the viability of this algorithm, several areas must be improved to obtain a fleet of autonomous chasing drones. The first one is perception. Pursuers must be able to extract the bearing angle information using only the embedded sensors, which are commonly the front camera and the Lidar. However, using a camera-type sensor with limited FOV arouses the interest of another field of investigation, the effect of limited visual sense on collective behavior. Recently, works, as in ref. [START_REF] Soria | The influence of limited visual sensing on the reynolds flocking algorithm[END_REF], have been dedicated to evaluating the impact of this non-linearity in the observation, which could improve our work results.

Fig. 1 .

 1 Fig. 1. The drone formation (left) mimics the behavior of lionesses hunting in group (center and right).

Fig. 2 .

 2 Fig. 2. Agent performance in the plane x, y. Its perception is limited for a vision field. The longitudinal velocity vx and lateral vy are defined in the body frame.

Fig. 3 .

 3 Fig. 3. Paths of deviated pursuit agents according to different angle offsets α. The agents begin the trajectory at a fixed position, given by the black dot and keep a constant velocity until reaching the target.

Fig. 5 .

 5 Fig. 5. Virtual wall projected when the drones approaching the border of the square workspace. Virtual agents are projected into the virtual cage

Fig. 8 .

 8 Fig. 8. Case of 15 pursuers and an intruder. The pursuers intercept the target early in their trajectory.

Fig. 11 .

 11 Fig. 11. Schema for the control levels for a given agent i.

Fig. 12 .

 12 Fig. 12. Real-time performance when three aerial drones pursuit a fixed target. The first image (left) shows the displacement of the drones in the flight arena. The second one is a picture of the experiment.

Fig. 13 .

 13 Fig. 13. Real-time performance when three aerial drones pursuit an aerial intruder drone with random movements.

Fig. 14 .

 14 Fig. 14. Inter-distance between pursuers during the experimental test.

Table I .

 I . Control parameters used in the real-time scenarios..

	Parameters	Scenarios 1 and 3	Scenario 2
	α	20 o	10 o
	u max	1.3	1.3
	R cap	1.6	1.6
	F ield -vision	360 o	120 o
	R int	5	2
	R col	1.5	0
	a 0	1	0

https://devel.hds.utc.fr/software/flair

Others controllers can be also used for this purpose.

Acknowledgement

R#2Q1 We thank Rhys Newbury (Monash University -Melbourne, Australia) for the valuable effort on the full text revision.