Dario Sanalitro
email: dario.sanalitro@laas.fr

Marco Tognon
email: mtognon@ethz.ch.

Jimenez Cano

Juan Cortés

Antonio Franchi
email: antonio.franchi@laas.fr

A E Jimenez Cano

Indirect Force Control of a Cable-suspended Aerial Multi-Robot Manipulator

Keywords: Aerial Systems: Mechanics and Control, Multi-Robot Systems, Compliance and Impedance Control

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I. INTRODUCTION

UAVs have shown their interest to perform various tasks, mostly for contact-less applications such as monitoring and mapping, but recently, also for physical interactions with the environment or with humans [START_REF] Ollero | Past, present, and future of aerial robotic manipulators[END_REF]. Indeed, possible applications requiring physical contact are numerous: structure assembly, contact-based inspection, transportation, harvesting, etc. The most frequent approach to enable a UAV with manipulation capabilities is based on the installation of dedicated equipment such as grippers or robotic arms [START_REF] Ollero | Past, present, and future of aerial robotic manipulators[END_REF]. Alternatively, the manipulation/transportation of objects can be performed using cables [START_REF] Palunko | Agile load transportation: Safe and efficient load manipulation with aerial robots[END_REF] or tethers [START_REF] Tognon | Theory and Applications for Control of Aerial Robots in Physical Interaction Through Tethers[END_REF]. Such a setting has the advantage of simplicity. Nevertheless, stabilizing the oscillation of suspended objects can be difficult, and may imply a significant loss of accuracy. In general, the use of two or more UAVs cooperatively allows better control of the load and larger payload overall. Note however that, systems composed of multiple vehicles rigidly attached to the load [START_REF] Tan | Cooperative control of multiple unmanned aerial systems for heavy duty carrying[END_REF], as well as systems with multiple robots attached through an insufficient number of cables [START_REF] Sreenath | Dynamics, control and planning for cooperative manipulation of payloads suspended by cables from multiple quadrotor robots[END_REF] cannot control the full pose (position and orientation) of the load. Conversely, it has been demonstrated that a system with three robots, connected to three noncollinear points by means of six cables can achieve such a task if one employs statically-rigid designs [START_REF] Prattichizzo | Grasping[END_REF]. One of these systems, called the Fly-Crane, was first presented in [START_REF] Manubens | Motion planning for 6-D manipulation with aerial towed-cable systems[END_REF]. Subsequently, a robust controller and a method for the inertial parameters estimation of such a system were introduced in [START_REF] Sanalitro | Fullpose manipulation control of a cable-suspended load with multiple UAVs under uncertainties[END_REF] and [START_REF] Petitti | Inertial estimation and energy-efficient control of a cable-suspended load with a team of UAVs[END_REF], respectively.

To the best of our knowledge, none of the previous work on Cable-Based Aerial Multi-Robot Manipulators (CS-AMRMs) has provided a controller that can be safely used when the load is in contact with the environment. With this work, we fill such a gap by presenting a method for the interaction control of the Fly-Crane system with the surrounding environment and possibly also with humans. In the literature, three classes of interactions can be identified:

1) Unexpected-collision interactions: the transportation or the manipulation of objects in adverse circumstances may arise in unexpected collisions that have to be carefully managed. 2) Expected-collision interactions: situations where it is known in advance that contacts will take place. As a consequence, such interactions have to be carefully regulated.

Examples are: pick-and-place operations, inspection by contact, or peg-in-hole tasks. 3) Active interactions: the transportation or the manipulation can be reinforced through direct collaboration with an active external agent such as another robot or a human. In the aerial robotics domain, one of the critical issues for robot interactions is the measurement/estimation of the interaction wrench 4 . A reliable solution proposed for single-UAV manipulators [START_REF] Nava | Direct force feedback control and online multi-task optimization for aerial manipulators[END_REF] is to make use of force/torque sensors to measure the interaction forces. However, the costs and weight of such devices cannot be considered negligible for aerial platforms. Moreover, a force/torque sensor can only measure the interaction punctually, where it is mounted, while collisions can occur at any point of the platform. Therefore, alternative solutions based on wrench estimation have to be investigated. In [START_REF] Meng | Contact force control of an aerial manipulator in pressing an emergency switch process[END_REF], the authors showed a contact force control for an aerial manipulator where the UAV acts as the sensor of the exerted force on the environment. In [START_REF] Ryll | 6D interaction control with aerial robots: The flying end-effector paradigm[END_REF], an IMU-enhanced momentum-based observer was presented. Conversely, in the context of multi-robot systems where the cooperative manipulation has been largely investigated [START_REF] Ollero | Past, present, and future of aerial robotic manipulators[END_REF], wrench estimators have mainly been employed in evaluating the role of the internal forces in co-manipulation tasks, as in [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF] while, to the best of our knowledge, wrench estimators have never been employed in the identification of possible external forces/torques acting on a CS-AMRM.

After the identification phase, the aerial multi-robot system has to properly react according to the forces resulting from an interaction event. Only a few works have considered this aspect. In [START_REF] Nguyen | A novel robotic platform for aerial manipulation using quadrotors as rotating thrust generators[END_REF], the authors firstly investigated the problem of cooperatively controlling the six DoFs (Degrees-of-freedom) of a flexible platform rigidly attached to three quadrotors by means of three movable bars. In addition, they proved its efficacy in rejecting external disturbances, also showing an example of object telemanipulation. Nevertheless, the authors' main focus was the analysis of the effectiveness of the compliant properties of the system. Therefore, intensity and directional information of the external disturbances, as well as a deeper characterization of the compliant behavior were not entirely described.

In this paper, we show that a well-thought-out selection of methods together with a careful implementation, allows us to solve the aforementioned problems for the control of a CS-AMRM, the Fly-Crane, while physically interacting with the environment. Although the employed algorithms are well known in the robotics community, they were mainly conceived for ground manipulators. They have rarely been implemented and evaluated on aerial multi-robot manipulators, and, to the best of our knowledge, never with CS-AMRMs. More precisely, for the first time: i) we investigate the reliability of an external wrench estimation method for the control of a CS-AMRM; ii) we integrate a 6D impedance shaping algorithm within the control architecture of such a complex system. Moreover, the performance of the proposed approach is evaluated through a set of experiments illustrating the system in one of the three different classes of interactions mentioned above: unexpected collisions. In fact, these are the most common type of interactions that can occur in the envisioned applications of the CS-AMRM under exam.

The paper is organized as follows: Section II introduces the Fly-Crane system and its modeling. The control architecture is detailed in Section III. The experimental results are described in Section IV, and Section V presents conclusions and future works.

II. SYSTEM MODELING

The Fly-Crane is an aerial manipulator that consists of three aerial vehicles attached to a platform by six cables, two per robot. With this configuration, the robots can exert a 6D wrench to the platform, allowing for 6D motions. The dynamic model has been presented in [START_REF] Petitti | Inertial estimation and energy-efficient control of a cable-suspended load with a team of UAVs[END_REF]. For the sake of completeness, here we provide a brief summary. We provide a schematic description of the Fly-Crane in Fig. 1. Let us define an inertial frame F W = {O W , x W , y W , z W }, a body frame F Ri = {O Ri , x Ri , y Ri , z Ri } rigidly attached to the center of i-th robot, and a body frame F P = {O P , x P , y P , z P } rigidly attached to the platform, where O and {x , y , z } are the origin and unit axes of F . F P is placed such that O P is in the geometric center of the platform. The vector W p P ∈ R3 describes the position of O P with respect to F W and W R P ∈ SO(3) describes the orientation of F P with respect to F W1 . Applying the Newton-Euler formalism, the dynamics of the platform can be written as m P pP = -m P gz W + f P + f e ṘP = S( P ω P )R P J P P ωP = -S( P ω P )J P P ω P + P τ P + P τ e ,

where m P ∈ R >0 is the mass of the platform, g ∈ R >0 is the free-fall acceleration, f P , P τ P ∈ R 3 , are the forces and moments acting on the platform as a consequence of the pulling action of the robots, f e and P τ e ∈ R 3 are the external forces and moments perturbing the platform, S( ) is the skew operator2 , P ω P ∈ R 3 is the platform angular velocity, and J P ∈ R 3×3 >0 the inertia of the platform. As shown in Fig. 1(a), each cable is attached to the platform at point B i and to an aerial vehicle at point O Ri where

O R1 = O R2 , O R3 = O R4 and O R5 = O R6 .
Their position with respect to F W is described by the vectors b i ∈ R 3 and p Ri ∈ R 3 , respectively. As commonly done, we consider each cable with a constant length, l i ∈ R >0 (no elastic deformations), and negligible mass and inertia. Furthermore, we assume no rotational constraints at the anchoring points.

We denote with f i ∈ R ≥0 the intensity of the internal force along the i-th cable. Notice that if f i > 0 the cable is taut, while it is slack if f i = 0. The aim is to move the system always preserving the tautness of each cable, such that f i > 0 for all i = 1, . . . , 6. The force on each cable can be defined as f Li = f i n i ∈ R 3 , where n i ∈ R 3 is the unit vector along ----→ B i O Ri which describes the orientation of each cable with respect to F W . All the forces can be then 18 and can be used in the definition of all the wrenches applied to the platform w P = [f P P τ P ] as follows

collected in f = [f L1 , f L2 , ..., f L6 ] ∈ R
w P = G(R P )f , (2)
where G is a matrix defined as

G = I 3 ... I 3 S( P b 1 )R P ... S( P b 6 )R P . (3)
The thrust generation units of the system are the aerial vehicles attached to the platform by means of the pair of cables (i, j). f Rij ∈ R 3 is the 3D controllable total thrust vector of each vehicle, after gravity compensation:

f Rij = R Ri   0 0 4 k=1 c f ω 2 ki   -m Ri g (4)
where ω ki is the spinning rate of each propeller of the i-th quadrotor, c f ∈ R >0 represents a propeller-dependent constant parameter, m Ri ∈ R >0 is the mass of each robot and finally g = [0 0 g] is the gravity vector. Assuming that the cables are always taut, it is possible to define a thrust intensity vector which lies on the plane formed by each pair of cables (see Fig. 1(b)), such as

f Rij = I 3 -n ij n ij f Rij = f i n i + f j n j , (5)
where I n ∈ R n×n is the identity matrix of dimension n, and n ij is the unit normal vector of the plane generated by n i and n j such that

n ij = (n i × n j ) / n i × n j . (6)
Finally f i and f j are the solutions of the following system

f i (n i • n i ) + f j (n i • n j ) = f Rij • n i f i (n j • n i ) + f j (n j • n j ) = f Rij • n j . (7)
In the end, the force along each cable pair turns out to be

f Li = f i n i f Lj = f j n j . (8)
The configuration of the pair of cables (i, j) is given by the angle α ij ∈ R between the plane where the cables (i, j) lay and the one composed of the axis {x P , y P }. Therefore, the platform configuration can be entirely described by

(p P , R P , α) ∈ C = SE(3) × R 3 , where α = [α 12 α 34 α 56 ] ∈ R 3 .
Let us denote with q = p P η P α ∈ R 9 the generalized coordinates of the system, where η P = [φ θ ψ] ∈ R 3 is an Euler angle parametrization of R P . By direct kinematics, the position of the aerial vehicles p Ri , can be computed:

p Ri (q) = p P + R P P p Ri (α ij ), (9)
where

P p Ri = P b i + l i RP bij (α ij )R z P (β i ) P bij P bij , P b ij is the vector ---→ B i B j expressed in F P and β i ∈ T is the angle between ---→ B i B j and ----→ B i O Ri .
Notice that β i is a constant of the system that depends on the geometry of the platform and on the length of the cables only. The rotation matrices RP bij (α ij ) and R z P (β i ) represent the rotation of α ij about axis P b ij and the rotation of β i about axis z P , respectively.

Let

v R = v R1 v R2 v R3
be the velocities of the aerial vehicles. The relation between the system configuration time derivative, q, and the robot velocities, v R , is defined as:

v R := ṗR = J (q) q, ( 10
)
where J ∈ R 9×9 is the Jacobian matrix which describes the mapping between the joint velocities (i.e. the velocities of the three UAVs) and the corresponding platform linear and angular velocity, q = [ ṗ P η P η P ] ∈ R 9 , and

p R = [p R1 p R2 p R3 ] .
The generalized momentum of the platform p ∈ R 6 is p = M P (q P ) qP ,

where q P = [p P η P ] and qP = [ ṗ P η P ] are the configuration of the platform and its derivative. From (1), the time evolution of p can be written as ṗ = τ + ṀP (q P ) qP -C P (q P , qP ) qPg P (q P ), [START_REF] Meng | Contact force control of an aerial manipulator in pressing an emergency switch process[END_REF] where τ = w P + w e ∈ R 6 represents the total wrench applied on the platform and it is the sum of the active force and torques, w P ∈ R 6 , and the external wrenches, w e = [f e P τ e ] ∈ R 6 ; C P (q P , qP ) ∈ R 6 is the centripetal and Coriolis vector, and g P (q P ) ∈ R 6 is the platform gravity vector.

The dynamics of the whole system, composed by the platform and the aerial robots, can be summarized as M (q)q +C(q, q) q +n(q) = J (q)

  f R12 f R34 f R56   + we , ( 13
)
where M ∈ R 9×9 is the generalized inertia matrix, C(q, q) ∈ R 9 is the centripetal and Coriolis vector, we = [w e 0 3 ] ∈ R 9 and n(q) ∈ R 9 is the gravitational effect such as

n(q) = m P   g 0 0   + m R J (q)   g g g   , (14)

III. CONTROL

Here we present the proposed control architecture, schematically depicted in Fig. 2. It is composed of three nested loops:

1) The outer control loop takes care of the interaction task by generating compliant system trajectories based on the external wrenches acting on the platform; 2) The intermediate control loop is a centralized controller that generates the velocity reference for each aerial vehicle, given the desired platform pose. At this level, the redundancy of the Fly-Crane with respect to the 6D positioning task is exploited to balance the robot efforts; 3) The inner control loop running on each aerial vehicle is a position controller computing the motors' commands given the desired robot trajectory.

A. Intermediate and Inner Control Loop

The intermediate loop generates the three aerial vehicles reference velocities v R ∈ R 9 to track the desired trajectory for the system configuration q d (t). For this we rely on a kinematic controller presented in [START_REF] Sanalitro | Fullpose manipulation control of a cable-suspended load with multiple UAVs under uncertainties[END_REF]. In particular, given the system desired configuration q d = [p d P η d P α d ] , the corresponding generalized velocities qd = [ ṗd P ηd P αd ] and the measured configuration q, the robot reference velocities can be computed as where

v R = J (q)(K q e q + qd ), (15)
K q = k q I 9 ∈ R 9×9
>0 is a positive definite matrix and e q = q dq. Considering the task of controlling the 6D platform pose, the desired angles of the cables, α d , are computed with a local optimizer similar to [START_REF] Petitti | Inertial estimation and energy-efficient control of a cable-suspended load with a team of UAVs[END_REF] which allows to resolve the redundancy of the system. The objective is to equally share the effort among the robots, considering the platform weight and the external wrenches acting on it. Moreover, the algorithm allows to maintain the tautness of the cables at a steady-state after a settling phase during the transient.

B. Outer Control Loop: Admittance Framework

An admittance framework, composed of a wrench observer and an admittance filter, allows to handle the physical interaction between the aerial system and the environment. The wrench observer allows to estimate and characterize interaction forces and torques. The admittance filter provides the reference to the intermediate control pool (motion control of the Fly-Crane) to be compliant against the interaction wrench.

1) Contact wrench estimation: Interactions, collisions or contacts knowledge is fundamental when a robot moves in cluttered environments. Particularly relevant are the direction and the intensity of the interaction forces. Force-torque sensors could be integrated into the architecture. However, their main limitation of providing only local information makes them not suitable with wide-ranging end-effectors. As a consequence, a monitoring method based on the generalized momentum [START_REF] Haddadin | Robot collisions: A survey on detection, isolation, and identification[END_REF] has been considered such that collision detection and collision identification can be always guaranteed at any platform point, providing sufficiently accurate estimations. Additionally, the generalized momentum method represents a less expensive solution and avoids the need of equipping the aerial manipulator with additional weight.

Following the approach introduced in [START_REF] De Luca | Sensorless robot collision detection and hybrid force/motion control[END_REF] for robotic manipulators, the momentum observer dynamics derived from ( 12) is ṗ = w P -C P (q P , qP ) qPg(q

P ) + ŵe ẇe = K O ( ṗ -ṗ) , (16)
where the property ṀP (q P ) = C P (q P , qP ) + C P (q P , qP ) has been used and

K O = diag{k O } > 0 ∈ R 6×6
is the diagonal observer gain matrix. For the sake of compactness, let us write β P = C P (q P , qP ) qP + g(q P ). The signal ŵe (t), which represents the estimated external disturbance, is obtained integrating [START_REF] De Luca | Sensorless robot collision detection and hybrid force/motion control[END_REF] as follow

ŵe = K O p(t) - t t0 ṗ(s)ds -p(0) (17) = K O p(t) - t t0 w P -β P + ŵe ds -p(0) , ( 18
)
where t and t 0 are the current and initial time instant. In ideal conditions, the residual vector ŵe can be seen as a virtual sensor for external wrenches acting on a rigid body. As a matter of fact, the relation between w e and ŵe is

ẇe = K O (w e -ŵe ), (19)
and represents a first-order low-pass filter with the property that ŵe → w e when t → ∞ and ŵe w e when K O ∞.

2) External wrench compliance: To handle the physical interactions and make the platform of the Fly-Crane compliant to interaction wrenches with the environment, we choose the admittance control framework over the impedance one. This choice grounds on the fact that the admittance controller shows better performance for trajectory tracking in free space and disturbance rejection [START_REF] Villani | Force control[END_REF].

The admittance framework generates desired motions for the system based on the measured wrench. In our case, given a platform desired trajectory q d P (t) = [p d P (t) η d P (t) ] , the admittance filter generates a modified reference trajectory q r P (t) based on the effect of the external disturbances ŵe . q r P (t) is then provided as reference to the Fly-Crane inverse kinematic motion controller. The reference trajectory is computed such that the controlled platform behaves similarly to an admittance model characterized by the following equation

M A (q d P -qr P ) + D A ( qd P -qr P ) + K A (q d P -q r P ) = ŵe , (20)
where

M A = diag{M A } ∈ R 6×6 >0 , D A = diag{D A } ∈ R 6×6 >0 and K A = diag{K A } ∈ R 6×6
>0 are matrices representing the desired (apparent) inertia, damping and stiffness, and

q d P -q r P = p d P -p r P 1 2 R d P R r P -R r P R d P ∨ . (21)
The operator [•] ∨ represents the mapping from SO(3) to R 3 . Regarding the stability of the overall system, the bandwidth of the admittance controller has to be lower than the equivalent bandwidth of the inverse kinematic motion control loop [START_REF] Villani | Force control[END_REF].

IV. EXPERIMENTS

In this section, we present experimental results that show the effectiveness of the presented control framework. First, we highlight the variety of behaviors that the system is capable of performing changing the physical properties of the admittance filter in Sec. IV-B1. Then we test the multi-robot aerial manipulator system in an unexpected-collision scenario, confirming its suitability in executing tasks while being in contact with the environment (see Sec. IV-B2). The interested reader can watch the attached video to see the system in action. A. Experimental Setup 1) Hardware: The Fly-Crane consists of three Quadrotor UAVs connected by six cables to a platform made of carbon fiber bars (see Fig. 3). The platform weights 0.737 [Kg], each cable length is 1 [m] and the weight of the aerial vehicles is 1.03 [Kg] each. The aerial vehicles are equipped with a standard flight controller, four ESCs (Electronic speed controllers) regulating the propeller speed in closed-loop [START_REF] Franchi | Adaptive closed-loop speed control of BLDC motors with applications to multi-rotor aerial vehicles[END_REF], and an onboard PC.

2) Software: The control framework runs partially onboard and partially on a Desktop PC. Each UAV runs onboard its state estimator and position controller [START_REF] Lee | Geometric tracking control of a quadrotor UAV on SE(3)[END_REF]. The desktop PC runs a Matlab/Simulink code which implements the inverse kinematics controller, the admittance filter, the wrench observer, and the redundancy resolution. The communication between the desktop PC and the onboard computers passes through a WI-fi connection and runs at 100 [Hz]. The intermediate and inner control loops are then closed based on the estimated state of the vehicles and the platform. The outer loop is closed through the thrust produced by the propellers computed from their measured speed, properly projected along the cables. State estimates are computed onboard at 1 [kHz] by an UKF (Unscented Kalman filter) that fuses the Motion Capture (Mo-Cap) System measurements (at 120 [Hz]) with the IMU measurements (at 1 [kHz]). Before the start of each experiment, the system goes to a hovering phase reaching a singularity-free configuration, then the admittance framework is activated.

B. Experimental Results

1) Admittance Shaping: This section illustrates the different behaviours of the system obtained by applying a step response when changing the admittance parameters.

In all the experiments, the equilibrium pose for the platform, in absence of external disturbances, is p d P = [0 0 1.28] [m] and R d P = I 3 . Two sets of experiments are presented: Set-1 (Fig. 4(d)-4(f)) and Set-2 (Fig. 4(g)-4(i)).

• Set-1: at the equilibrium pose and in absence of external force, a constant virtual external force of f n,x = -1.7 [N] is suddenly applied along x W , as shown in dashed lines in Fig. 4(a)-4(c). • Set-2: at the equilibrium pose and in absence of external force, a real external force is suddenly applied along x W using a weight suspended by a pulley-cable system. The reached final force depends only on the weight of the mass,x and it is about -1.7 [N] (see Fig. 3(b)) for the case under exam. However, the system is unaware of the mass amount, and the force is estimated using the wrench observer described in [START_REF] Lee | Geometric tracking control of a quadrotor UAV on SE(3)[END_REF]. The behavior through the different experiments is shown in Fig. 4(a)-4(c) in solid lines. In the practical realization, a constant bias in ŵe has been noticed. We believe that this is caused by modeling errors in the aerial vehicles and platform manufacturing, as well as model uncertainties like rotors and cables models. To mitigate the effect of the bias, a simple calibration is done in order to compensate for the steady-state error at rest.

Testing the system both with a virtual disturbance (perfectly known) and a real force estimated through the wrench observer, let us assess the impact of the wrench observer and other non-modeled effects, such as, e.g., the friction of the pulley, on the entire closed loop architecture.

The three parameters M A , D A , K A , which define the virtual admittance, have been individually manipulated in the different experiments. Additionally, to make further comparisons, we also show the nominal output of an ideal mass-springdamper system with the same characteristic and subjected to the same external input. As reference values we chose

M A = 4I 3 [Kg], D A = 10I 3 [Ns/m], K A = 7I 3 [N/m
] and all the other values have been generated by increasing or decreasing such values by suitable deltas. In the first set of comparisons, see Fig. 4(d) and Fig. 4(g), the focus is on the stiffness constant K A . The aim was to evaluate the behavior of the platform at steady-state. For this purpose we selected

K A = 4I 3 [N/m], the reference K A = 7I 3 [N/m] and K A = 10I 3 [N/m].
The performances of the system, when the external force was both applied (charging phase) and removed (discharging phase), are shown in the graphs in their left and right halves, respectively. The admittance scheme suitably performs at steady-state, where the position of the platform remains close to the nominal one (i.e. p n,x ) according to the corresponding stiffness value. The maximum error (e Px = p n,x -p P,x ) between the nominal position and the actual one does not exceed 0.01 The graphs show two phases: on the left of the black line we apply the constant force (charging phase), right after the black line the constant force is removed (discharging phase). Since the force has only been applied along xW , we show the first component of the position/velocity/acceleration vector (solid lines). We make the comparison with the ideal mass-spring-damper system (dashed lines) to underline the admittance shaping capability. In Fig. 4(d) the stiffness KA has been changed, obtaining three different steady-state positions. On the contrary, in Fig. 4(e) the DA has been the subject of the variation. Fig. 4(f) shows the effects of changing the virtual mass MA.

consequence of three different stiffness values. On the other side, in the transient of the charging phase of Set-2 the filtering action of the wrench observer has a non-negligible role in slowing down the reactivity of the system, as it can be seen comparing Fig. 4(g) with Fig. 4(d).

The second set of comparisons (Fig. Moreover, the nominal velocity reference ṗn of the ideal mass-spring-damper system and the actual platform velocity ṗP have been examined side by side. Two major behaviors can be extrapolated from the observation of the graphs. At first, in Set-2 (Fig. 4(h)), the ideal mass-springdamper system is better emulated during the discharging phase rather than the charging one. Unavoidable sticky-slips effects and friction in the pulley-cable significantly affect the natural evolution of the system. As a matter of fact, they do not arise during the discharging phase. Secondly, there is a small tracking delay of ṗP with respect to ṗn which is much more accentuated in Set-2 when D A = 1I 3 [Ns/m]. This latter result is a direct consequence of using the wrench estimator. No differences can be appreciated in the reached steady-state value of the position ( -0.2 [m], not shown in the plots), which remains unchanged in all three cases.

In the last experimental tests, Fig. 4(f) and Fig. 4(i), we compare the effects of the variations of the virtual inertia from the reference value M A = 4I 3 [Kg], with a decreased mass M A = 2I 3 [Kg] and an increased one M A = 6I 3 [Kg]. The aim is to analyze the performance of the system when high dynamics assignments are requested. In the examined cases, results show the impossibility for the system to satisfactorily shape the inertia, i.e., to track the ideal acceleration output corresponding to different virtual inertia. Several plausible reasons can be given. A good tracking of the virtual mass requires a very precise and theoretically instantaneous control of the thrust, which is impossible in our setup. In fact, the thrust is indirectly controlled through the rotor speeds, which are also subject to their first-order dynamics. On the other hand, the developed IKC controller is designed for tracking velocities rather than accelerations, and in general, the controller is purely based on feedback and it lacks the proper feed-forward term that is required to compensate for the inertial dynamics. A precise knowledge of such a term is indeed very difficult to attain in practice.

2) Unexpected-collision interaction: This experiment is designed to test the capability of the proposed aerial manipulation system to stably react to unexpected collisions and to adapt its behavior to such interaction. Additionally, the experiments also test the capabilities of the three aerial vehicles to maintain the cables' tightness at steady-state despite the contacts. To this end, a surface is placed in the middle of the arena, obstructing the way to the platform. The center of the surface is positioned at [-1.55 0 0.50][m] and its inclination about x W is 8 [deg] (see Fig. 3(c)). The admittance parameters have been set as follows * :

K Ap = 15I 3 [N/m] D Ap = 50I 3 [Ns/m], K Aη = 2I 3 [N/rad], D Aη = 14I 3 [Ns/rad], M Ap = 2I 3 [Kg] and M Aη = 20I 3 [Kgm 2 ].
The experiment starts with the platform positioned at p P = * The subscripts p and η refers to the gains in position and orientation respectively 5 and 6 present the crucial phases of the experiments. The area in which contacts occur has been highlighted in green. Fig. 5(a) shows the z components of the platform position, the desired trajectory, and the reference generated by the admittance filter. It can be appreciated how the admittance filter lets the reference deviate from the desired trajectory during the contact phase, thus indirectly keeping the pushing force limited in the direction of motion. In Fig. 5(b), we illustrate the position error e P = p r P -p P between the reference trajectory and the actual position of the robot. The norm ||e P || stays bounded around ±4 [cm] in position during the experiment assuring, in such a way, a reliable position tracking for the task under exam. With Fig. 5(c), we show the first three components of ŵe , namely the forces resulting from the external wrench estimation. In this graph, the contact, happening at 140[s], is fully recognizable in f z , as expected.

Figures 6(a achieved despite the already discussed delay introduced by the wrench observer. The estimated torque experienced by the platform is shown in Fig. 6(c), and the orientation error between the pre-planned trajectory and the trajectory produced by the admittance filter is presented in Fig. 6(d). Similarly to the translation case, during the contact phase, the multi-robot system is capable of adapting the platform orientation with considerable accuracy.

The experiment has been repeated to show the effect of a different gain choice on the platform rotational compliance about x W . In particular, the gain has been lowered to K Aη = diag(1.4, 2, 2) [N/rad]. The aim is to obtain a more compliant behavior. Fig. 7 shows the difference between the two experiments. The previous one, characterized by a higher stiffness gain, is depicted in dotted lines. The last one is in dashed lines. As can be seen from the higher deviation between the desired and the reference trajectory in roll (φ d -φ r ), the platform reaches a more compliant behavior when the stiffness gain is lower. Such a choice allows the platform to lean completely against the tilted surface. The effect of a different gain choice on the platform rotational compliance about z W has also been evaluated, leading to similar conclusions. The interested reader is referred to the attached multimedia material to appreciate the behavior of the system in action.

V. CONCLUSIONS

We have addressed the challenging problem of controlling a complex cable-based aerial multi-robot manipulator while performing tasks in physical interaction with the environment. Our work shows that the problem can be solved using a suitable control architecture that combines algorithms at different levels. We have also shown that the approach is effective and reliable despite the absence of dedicated force/torque sensors, thanks to the implementation of a state-of-the-art wrench observer adapted to the Fly-Crane, our cable-based aerial multi-robot manipulator.

However, the method presents some limitations here summarized. First of all, the system has not been examined when collisions could demand agile moves far from quasistatic motions. Besides, the cable tautness has been only guaranteed at steady-state in the presented experiments. In the future, we intend to evaluate the system performances in the other two identified classes of interactions (i.e. Expected collisions and Active interchanges). Moreover, we plan to extend the capabilities of the proposed control approach to perform outdoor experiments. Less accurate robot localization in outdoor conditions, as well as wind disturbances, will represent additional difficulties to be faced for the validation of control methods in real-world conditions. Finally, we intend to demonstrate the performances of our control framework in dealing with several tasks involving physical interactions such as pick-and-place manipulation or structure assembly in outdoor scenarios.

Fig. 1 :

 1 Fig. 1: Schematic representations of the Fly-Crane system: (a) shows the relevant variables of the system; (b) shows the forces exerted by each quadrotor and projected on each pair of cables.

Fig. 2 :

 2 Fig. 2: Control architecture of the admittance framework of the Fly-Crane system. The outer loop in green, the intermediate loop in blue, and the inner loop in red.

Fig. 3 :

 3 Fig. 3: The Fly-Crane involved in three different scenarios: (a) shows the Fly-Crane in a contact-free scenario; (b) illustrates the pulley-cable system which has been employed for the admittance property shaping; (c) shows the real-case scenario where the Fly-Crane has to approach a tilted surface being compliant with it.

  [m]. As expected, three different steady-state positions, -0.1 [m], -0.2 [m], -0.4 [m], have been obtained during the experimental campaign as a Preprint version, final version at http://ieeexplore.ieee.org/ : Stiffness tests: virtual and estimated force

  : Stiffness tests with perfectly known force: nominal and real position.

6 (

 6 e): Damping tests with perfectly known force: nominal and real velocity.

  : Inertia tests with perfectly known force: nominal and real acceleration.

  : Stiffness tests with estimated force: nominal and real position.

  : Damping tests with estimated force: nominal and real velocity.

  : Inertia tests with estimated force: nominal and real acceleration.

Fig. 4 :

 4 Fig.4: Admittance Shaping step responses: The graphs show two phases: on the left of the black line we apply the constant force (charging phase), right after the black line the constant force is removed (discharging phase). Since the force has only been applied along xW , we show the first component of the position/velocity/acceleration vector (solid lines). We make the comparison with the ideal mass-spring-damper system (dashed lines) to underline the admittance shaping capability. In Fig.4(d) the stiffness KA has been changed, obtaining three different steady-state positions. On the contrary, in Fig.4(e) the DA has been the subject of the variation. Fig.4(f) shows the effects of changing the virtual mass MA.

  4(e) and Fig. 4(h)) shows the effects of the damping constant variation D A . The purpose is to assess the behavior of the platform in the transient phase. The chosen values for the comparison are D A = 1I 3 [Ns/m], the reference D A = 10I 3 [Ns/m], and D A = 20I 3 [Ns/m].

Fig. 5 :

 5 Fig. 5: Unexpected-collision interaction: 5(a) vertical actual platform position pP in solid lines, desired position p d P and admittance reference p r P in dashed lines. 5(b): position error eP . 5(c): forces applied to the platform before and throughout the collision. 5(d): difference between the pre-planned trajectory and the references produced by the admittance filter.

Fig. 6 :

 6 Fig. 6: Unexpected-collision interaction: 6(a): actual platform orientation ηP in solid lines, desired orientation η d P and admittance reference η r P in dashed lines. 6(b): orientation error eη P . 6(c): torques applied to the platform before and throughout the collision. 6(d): difference between the pre-planned trajectory and the references produced by the admittance filter.

Fig. 7 :

 7 Fig. 7: The effect of changing the rotational stiffness about xW between two experiments.

  Preprint version, final version at http://ieeexplore.ieee.org/ 6 IEEE Robotics and Automation Letters 2022

An interesting framework for collision management for robotic manipulators was introduced in[START_REF] Haddadin | Robot collisions: A survey on detection, isolation, and identification[END_REF]

The left superscript indicates the reference frame. From now on, F W is considered as reference frame when the superscript is omitted.

S( ) : R

→ R 3×3 is such that S(v)u = v × u ∀v, u ∈ R 3Preprint version, final version at http://ieeexplore.ieee.org/ 2 IEEE Robotics and Automation Letters 2022

Preprint version, final version at http://ieeexplore.ieee.org/ 4 IEEE Robotics and Automation Letters 2022

This work has been funded by the ANR, under the Project ANR-17-CE33-0007 MuRoPhen, and by the European Union's Horizon 2020 research and innovation programme under grant agreement ID: 871479 AERIAL-CORE


Dana Al Kukhun
email: danakukhun@gmail.com

Bouchra Soukkarieh
email: soukkari@irit.fr

Florence Sedes
email: sedes@irit.fr

ADMAN: an Alarm-based mobile Diabetes MANagement system for mobile geriatric teams

In this article, we introduce ADMAN an alarm-based diabetes management system for the disposal of Mobile Geriatric Teams MGT. The system aims at providing a form of remote monitoring in order to control the diabetes rate for elder patients. The system is multidimensional in a way that it resides at the patient mobile machine from a side, the doctor's mobile machine from another side and can be connected to any other entity related to the MGT that's handling his case (e.g. dietitian).

Introduction

One of the main objectives behind the initiation of Mobile Geriatric Teams is to provide specialized services for elder patients at hospitalization phases and to minimize the number of hospitalization days especially at the recovery phase. Since Mobile Geriatric Teams MGT can follow the patient wherever he is, a special follow up technique usually takes place in order to return patients to their homes or elder centers and monitor their stability with the help of their families or the centre's staff. The Mobile Geriatric Teams then make follow up visits and trace the evolution of each case.

Diabetes [23] is a chronic disease that occurs either when the pancreas does not produce enough insulin or when the body cannot effectively use the insulin it produces. Hypoglycemia, or raised blood sugar, is a common effect of uncontrolled diabetes and leads to serious damage to many of the body's systems over time, especially the nerves and blood vessels. This is why it is important to keep a good glucose levels.

Not controlling diabetes can also complicate medical cases and influence patient's recovery. Usually diabetic patients need extensive care and daily follow up in order to stabilize the case and ensure recovery. Many works discussed the economic burden of hospitalization costs when other solutions could exist. For instance, the works of [START_REF] Allan | Admissions avoidance and diabetes: guidance for clinical commissioning groups and clinical teams[END_REF] introduced a study dedicated to avoid the admission of diabetes patients in UK hospitals, other works like [START_REF] Launay | Mobile Geriatric Team and Length of Hospital Stay Among Older Inpatients: A Case-Control Pilot Study[END_REF] discussed the influence of Mobile Geriatric teams on the length of hospital stay among elder patients.

In France, the MGT were the core of many studies, such as [START_REF] Arthus | Quels outils pour piloter la performance d'une Equipe Mobile de Gériatrie[END_REF] that aimed at evaluating the performance of the MGT Service at the Midi Pyrénées region and their efficiency in taking care of elder patients in charge. Other research works like [START_REF] Kukhun | La mise en oeuvre d'un modèle de contrôle d'accès adapté aux systèmes pervasifs. Application aux équipes mobiles gériatriques[END_REF] have proposed adaptive accessibility while accessing to information sources during the mobility of the MGT members.

In this paper, we aim to help in solving the remote monitoring problem in another direction where we propose a web-based mobile application that helps doctors in following up with their diabetes patients during the recovery period and that alerts them in emergent cases. The application is multi-dimensional in a way that serves the patient from a side, the medical team from another and other sides whenever needed (dietitian, etc.). The application can be downloaded at the patient side where he, a family member or a personal assistant could enter his daily diabetes rates and at the MGT's side where they could monitor the patient's progress or retreat. Other sides such as family members, care specialists and the dietitian can also have access to the system through their mobiles.

The remainder of this paper is structured as follows: in Section 2, we introduce a state of the art about the interoperability of web services then, we present some examples of web applications and multi-tier systems for diabetes management. In section 3, we describe the architecture of our ADMAN system detailing its database and the functionality of its Web services. Finally, section 5 concludes the paper with brief remarks.

2

State of the art

In this section we will start by highlighting the importance of accomplishing our idea by using an interoperable and powerful tool as the web service. First, we will present the web service, its definition, components and advantages like interoperability. Then, we will preview the different research works that presented services to solve the problem of diabetes monitoring, starting with server based mobile applications and continuing with multi-tier systems that connect the patient with the doctor.

Interoperability & web services

The increasing number of heterogeneous applications and devices used for communication in the internet claims numerous efforts to improve the interoperability [START_REF] Moritz | Web Services to Improve Interoperability of Home Healthcare Devices[END_REF]. Interoperability is a key feature enabling the interaction between distributed components. In the domain of healthcare, interoperability is the ability of different information technology systems and software applications to communicate, exchange data, and use the information that has been exchanged [START_REF][END_REF]. Interoperability means the ability of health information systems to work together within and across organizational boundaries in order to advance the health status of, and the effective delivery of healthcare for, individuals and communities [8].

Web services are a natural consequence of the evolution of the web, they represent one of the best technological solutions that can provide a new assured level of interoperability between different applications [START_REF] Shetty | Interoperability issues seen Web Services[END_REF].

Web Services, at a basic level, can be considered a universal client/server architecture that allows disparate systems to communicate with each other without using proprietary client libraries [START_REF] Myerson | Web Services Architectures[END_REF].

"Web Services are self-contained, modular, distributed, dynamic applications that can be described, published, located, or invoked over the network to create products, processes, and supply chains. These applications can be local, distributed, or Web based. Web services are built on top of open standards such as TCP/IP, HTTP, Java, HTML, and XML" [START_REF]WebSphere Business Integration Adapters[END_REF].

The classical architecture of Web services (c.f. fig. 1) is composed of three entities (provider, user, and registry); the service provider builds the service and publishes its description in a registry [START_REF] Soap | Recommandation W3C 27[END_REF] [START_REF]WSDL, W3C, Web Services Description Language (WSDL), the last version[END_REF]. The user needs are translated into queries that are carried on by the Web Services registry [START_REF]W3Schools[END_REF]. Once the service is found, the user will obtain direct interaction with the service [START_REF] Melliti | Interopérabilité des Services Web complexes. Application aux systems multiagents[END_REF].

Fig. 1. The Classical Architecture of Web Services

Many research works studied Web services and based their works on their classic architecture such as [10] [20]. Also, in the healthcare area, several works used web service technology in order to solve specific problems such as in [START_REF] Mrissa | A mediation framework for Web services in a distributed healthcare information system[END_REF] that proposed an approach based on Web service technology to solve the heterogeneity of data problems within healthcare information systems.

[17] described a Web service middleware-based framework that supports the interoperable information needs of healthcare organizations.

Another example is presented in [START_REF] Anzböcka | Modeling and implementing medical Web services[END_REF] who proposed a modeling process for medical e-service.

The interoperability ensured by the Web service is the principal reason that motivated us to use it in our system. Now, after introducing web services and their capabilities of interoperability, we'll present some of the web applications that were introduced in previous research works for monitoring diabetes from the patient side.

Mobile Applications for Diabetes Management

With the evolution of mobile health, nowadays, we can find more than 1,000 iOS and Android applications listed on the Apple App Store and Google Play. These applications are specifically designed for diabetes patients or for healthcare professionals treating diabetes.

Through an analytical study, we have followed examples of the evolution of mobile diabetes management software, the first type of mobile applications dedicated for diabetes is concerned in tracking the diabetes measurements through time in order to improve the patient's consciousness and self-management of diabetes glucose readings.

Glucose Buddy [START_REF]Glucosebuddy[END_REF] is an example of these applications where it allows users to manually enter their glucose measurements, carbohydrate consumption, insulin dosages, and other activities. The system allows entering contextual information related to the measurement time (e.g. before breakfast, after breakfast, etc.).

iGBStar [25] is another example of diabetes management applications that is connected with a hardware solution, where you can connect your mobile to the blood glucose meter and the application makes it easier to collect, view, share blood glucose readings and store their values.

Glooko [START_REF][END_REF] is another mobile, cloud based diabetes management system that serves patients by seamlessly, easily and quickly downloading blood glucose readings into their mobile devices and accounts. It Saves time and eliminates errors from manual entry.

We have noticed that the presented examples were mainly interested in monitoring diabetes at the patient side for enhancing the patient's self control. These systems have taken into consideration the patient and his diabetes level's evolution but they haven't classified them or connected them in a seamless way with the doctor or any other specialist (e.g. dietitian). Also, these systems didn't consider cases and scenarios where a family member could be interested to be connected and alarmed in the cases of Hypoglycemia or severe insulin increase.

Next, we'll present some of the systems dedicated for connecting the patient results with the doctor in order to ensure diabetes rate monitoring.

Multi-tier systems for Diabetes Management

In this section, we overview another type of systems for diabetes management that were present in the literature. Their objective is to share the glucose blood readings with the doctor and healthcare professionals in order to enable him to monitor the patient case remotely.

The works of [START_REF] Soukkarieh | L'adaptation au contexte dans un système d'information web[END_REF] have proposed a Patient monitoring framework that is dedicated for the assistance of diabetes patients. They present a module-based application, divided in 2 main parts, the specialist and the patient. The platform provides statistics created to offer the doctor the progress of each patient and suggestions, in charge of giving the doctor some pieces of advice based on the statistics of the patient.

The application at the patient's side enables communications, diet enhancements and suggestions. The suggestion module lays on the diet module to present suggestions about beneficial food.

The works of [START_REF] Nachman | Pervasive[END_REF] have presented a pervasive healthcare platform for diabetes management and that employs sensors for monitoring the patient's heart rate and physical activity. The main goal of the system is to empower patients to manage their life style with respect to diet and exercise hence lowering their risk factors for metabolic syndrome, and to enable physicians to be more effective in helping their patients reach their goals.

The prementioned works have connected the patient results to the doctor using conventional internet applications. Knowing that nowadays, people mostly interact using their mobile devices and that these applications did not adapt with the mobility of the Mobile Geriatric Teams, we have proposed a mobile application that aims at monitoring the patient's diabetes rates remotely by the doctor or specialist.

Contribution

In this part, we will start by detailing the architecture of our system ADMAN then, we will describe the components of the ADMAN xml database and finally, we will explain the functionalities of the ADMAN web services.

Architecture of our application ADMAN

In this section, we will present our system architecture (see fig. 2) that aims to ensure that each measured insulin result is connected between the MGT and the patient.

Our architecture is composed of three main entities:

• Patient: This entity is an elder patient who needs to be taken care of either by a family member or an assistant. The surrounding environment of the patient is important; it helps the family member or nurse to deliver his insulin measurements by using the diabetes measurement machine and his mobile device that contains our ADMAN application. • Services: this entity is the core of our system. It is responsible for all the computational activities needed to be done in order to connect between the patient and the MGT in a real time basis. The service can analyze the measured results and sort them within predefined rates in order to judge the condition of the patient (hypoglycemia, normal, severe increase). • MGT: this entity contains the Mobile Geriatric Team that is composed of a secretary, nurse, nursing assistant and physician (generalist or specialist). In our case, the system is mainly interested in being downloaded to the doctor's machine and could also be downloaded to external parties (e.g the dietitian's machine and any other family member that is interested to monitor the patient's case).

The functionality within our architecture goes as follows:

Fig. 2. ADMAN Architecture (1)
As the patient wakes up, he should measure his diabetes rate (1a), this measurement can also be taken by a family member or an assistant (1b). ( 2) The rate value is then explicitly inserted to our system by the patient (2a)

or by the care specialist or any family member (2b). In an advanced scenario, this rate will be implicitly inserted from the machine. (3) The measured rate will be transferred to an xml database (c.f. fig. 3). The file contains the reading information for each patient, for more details, see section 3.2. (4) The transferred results will be uploaded to the web service (Service1) in order to be shared with the doctor's machine (3). (5) This web service is a black box that takes the measured rate and compares it with different intervals (c.f. fig 4 and section 3.3 for the algorithm) in order to return a textual value that describes the patient's status. The returned value is shared with both sides (4a, 4b) (the doctor and the patient who will view the analysis of the measured result). For more details, see paragraph 3.3 (6) Our first web service will enable both sides to view other analytical reports of the calculated values over time (weekly report, monthly report and a 3 months report). [START_REF][END_REF] In the case of Hypoglycemia or severe insulin rate, the doctor and patient will receive push notifications to alarm them for the danger of the case; this alarm will be placed in the message area (4a, 4b).

Patient MGT(Doctor, Dietitian, etc.)

<name… </name> <reading> <date>..</date> … </reading> <message>…</message> <notification> …</not (1a) (1b) (2a) (2b) Services (3) (4a) (4b) (5a) (5b) (2a) (2b)
<name… </name> <reading> <date>..</date> … </reading> <message>…</message> <notification> …</not Service2 Service1 (8) After receiving the alarm, the Mobile Geriatric Team would make the appropriate action and contact the patient through a notification (5a, 5b) this aims at dealing with the patient's case.

The ADMAN Database

As shown in fig. 2, the database that our ADMAN system employs is an xml database [START_REF]XML Base W3C Recommendation 28[END_REF] which is interoperable and light in size. This file contains information about the patient such as his name, reading (date, time and value), massage and notification.

• The first element (name) is the account name of the patient.

• The second element represents the glucose measurement (reading) that the user enters its value explicitly; the system enters the date and time implicitly.

• The third element is the (message) which is filled by the web service side.

• The forth element is the (notification), which is inserted to the system by the Geriatric Team specialist.

<patient id = ""> <name> … </name> <reading> <date> … </date> <time> …</time> <value> … </value> </reading> <message alarm = ""> … </message> <notification> … </notification> </ patient >

Fig. 3. XML File of the ADMAN database

The database existing at the patient's side will be also residing at the Mobile Geriatric Team's side, the MGT will eventually manage different accounts for different patients.

Web Services Functionality

Our ADMAN system depends on 2 web services that have different functionalities: Service1: It analyzes the measured results. This service will take the measured insulin value as input and will compare it to a group of values in order to output the rating of this result. For more details, see fig. 4 that features the algorithm of rating the measured insulin value. This service will also pass a message containing the analyzed measured result to both sides the MGT and the patient. Finally, this service will also alarm the MGT and the patient in extreme cases. Service2: In case of extreme cases, it will pass a message (notification) from the doctor to the patient to help him to react.

Use Case

Taking the example of the patient "John Smith" who's aged 75 years and who's recovering from hospitalization. A nurse will be passing by him on a daily basis in order to check up his insulin rate measurements and to do some other checkups.

Our application is downloaded on different machines related to different people who are interested in following up the patient's case (Doctor, dietitian, family member).

On the first day, the nurse measures the patient's diabetes rate. Supposing that the inserted measurement of his insulin value is equal to 90, then the ADMAN's web service will rate it as a normal value and will include this result within the xml file that is sent to both sides.

On the second day, the insulin value out to be equal to 57, the ADMAN's web service will rate it as a Hypoglycemia and will send this result and an alarm to the different entities related to the patient.

Here, the doctor will order the nurse to give the patient some sugar and tell her to re-measure the diabetes rate after 15 minutes. Also, he might think of rescheduling the insulin dose taken by the patient. The dietitian can then take an action and send a notification including some directions to the patient. The family member would call in order to check up on his case.

Any action taken can be included within the notification tag of our database. This way, the notes taken from this experience would be shared with the family member who will make sure that they will be applied. <patient id = "777"> <name> John Smith </name> <reading> <date> 2015-04-14 </date> <time> 08:00:00</time> <value> 57 </value> </reading> <message alarm = "on"> "Hypoglycemia" </message> <notification id = "1"> "Please take some sugar and re-measure the diabetes rate after 15 minutes, Please take a lower dose of insulin at night" </notification> <notification id = "2"> "Please make sure that you eat some carbohydrates at dinner " </notification> </ patient > Fig. 5. An XML file completed by our ADMAN service.

Conclusion

Many systems were proposed to encounter the challenge of diabetes monitoring but none of these systems considered to channel the doctor with the patient in the form of a simple mobile application. In this paper, we have presented the ADMAN system that helps the mobile Geriatric Team members to remotely monitor the daily insulin measurements for diabetic patients. Our objective is to minimize the hospitalization period for elder people while providing them with distant medical surveillance. Our system will also help patients to optimize their self-control and monitor themselves.

One of the limits to the success of our systems is the accuracy of the measured result taken from the machine (normally the MGT depends on values checked up at the hospital).

In future works, our system will take into consideration different doctors can treat same patient. Another is taking into consideration the user profile and his medical doses in order to automatically measure the insulin doses to be taken after each measurement.

Fig. 4 .

 4 Fig. 4. Algorithm for analyzing the patient's diabetes measurements.

  Fig 5 shows an example of a filled xml file coming from the doctor's side and the dietitian side and that is being received by the patient and other users.


est destinée au dépôt

Virtual reality and interactive and immersive planning for the assistance to manipulation or navigation

Philippe Fillatreau

To cite this version: Open Archive Toulouse Archive Ouverte

  Philippe Fillatreau. Virtual reality and interactive and immersive planning for the assistance to manipulation or navigation. 2021 IEEE International Workshop of Electronics, Control, Measurement, Signals and their application to Mechatronics (ECMSM), Jun 2021, Liberec, Czech Republic. pp.1-1. hal-03664914 OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible Any correspondence concerning this service should be sent to the repository administrator: tech-oatao@listes-diff.inp-toulouse.fr This is an author's version published in: http://oatao.univ-toulouse.fr/28331 To cite this version: Fillatreau, Philippe Virtual reality and interactive and immersive planning for the assistance to manipulation or navigation. (2021) In: 2021 IEEE International Workshop of Electronics, Control, Measurement, Signals and their application to Mechatronics (ECMSM), 21 June 2021 -22 June 2021 (Liberec, Czech Republic).


J S Leontini

D Lo Jacono

M C Thompson

Three-Dimensional Transition in the Wake of an Ellipse

The transition to three-dimensional flow from the nominally two-dimensional Kármán vortex street in the wake of bluff bodies is a problem of fundamental importance as it marks the first step on the path towards fully developed turbulence. Here, this transition is studied in the wake of an elliptical cross-section using Floquet stability analysis. A number of modes of instability are identified as a function of the aspect ratio of the ellipse. Three-dimensional simulations confirm the importance of the identified instability modes.

Introduction

This paper reports on the results of a study of the linear stability of the Kármán vortex street in the wake of an elliptical crosssection. The aim is to identify the instability modes that cause the wake to cease being uniform in the spanwise direction (twodimensional) and to become three-dimensional.

The flow is governed by two parameters: the Reynolds number Re = UD/ν, where U is the freestream velocity, D is the width of the cross-section across the flow, and ν is the kinematic viscosity; and the elliptical cross-section aspect ration Γ = L/D, where L is the length of the body in the flow direction. When Γ = 1, the body is a circular cylinder. As Γ is increased beyond Γ > 1, the body becomes a streamlined ellipse. Here, flow for bodies 1 ≤ Γ ≤ 2.4 are studied. Two-dimensional direct numerical simulations have been performed for a series of values of Γ and Re. The stability of these base flows has then been established using Floquet stability analysis . Five instability modes have been identified, and their dependence on Γ ascertained. Of these five modes, two are only found above a certain value of Γ, meaning they do not occur in the wake of a circular cylinder. Fully three-dimensional direct numerical simulations have also been performed, confirming the existence of the saturated state of these two new modes.

Methodology

Floquet Stability Analysis

In general, Floquet stability analysis is a technique to ascertain the stability of periodic solutions. Here, the periodic solution is the two-dimensional Kármán vortex street. To assess the stability, first linear equations are formed that govern the evolution of a perturbation to the periodic solution. The perturbation equations are integrated forward in time one period, and the size of the perturbation is compared to its size one period prior. This process is repeated over many iterations. The ratio of the size of the perturbation at the current period to that at the previous pe-riod is known as the Floquet multiplier, µ f . If the perturbation has grown over one period, |µ f | > 1, the original periodic solution is unstable; if the perturbation decays, |µ f | < 1, the original solution is stable. Note that µ f can be a complex number. If it is complex, this implies that the instability mode varies with a frequency that is incommensurate with the periodic base flow.

A further extension to the method is to decompose the perturbation into Fourier components in the spanwise direction. If this is done, it can be shown that the various Fourier modes decouple, and therefore the stability with respect to a perturbation of a specified spanwise wavelength λ can be ascertained. Therefore, the Floquet multiplier µ f is a function of the flow variables (in this case Re and Γ) and the spanwise wavelength λ. A full explanation of this technique can be found [START_REF] Barkley | Three-dimensional Floquet stability analysis of the wake of a circular cylinder[END_REF].

Each instability mode will have a characteristic λ. At this λ, µ f will be maximum for that mode. Therefore, when a point of marginal stability is found (a point in the Γ,Re plane where µ f = 1), there is a corresponding value of λ. So, the main task of the analysis is to find the curves of marginal stability (µ f = 1) in the Γ,Re plane for each mode and the wavelength λ at which this occurs.

Simulation Technique

Calculations were performed using a high-order spectral element method to solve the weak form of the incompressible Navier-Stokes equations for the base flows, and a linearized version of these same equations for the perturbation fields. Full details of the spectral-element method can be found [START_REF] Karniadakis | Spectral/hp element methods for computational fluid dynamics[END_REF]. Validation of the code employed here for the base flows and the stability analysis is given in a series of papers [START_REF] Thompson | Hydrodynamics of a particle impact on a wall[END_REF][START_REF] Leontini | Threedimensional transition in the wake of a transversely oscillating cylinder[END_REF][START_REF] Rao | Three-dimensionality in the wake of a rotating cylinder in a uniform flow[END_REF]. The generally complex Floquet multipliers (and the corresponding instability modes) were calculated using Arnoldi decomposition.

Results and Discussion

Stability Analysis

Five distinct modes were found to become unstable in the wake of the ellipse, and representative images of each of them are shown in figure 1. The images show the spanwise perturbation vorticity as colour contours, and the base flow vorticity (the vortex street) as solid lines. The modes are shown in order of increasing characteristic wavelength, from top to bottom. Three of the modes shown in figure 1 are known from studies of the circular cylinder, namely modes A, B, and QP. Mode A is thought to be due primarily to an elliptic instability of the wake vortices [START_REF] Leweke | Three-dimensional instabilities in wake transition[END_REF][START_REF] Thompson | The physical mechanism of transition in bluff body wakes[END_REF], while mode B appears to be an instability of the shear layers in between the wake vortices. Mode QP is a quasiperiodic mode, meaning it has a frequency incommensu- rate with the base flow, and therefore a complex Floquet multiplier [START_REF] Blackburn | Symmetry breaking of two-dimensional time-periodic wakes[END_REF].

Two new modes have been discovered that only occur for a range of Γ, and therefore do not occur in the circular cylinder wake. These are modes A and B. Mode A has a structure very similar to mode A, until a few diameters downstream. In figure 1, the first vortex where a discernible difference between the structures of modes A and A are circled with a dashed line. Mode B has a spatio-temporal symmetry the same as mode B, however its spatial structure is quite distinct. It bears some resemblance to the mode named B ′ found in the wake of elliptical plates [START_REF] Ryan | Threedimensional transition in the wake of bluff elongated cylinders[END_REF].

The dependence of the critical Reynolds number Re c on Γ for each of these five modes is shown in figure 2. Modes A and B are shown to be stabilized by increasing Γ, with Re c increasing. Mode QP is initially destabilized for Γ < 1.2, but it stabilized with further increases in Γ. The instability scenario for Γ < 1.2 is basically the same as for the circular cylinder; with increasing Re, mode A first becomes unstable, then mode B, then mode QP.

For Γ > 1.2, the transition scenario is different from the circular cylinder. Mode A is resolved, and shown to be the first mode to become unstable with increasing Re for 1.2 < Γ < 1.9. For Γ > 1.9, mode B is the first to become unstable, followed by mode A.

Other features of figure 2 are of note. First, mode B is found to be unstable in a closed region of the parameter space, i.e, there is an upper limit to the range of Re where this mode is unstable. This upper limit is shown by the dashed line in figure 2. Second, mode A appears to almost branch off from mode A around Γ = 1.2.

Further evidence of mode A branching off from mode A is found by plotting the characteristic wavelength λ c of modes A and A, as found in figure 3. The figure shows that while the wavelength of mode A is weakly affected by increasing Γ, mode A is strongly affected, with λ c almost linearly with Γ. At Γ = 1.2, the wavelengths of modes A and A coincide, and for Γ < 1.2 mode A is not resolved. This indicates that the instability mechanism of mode A is likely to rely on the structure of the wake close to the body in a similar way to mode A.

Three-Dimensional Simulations the linear stability analysis can give some indication of the parameter values for which the base flow is unstable, and the structure of the linear instability mode, it does not necessarily give a clear picture of the form of the flow beyond the bifurcation. This is because nonlinear effects are not captured in the stability analysis, which play a role in the development of the final fully saturated state. To capture this, fully three-dimensional direct numerical simulations are required.

The saturated state in the wake of a circular cylinder is well documented [START_REF] Williamson | The existence of two stages in the transition to three-dimensionality of a cylinder wake[END_REF][START_REF] Thompson | Threedimensional instabilities in the wake of a circular cylinder[END_REF], and depends primarily on modes A and B. However, the impact of the new modes A and B has not been previously reported.

Figure 4 presents results from simulations for Γ = 2, for Re = 300 and Re = 350. In both cases, both modes A and B are predicted to be unstable according to figure 2.

For the simulations at Re = 300 shown in 4a, the spanwise domain was restricted to 9.6D. On this restricted domain, clear wavelength of 2.4D develops, attributable to mode B. The three-dimensionality is relatively weak, restricted to a moderate waviness of the shear layer separating from the body and the vortex cores downstream.

For the simulations at Re = 350 shown in figure 4b, the spanwise domain was extended to 22D. In this case, a mode with a wavelength around 7.5D becomes dominant in the saturated state. This is attributable to mode A. This is despite mode B becoming unstable at a lower Re. The flow state is quite complex, with an initial waviness of the separating shear layer, leading to a structure of essentially streamwise vortices developing further downstream. These streamwise structures appear to undergo a secondary instability around 25D downstream, moving across the wake towards each other.

Conclusions

stability analysis, and supporting three-dimensional direct numerical simulations, have been conducted of the wake of an elliptical cross-section. It has been shown that for small aspect ratios, the instability scenario is similar to that for a circular cylinder. However for larger aspect ratios, new modes of instability are resolved. These new modes have been shown to play a leading role in the fully saturated three-dimensional state of the flow. the National Computational Infrastructure (NCI).

Figure 1 :

 1 Figure 1: Images of the different modes at Γ = 1.8 (except for mode B shown at Γ = 1.4) shown in order of increasing characteristic wavelength. The red and blue contours represent spanwise perturbation vorticity over normalized levels of ±0.1. The solid lines show contours of the base flow vorticity at levels ±1. Mode B has a distinct structure, but shares the same spatiotemporal symmetries as mode B. Mode A has a similar structure to mode A in the two vortices closest to the body, but is distinct from the third vortex (circled with a dashed line) and further downstream.

Figure 2 :

 2 Figure 2: Re c , the value of Re at marginal stability for all the modes as a function of Γ. Points are measured values for for mode A (•), mode B ( ), mode QP (•), mode A ( ), mode B (×). Lines are best fits to the measured values. The upper limit of instability for mode B is marked with a dashed line as it is the result of extrapolation.

Figure 3 :

 3 Figure 3: The wavelength at marginal stability as a function of Γ for modes A and A. Mode A appears to bifurcate from mode A around Γ = 1.2.

Figure 4 :

 4 Figure 4: Visualization of the flow calculated from threedimensional DNS at (a) Γ = 2, Re = 300, mode B, and (b) Γ = 2, Re = 350, mode A. Isosurfaces show the λ 2 = -0.001 criterion [12]. The total length of the domain, and one wavelength of the dominant mode, are marked for both cases.

Acknowledgements

The authors acknowledge the financial support of the Australian Research Council through Discovery Project grant DP110102141, and the provision of computational resources and expertise of the Monash University e-research centre, and (a)