Member, IEEE Marco Tognon
email: mtognon@ethz.ch.

Member, IEEE Rachid Alami
email: rachid.alami@laas.fr.

Fellow, IEEE Bruno Siciliano
email: bruno.siciliano@unina.it.

Application to a Force-based Human Guiding Problem

Keywords: Aerial Systems: Mechanics and Control, Physical Human-Robot Interaction, Motion Control, Force Control. I. INTRODUCTION

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tasks like pushing, sliding [START_REF] Tognon | A truly redundant aerial manipulator system with application to push-andslide inspection in industrial plants[END_REF] and transportation, allowing new applications as contact-based inspection [START_REF] Ollero | The AEROARMS project: Aerial robots with advanced manipulation capabilities for inspection and maintenance[END_REF] and assembly.

It is natural that the presence of such robots in our daily life will grow as well. This requires the next generation of aerial robots to be able to safely and reliably interact with the environment, but also with humans. So far, Human-Robot Interaction (HRI) has been rarely addressed in the aerial robotics community. Most of the works focus on remote control proposing bilateral teleoperation methods to help humans control single and multiple aerial vehicles [START_REF] Mersha | On bilateral teleoperation of aerial robots[END_REF]. Recently, the teleoperation problem has been extended to APhI as well [START_REF] Gioioso | A forcebased bilateral teleoperation framework for aerial robots in contact with the environment[END_REF].

In these cases, humans and robots are in separate environments and the interaction is only virtual. The study of scenarios where aerial robots share the same environment of humans is still poorly addressed and limited to contactless HRI [START_REF] Garrell | Teaching a drone to accompany a person from demonstrations using non-linear asfm[END_REF], [START_REF] Yeh | Exploring proxemics for human-drone interaction[END_REF]. In [START_REF] Peshkova | Natural interaction techniques for an unmanned aerial vehicle system[END_REF], [START_REF] Cauchard | Drone & me: an exploration into natural human-drone interaction[END_REF], the authors propose a survey on the use of natural communication means like gestures, speech, and gaze direction to let the human interface with the robot.

Another important topic recently explored is the social acceptance of aerial robots [START_REF] Acharya | Investigation of human-robot comfort with a small unmanned aerial vehicle compared to a ground robot[END_REF]. In [START_REF] Liew | Quadrotor or blimp? noise and appearance considerations in designing social aerial robot[END_REF] the problem is addressed from a design point of view, analyzing how the propellers of multi-rotor platforms affect the comfort of humans. In a different direction, a few papers [START_REF] Wojciechowska | Collocated human-drone interaction: Methodology and approach strategy[END_REF], [START_REF] Duncan | Comfortable approach distance with small unmanned aerial vehicles[END_REF] recently started to investigate the social constraints (distance, speed, direction of approach, etc.) that should be applied to the motion of an aerial robot in the proximity of a human. While human-aware motion planning has been substantially studied for ground robots [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF], it is still very preliminary for aerial robots.

Therefore, the works presented so far focus only on contactless HRI for aerial robots. On the other hand, physical Human-Robot Interaction (pHRI) for aerial vehicles is still an unexplored domain. To the best of our knowledge, the only work addressing this topic is [START_REF] Augugliaro | Admittance control for physical human-quadrocopter interaction[END_REF] where a compliant controller based on an admittance filter is proposed. Compliance with respect to external forces is shown, but the interaction with the human is limited to simple contacts. The acceptance of touch-based HRI with aerial platforms is studied in [START_REF] Abtahi | Drone near me: Exploring touch-based human-drone interaction[END_REF].

Differently, research done on pHRI with ground manipulators is already quite extensive [START_REF] Santis | An atlas of physical human-robot interaction[END_REF], [START_REF] Ajoudani | Progress and prospects of the human-robot collaboration[END_REF]. Many control methods, most of them based on admittance [START_REF] Ficuciello | Variable impedance control of redundant manipulators for intuitive human-robot physical interaction[END_REF] and passivitybased methods [START_REF] Colgate | Robust control of dynamically interacting systems[END_REF], have been proposed.

To advance the study of pHRI methods for aerial robots, in this work we face one of the first systems in which a human and an aerial robot are tightly coupled by a physical means. In particular, we consider a human holding a handle which is in turn connected to an aerial vehicle by a cable (see Fig. 1). In this manuscript, we want to design a control strategy that allows the robot to safely "physically guide" the human toward a desired position (or along a predefined path) exploiting the cable as a force-based communication means. Previous works [START_REF] Tognon | Dynamics, control, and estimation for aerial robots tethered by cables or bars[END_REF]- [START_REF] Sandino | On the use of tethered configurations for augmenting hovering stability in smallsize autonomous helicopters[END_REF] already showed that tethered aerial vehicles are very suitable for APhI. In [START_REF] Tognon | Theory and Applications for Control of Aerial Robots in Physical Interaction Through Tethers[END_REF] a complete study and literature review of the modeling and control for tethered aerial vehicles is provided. However, previous solutions cannot be directly applied because they consider the cable always taut. This would limit the validity of the proposed results. On the other hand, the modeling and control designs have been inspired by the study in [START_REF] Tognon | Aerial comanipulation with cables: The role of internal force for equilibria, stability, and passivity[END_REF] showing how the force along a cable can be used for cooperative-manipulation as indirect communication means among robotic agents.

The problem of guiding humans by aerial vehicles has indeed been already addressed in the literature but proposing contactless approaches only, e.g., based on sound [START_REF] Avila | Dronenavigator: Using drones for navigating visually impaired persons[END_REF] or vision [START_REF] Mueller | Jogging with a quadcopter[END_REF]. In this work, we rather exploit the physical interaction using forces as more direct feedback. This solution particularly suits for persons with visual or hearing limitations. There have been a number of contributions of ground robots guiding humans [START_REF] Clodic | Rackham: An interactive robot-guide[END_REF], [START_REF] Triebel | SPENCER: A socially aware service robot for passenger guidance and help in busy airports[END_REF]. However, while they have considered a variety of capabilities to take into account human constraints and a variety of interaction modalities, only a small number of contributions considered pulling mobile robots or walking helpers for assisting elderly [START_REF] Morris | A robotic walker that provides guidance[END_REF] or visually impaired people [START_REF] Tachi | Electrocutaneous communication in a guide dog robot (meldog)[END_REF], [START_REF] Melvin | Rovi: a robot for visually impaired for collision-free navigation[END_REF]. However, the use of aerial vehicles in physical interaction with humans demands a careful treatment of the interaction forces and the stability of the system.

Inspired by the extensive literature of pHRI for ground manipulators, we firstly propose the use of an admittancebased strategy to make the robot comply with the human. We shall show that exploiting this approach we can impose a desired cable force that has the function to pull the human toward the desired point. The design of such input requires particular attention due to the flexible nature of the physical link connecting the human to the robot. Taking inspiration from the theory on control of manipulators with elastic joints [START_REF] De Luca | Robots with flexible elements[END_REF], we design a robot-feedback solution together with a feedforward term. The latter enforces the tautness of the cable. Approximating the human behavior as a mass-spring-damper system and considering the overall closed-loop system, we formally prove (though Lyapunov theory) that the proposed control method allows the robot to asymptotically steer the human to the desired position.

We also show that the system is output-strictly passive. This guarantees the stability of the system even in case of i) model mismatches, disturbances and tracking errors, ii) time-varying reference for the human position, and iii) additional forces applied by the human to the cable (e.g. if he/she wants to stop). This property grants robustness and safety of the method.

To address the problem of path following, we design a maneuver regulation method based on the works in [START_REF] Spedicato | Aggressive maneuver regulation of a quadrotor UAV[END_REF], [START_REF] Hauser | Maneuver regulation from trajectory tracking: Feedback linearizable systems[END_REF]. This avoids the excessive growth of the tracking error which may lead to input saturation and thus instability, or too big deviations from the desired path.

The control performance and stability of the system are validated by real experiments firstly showing (from the best of our knowledge) safe and reliable physical interaction between a human and an aerial vehicle.

The manuscript is organized as follows: In Sect. II we model the entire system. In Sect. III and Sect. IV, we describe the proposed control strategy and we analyze the stability of the closed-loop system. The additional passivity property is shown in Sect. V. In Sect. VI, we describe the path following approach. The experimental results are reported in Sect. VII. Section VIII ends the manuscript with some final discussions.

II. MODELING

The system (see Fig. 1) is composed of an aerial vehicle tethered by a cable to a handle held by the hand of a human. To describe it, we define an inertial frame, F W = {O W , x W , y W , z W }, where O W is the arbitrary origin and {x W , y W , z W } are its unit axes, with z W opposite to gravity.

We consider the position of the human as the position of his/her hand holding the handle. A body frame, F H = {O H , x H , y H , z H }, is rigidly attached to the handle. Its origin O H is centered in the point where the cable is attached to the handle. The state of the human is then given by the position of O H and its linear velocity, defined by the vectors p H = [p Hx p Hy p Hz ] ∈ R 3 and v H = [v Hx v Hy v Hz ] ∈ R 3 , respectively, both with respect to F W .

For the sake of designing a human-friendly robot controller, it is important to consider the human's behavior when interacting with the environment. In [START_REF] Ikeura | Cooperative motion control of a robot and a human[END_REF], the authors analyzed the interaction forces during a task in which two humans cooperatively moves an object. It turned out that, for this kind of task, the human's dynamics can be approximated with a mass-spring-damper system. The validity of such a model has been confirmed in other works like [START_REF] Speich | Modeling the human hand as it interacts with a telemanipulation system[END_REF]. In [START_REF] Ikeura | Control of robot cooperating with human motion[END_REF]- [START_REF] Duchaine | General model of human-robot cooperation using a novel velocity based variable impedance control[END_REF], this impedance model has been used as a basis to develop human-robot cooperative or teleoperated manipulation tasks. We consider the human's dynamics as:

m H vH = -g H -B H v H + f c + f g , (1)
where m H ∈ R >0 and the positive definite matrix B H ∈ R 3×3 >0 are the mass and apparent damping, respectively;

f c = [f cx f cy f cz ] ∈ R 3 is the cable force applied to the human at O H ; g H = m H gz W ; g ∈ R >0 is the gravitational constant.
Without loss of generality, we can assume the ground being flat and horizontal. f g is the ground reaction force such that:

• f g x W = f g y W = 0, • f g z W > 0, and • v H z W = v H z W = 0, i.e.
, the human is constrained on the ground.

It follows that f g = f g z W , where f g ∈ R ≥0 is the intensity of the reaction force. Notice that it has to be f c z W < m H g. This prevents the human from being lifted by the robot. Practically, this is unfeasible for standard micro aerial vehicles. The human mass, m H , models the reactivity of the human to external forces. A low mass means a high willingness of following external forces. For a high mass, it is the opposite. The damping effect models the willingness of the human to stay still. Finally, in a mass-spring-damper system, the spring models the desire of reaching a specific point. However, in [START_REF] Ruggiero | Aerial manipulation: A literature review[END_REF] there is no spring term since, in our scenario, the human is not aware of the desired path. The human simply follows the external force applied by the robot through the cable.

Remark 1. The precise modeling of the human body dynamics is out of the scope of this work. Through model (1), we want to capture only the macro navigation behavior of a human when pulled by an external force.

Remark 2. In this work, we focus on the control of the human hand position rather than his/her head position. This is motivated by the fact that a handle integrating positioning sensors like GPS, IMU, and cameras, is much more comfortable and wearable with respect to a fully equipped helmet. Nevertheless, the use of a kinematic model could be employed to estimate the human's head position if required.

The cable is attached from one side to the handle, and to the other side to the aerial vehicle at point O R , which coincides with its center of mass. We define the body frame F R = {O R , x R , y R , z R }, rigidly attached to the robot. Its configuration is described by the position of O R with respect to (w.r.t.) F W denoted by the vector p R = [p Rx p Ry p Rz ] ∈ R 3 . We complete the state of the robot with the linear velocity v R = [v Rx v Ry v Rz ] ∈ R 3 . We neglect the rotational coordinates and relative velocity since it is not relevant for the following derivations.

We assume that the robot is controlled by a position controller (e.g., MPC or SQP-based [START_REF] Nava | Direct force feedback control and online multi-task optimization for aerial manipulators[END_REF]) which minimizes the tracking error with respect to a desired C 2 trajectory, considering the full dynamics of the vehicle, its input and state limits, as well as external disturbances. Thus, we can consider the robot closed-loop translational dynamics as the one of a double integrator plus a tracking error:

vR = u R + δ, (2)
where u R ∈ R 3 is a virtual input to be designed, while δ ∈ R 3 is a bounded variable that takes into account all the tracking errors due to model mismatches and uncertainties, input and state limits, external disturbances, and estimation errors. The model [START_REF] Bartelds | Compliant aerial manipulators: Toward a new generation of aerial robotic workers[END_REF] means that the robot can track any desired acceleration u R with a possibly non-zero error δ. This model is suitable for both multidirectional and unidirectionalthrust platforms. If we consider a multidirectional-thrust platform capable of controlling both position and orientation independently [START_REF] Ryll | 6D physical interaction with a fully actuated aerial robot[END_REF], (2) is a very good model of the closedloop system with δ very small. It is well known that a unidirectional-thrust vehicle is differentially flat w.r.t. to p R and the rotation along the thrust direction (also called yaw angle). Therefore, the system is feedback linearizable and perfect tracking of any desired position trajectory of class C 4 (continous up to the fourth time derivative) can be theoretically obtained. With δ we take into account all possible errors. According to (2), the platform might seem 'infinitely stiff' with respect to interaction forces. However, we shall suitably design the input u R so as to re-introduce a compliant behavior. Afterward, we model the cable that connects the aerial vehicle and the handle, or equivalently the human hand, as a unilateral spring along its principal direction. We define the cable force produced at O H on the handle as:

f c = t c ( l c )l c / l c , (3)
where t c ( l c ) represents the cable internal force intensity and l c = p R -p H . The force produced on the other end of the cable, namely on the robot at O R , is equal to -f c . In practice, the forces involved shall be very small (few newtons) and the distance between the human and the robot can be of few meters only. For these reasons, a thin and short cable can be used which has negligible mass and inertia w.r.t. the ones of the robot and of the human.

Defining by lc ∈ R >0 the constant nominal length of the cable, we say that the cable is: 1) Slack when the distance between the two ends is shorter than lc . In this case, the cable force intensity is zero; 2) Taut when the distance between the two ends is equal or greater than lc . In this case, the cable force intensity follows Hooke's law. In particular, t c ( l c ), is defined as:

t c ( l c ) = k c ( l c -lc ) if l c -lc > 0 0 otherwise , (4)
where k c ∈ R >0 is the constant elastic coefficient.

Remark 3. Notice that t c ( l c ) can be any continuous and differentiable monotonically increasing function such that:

t c ( l c ) ≥ l c + γ, if l c -lc > 0, (5)
for some ∈ R >0 and γ ∈ R. This allows modeling more complex behaviors of the cable during the initial stretching/unstretching. We also consider the system to be always far from the maximum feasible tension.

III. CONTROL METHOD

Since the robot physically interacts with the human, it is important that the interaction is done in a compliant way, so as to always preserve stability and prevent the human to be armed or subjected to uncomfortable forces. For this problem, and more in general for physical interaction problems, a very common solution consists of using an admittance control strategy [START_REF] Haddadin | Physical human-robot interaction[END_REF]. According to this method, the desired motion of the robot is computed as the one of an object with a specific admittance (mass, stiffness, and damping) subjected to the measured external force acting on the robot. In this way one can reshape the apparent mechanical characteristics of the system. We define the robot control input u R as:

u R = M -1 A (-B A v R -f c + u A ) , (6)
where the two positive definite diagonal matrices M A , B A ∈ R 3×3 are the virtual inertia and damping of the robot. u A ∈ R 3 is an additional input that will be defined in the following to achieve the desired control goal. Notice that to implement [START_REF] Ollero | The AEROARMS project: Aerial robots with advanced manipulation capabilities for inspection and maintenance[END_REF], only the state of the robot (p R , v R ), and the force applied by the cable f c are needed. The first can be retrieved with standard onboard sensors, while the second can be measured with a force sensor or estimated by a model-based observer [START_REF] Ryll | 6D physical interaction with a fully actuated aerial robot[END_REF].

It is useful to define the system state vector as 12 and to write the dynamics in a more compact state form as ẋ = f (x, u A ), where:

x = [p H v H p R v R ] ∈ X ⊂ R
f (x, u A ) =     v H 1 m H (-g H -B H v H + f c + f g ) v R M -1 A (-B A v R -f c + u A )     , (7)
and f c is computed as in [START_REF] Tognon | Dynamic decentralized control for protocentric aerial manipulators[END_REF]. For the sake of deriving the control law and proving the asymptotic stability in nominal conditions, we consider a zero modeling error. Later we shall show the robustness of the method when δ = 0. Remark 4. The system has a hybrid nature characterized by two different behaviors according to the state of the cable: 1) When x ∈ X S = {x ∈ X | l c -lc ≤ 0}, the cable is slack and the cable force is zero, i.e., f c = 0. In this condition, the robot and human evolve independently; 2) When x ∈ X T = {x ∈ X | l c -lc > 0}, the cable is taut and the cable force is non-zero, i.e., f c = 0. In this condition, the robot and human interact with each other through the cable force.

When the cable is taut the system appears to be a couple of mass-damper elements connected by a spring. This system is similar to an elastic manipulator where the actuator (the robot) is connected to the end-effector (the human) by an elastic element (the cable). For this system, we want to control the position of the end-effector (human position) by the input force u A generated by the actuator (robot). Given this parallelism, we naturally took inspiration from the stateof-the-art on control of manipulators with elastic joints [START_REF] De Luca | Robots with flexible elements[END_REF].

Without the elastic element, the most natural choice for the design of a linear controller is to apply a simple proportionalderivative (PD) feedback from the end-effector position and velocity. However, considering a single link rotating on a horizontal plane and actuated with a motor through an elastic joint coupling, it has been proven that a feedback action entirely based on the end-effector position and velocity leads to instability, no matter the gain values [START_REF] De Luca | Robots with flexible elements[END_REF].

The most preferred and grounded solution for manipulators with elastic joints is to apply a feedback completely based on the motor variables. If the proportional and derivative gains are strictly positive, then the closed-loop system will be asymptotically stable. Notice that, given a desired end-effector position, the desired motor value is computed inverting the dynamic model whose parameters need to be known.

p H p R p r H p r R f c f r c f r c K H e R u A u A = f r c
Path following To implement this control law, we only need a proportional feedback w.r.t. the robot position. The damping action w.r.t. the robot velocity is already present in [START_REF] Ollero | The AEROARMS project: Aerial robots with advanced manipulation capabilities for inspection and maintenance[END_REF]. Thus, we set

vR = u R u R vR = M -1 A (-B A v R -fc + u A ) u A p r H f
u A = K H e R + f r c , (8)
where

K H = diag (k H , k H , 0) with k H ∈ R >0 , e R = p r R -p R is the robot position error, p r
R is the robot position of reference, and f r c ∈ R 3 is a constant forcing input. In practice, f r c is the desired cable force. Figure 2 explains the working principle of the controller, while Figure 3 shows its block diagram.

IV. EQUILIBRIA AND STABILITY ANALYSIS

In the following, we shall show that, by properly choosing p r R and f r c , the control law (8) can steer the human position to the desired value.

Theorem 1. Let us define p r H ∈ R 3 as the human position reference such that z W p r H = 0. Considering the system (7) under the control law [START_REF] Gioioso | A forcebased bilateral teleoperation framework for aerial robots in contact with the environment[END_REF] where

f r c = f z z W (9) p r R = p r H + f r c 1 k c + lc f r c , ( 10
)
the zero velocity equilibrium x = [p r H 0 p r R 0 ] ∈ X T is asymptotically stable if 0 < f z < m H g.
Proof. Let us first compute the equilibrium state. Plugging (8) into (7) yields the final closed-loop dynamics:

f (x) =     v H 1 m H (-g H -B H v H + f c + f g ) v R M -1 A (-B A v R -f c + K H e R + f r c )     . ( 11
)
Imposing the stability condition into [START_REF] Peshkova | Natural interaction techniques for an unmanned aerial vehicle system[END_REF], i.e., ẋ = f (x) = 0, from the second row we get:

f c = (f g -m H g) z W . ( 12
)
This means that at the equilibrium the cable force, and thus the cable itself, have to be vertical. From the fourth row in [START_REF] Peshkova | Natural interaction techniques for an unmanned aerial vehicle system[END_REF] and replacing [START_REF] Cauchard | Drone & me: an exploration into natural human-drone interaction[END_REF] we obtain

0 = (f g -m H g + f z ) z W + K H e R . (13)
Expanding the components of K H e R along the axes of F W , it is easy to verify that ( 13) is satisfied if:

x W e R = 0 y W e R = 0 f g = m H g -f z , (14)
which implies:

p Rx = p r Rx p Ry = p r Ry f c = f r c . ( 15
)
We recall that the intensity of the ground reaction force should be equal or greater than zero, i.e., f g ≥ 0. Thus, it has to be f z ≤ m H g. Notice that there are no stable configurations for which f c = 0, i.e., such that l c ≤ lc .

To conclude the computation of the equilibrium configuration, we need to calculate the position of the human, p H , and the vertical position of the robot, p Rz . Replacing conditions (15) into (3), after a few simple calculations, we obtain:

  p r Rx p r Ry p Rz   +   p Hx p Hy 0   =   0 0 fz kc + lc   . ( 16
)
Replacing also ( 9) and ( 10), we finally obtain that, at the equilibrium, p H = p r H and p R = p r R . Thus, the only equilibrium state for the system [START_REF] Mersha | On bilateral teleoperation of aerial robots[END_REF]

is x = [p r H 0 p r R 0 ] ∈ X T .
Now we analyze the stability of the equilibrium state x employing a Lyapunov-based analysis. Let us define the following Lyapunov function:

V (x) = V 1 (x) + V 2 (x) + V 0 , (17)
where,

V 1 (x) = 1 2 m H v H v H + v R M A v R + e R K H e R (18)
V 2 (x) = lc(t) lc0 t c (τ )dτ -l c fc (19)
V 0 = f 2 z /k c + 2 lc f z . (20)
We defined l c0 = l c (0) . In the following we shall show that the defined Lyapunov function, V (x), is continuously differentiable and radially unbounded.

V (x) is continuously differentiable because V 1 , V 2 , V 0 are clearly continuously differentiable.
We then show that ( 17) is radially unbounded (also called coercive), i.e., lim x →∞ V (x) = ∞ with x ∈ X . We have that clearly lim x →∞ V 1 (x) = ∞ for x ∈ X . For V 2 , we firstly notice that, thanks to [START_REF] Tognon | A truly redundant aerial manipulator system with application to push-andslide inspection in industrial plants[END_REF], there exist some ∈ R >0 and γ ∈ R such that:

lc(t) lc0 t c (τ )dτ ≥ 2 l c 2 + γ l c . (21)
This implies that:

lim x →∞ V 2 (x) ≥ lim x →∞ l c 2 + γ l c -l c f r c = lim x →∞ l c 2 2 + γ l c - f r c l c = +∞. (22)
This is enough to show that ( 17) is radially unbounded.

In order to prove that V (x) is positive definite, we show that it has a unique global minimum in x, i.e., x = arg min x V (x) and V ( x) = 0. Since V (x) is continuous and radially unbounded, for Theorem 1.15 of [START_REF] Horst | Introduction to global optimization[END_REF], we can say that the Lyapunov function [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF] has a global minimum. Now we can look for this minimum among the stationary points, i.e., the x where the gradient ∇V (x) = 0.

It is clear that

∇V 1 (x) = 0 only if v H = v R = 0, p Rx = p r
Rx and p Ry = p r Ry . Regarding V 2 (x), let us consider its gradient w.r.t. the cable configuration l c . For every x ∈ X S :

∇ lc V 2 (x) = ∂V 2 (x) ∂l c = ∂ ∂l c lc lc0 t c (τ )dτ -f r c . ( 23
)
Let us notice that for the Leibniz integral rule we have that:

∂ ∂l c lc lc0 t c (τ )dτ = t c ( l c ) ∂ l c ∂l c -t c (l c0 ) ∂l c0 ∂l c = t c ( l c ) l c l c = f c , (24)
where we used (3). Plugging ( 24) into ( 23) yields

∇ lc V 2 (x) = f c -f r c . It follows that ∇ lc V 2 (x) = 0 if and only if t c ( l c )l c / l c = f c = f r c . (25)
Condition [START_REF] Lupashin | Stabilization of a flying vehicle on a taut tether using inertial sensing[END_REF]

holds only if a) t c ( l c ) = f r c and l c / l c = f r c / f r c ; or b) t c ( l c ) = -f r c and l c / l c = -f r c / f r c
. In these two cases, the cable produces the same force f r c , but in case a) the cable is stretched, while in case b) the cable is compressed. However, cables cannot be compressed and thus case b) is not feasible for definition. The only solution is:

l c = 1 k c + lc f r c f r c , (26)
which, recalling [START_REF] Yeh | Exploring proxemics for human-drone interaction[END_REF], leads to the conclusion that ∇ lc V 2 (x) = 0 only if p H = p r H and p R = p r R . This completes the proof that ∇V (x) = 0 only if x = x, and thus x = arg min x V (x). Finally, it is easy to verify that V 1 ( x) = 0 and V 0 has been defined such that V 1 ( x) + V 2 ( x) = -V 0 , in order to make V ( x) = 0. This proves that V (x) is positive definite.

We can now compute the time derivative of [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF]. Similarly to [START_REF] Tognon | Dynamics, control, and estimation for aerial robots tethered by cables or bars[END_REF], applying the Leibniz integral rule, we have that:

d dt lc(t) lc0 t c (τ )dτ = t c ( l c ) ∂ l c ∂l c lc = f c lc . ( 27
)
Therefore, the time derivative of ( 17) is equal to:

V (x) =v H (-g H -B H v H + f c + f g ) + v R (-B A v R -f c + K H e R + f r c ) -v R K H e R + lc f c -lc f r c . (28)
After few simple algebraic steps we get:

V (x) = -v H B H v H -v R B A v R . (29)
V (x) is clearly negative semidefinite, in X .

Since V (x) is only negative semidefinite, to prove the asymptotic stability we rely on LaSalle's invariance principle [START_REF] Khalil | Nonlinear Systems[END_REF]. The Lyapunov candidate V (x) is a continuously differentiable function. Let us define a positively invariant set 17) is radially unbounded. Notice that Ω 0 contains only x. Then we need to find the largest invariant set

Ω α = {x ∈ X | V (x) ≤ α with α ∈ R >0 }. By construction, Ω α is compact since (
M in E = {x ∈ Ω α | V (x) = 0}. A trajectory x(t) belongs identically to E if V (x(t)) ≡ 0 ⇔ v H (t) = v R (t) ≡ 0 ⇔ vH = vR = 0.
As we saw previously in the calculation of the equilibria, this is verified only if x = x. Therefore M contains only x. All conditions of LaSalle's principle are satisfied and x is globally asymptotically stable.

To get more hints on the behavior of the system let us analyze it for x ∈ X S (slack cable). Human's and robot's dynamics are independent and the first results to be:

m H vH = -B H v H . ( 30
)
Whatever is the initial condition, lim t→+∞ v H (t) = 0. Notice that the human will stop but his final position depends only on the initial conditions. For the robot dynamics, let us write first the dynamic equation along x W :

m A vRx = -b A v Rx + k H e Rx , (31)
where

m A ∈ R >0 and b A ∈ R >0 are such that M A = diag (m A , m A , m A ) and B A = diag (b A , b A , b A ).
It is easy to verify that the state (v Rx , e Rx ) = (0, 0) is asymptotically stable. The same for the robot dynamics along y W : the state (v Ry , e Ry ) = (0, 0) is asymptotically stable. On the other hand, the robot dynamics along z W results to be:

m A vRz = -b A v Rz + f z . (32)
Whatever is the initial condition, lim t→+∞ v Rz (t) = f z /b A > 0. This means that the robot horizontal position will tend to the desired one, while, along the vertical axes, it will tend to fly up until the cable will become taut. In fact,

d dt l c (t) = v R (t) -v H (t) → f z /b A > 0.
This means that l c will grow until l c > lc , i.e., until the cable becomes taut. After this instant, the robot interacts with the human through the cable and eventually pulls him/her to the desired position. According to the specific initial condition, it might be that the state trajectory might pass from X T to X S several times. However, the previous Lyapunov-based reasoning proofs that eventually the system will converge to x, no matter the initial conditions.

Remark 5. Although the system seems passive for every virtual admittance of the robot, the dynamics of the robot actuators and sensors, as well as the real implementation of the control framework, impose minimum values for the virtual inertia and damping. This is in accordance with the practical intuition that minimum compliance of the interaction is required. Notice that the mechanical flexibility of some elements, e.g., the cable, helps in meeting such requirements.

V. ROBUSTNESS OF THE SYSTEM

Previously, we formally proved that an admittance strategy together with controller (8) asymptotically steer the human position to a desired constant position p r H . This is valid if the human follows the force provided by the robot without exerting other forces except for the damping effect. In the following, we shall show the intrinsic robustness of the controller against time-varying position references, modeling errors, and possible additional human forces (e.g., if he/she wants to stop).

A. Time-varying position reference

We remark that with the control law (8), the resulting final path followed by the human to arrive at the goal only depends on the initial conditions. In a real application, this is not a desired behavior, e.g., when a blind human must be guided to a specific point in an environment with possible obstacles. In this scenario, it is more likely that the human position reference is not constantly equal to the final goal position, but is rather a time-varying path p r H (t) that leads the human to the goal, avoiding obstacles or possible sources of danger. To face this objective, taking into account a possibly time-varying path p r H (t), and time-varying desired cable force f r c (t), for t ∈ [0, T ], we modify the original control law (8) as:

u A (t) = K H e RT (t) + f r T + u A (t), (33)
where e RT (t

) = p r R (T ) -p R (t), f r T = f r c (T ) = f z z W , and u A (t) is such that u A (t) = K H (p r R (t) -p R (t)) + f r c (t) for every t ∈ [0, T ]
. This is a simple way to gather in u A the time-varying desired quantities. It results that u

A (t) = K H (p r R (t) -p r R (T )) + (f r c (t) -f r c (T )).
To ensure that the system remains stable even when the human position reference is time-varying, i.e., for u A (t) = 0, we shall prove that the system is output-strictly passive w.r.t. the input-output pair (u 1 , y 1 ) = (u A , v R ).

Theorem 2. If u A is defined as in (33) for a certain p r H (t) and f r c (t), with t ∈ [0, T ], then system (7) is output-strictly passive w.r.t. the storage function [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF] and the input-output pair (u 1 , y 1 ) = (u A , v R ).

Proof. In the proof of Theorem 1 we already showed that ( 17) is a continuously differentiable positive definite function. Replacing (33) into [START_REF] Mersha | On bilateral teleoperation of aerial robots[END_REF], and differentiating [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF] we obtain:

V (x) = -v H B H v H -v R B A v R + u A v R (34) ≤ -v R B A v R + u A v R = u 1 y 1 -y 1 φ 1 (y 1 ), (35)
with φ 1 (y 1 ) = B A v R . Since y 1 φ 1 (y 1 ) ≥ 0 for every y 1 = 0, we can conclude that the system (7) is output-strictly passive [START_REF] Khalil | Nonlinear Systems[END_REF].

Given the passivity of the system, we have that for a bounded input u A , i.e., bounded reference trajectories, the overall energy of the system remains bounded too. Furthermore, the system stabilizes to a constant value as soon as u A becomes constant again. This means that, while the robot moves following p r R (t) and f r c (t), the overall state of the system will remain bounded. It will then converge to another specific equilibrium configuration when u A becomes constant.

B. Modelling errors

As we mentioned in Sect. II, non-zero model errors might appear due to model uncertainties, non-linear effects, external disturbances, etc. With δ = 0, the system dynamics becomes:

f (x, u A ) =     v H 1 m H (-g H -B H v H + f c + f g ) v R M -1 A (-B A v R -f c + u A ) + δ     . (36)
We shall show that the system remains stable also when δ = 0.

Theorem 3. System (36) under the control law (8) is outputstrictly passive w.r.t. the storage function [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF] and the inputoutput pair (u 2 , y 2 ) = (δ, v R ).

Proof. We can rewrite equations ( 36) and ( 8) as [START_REF] Mersha | On bilateral teleoperation of aerial robots[END_REF] and

u A = K H e R + f r c + M A δ, (37)
respectively. Noticing the similarity between ( 37) and ( 33), the proof follows as the one of Theorem 2.

Given the passivity of the system, we have that for a bounded model error δ, the overall energy of the system remains bounded too. Passivity is a well known robust property, especially w.r.t. model uncertainties. This means that, to ensure the stability of the system, an accurate model of the system is not necessary, as far as the passivity is guaranteed.

C. Additional human force

In Sect. II we modeled the human as a simple massdamper system subjected to the force exerted by the cable only. Nevertheless, during the guidance, the human may want to stop or to deviate from the desired path, e.g., to pick up an object. In these cases, the human will generate additional forces that will modify his/her trajectory. It is important that, even in this condition, the system is guaranteed to be stable and that the state converges to the desired value as soon as the human forces go back to zero.

We define the vector u H ∈ R 3 as the sum of all the forces applied by the human that generate a translational motion. The closed-loop system dynamics, expressed in [START_REF] Mersha | On bilateral teleoperation of aerial robots[END_REF], becomes:

f (x) =     v H 1 m H (-g H -B H v H + f c + f g + u H ) v R M -1 A (-B A v R -f c + u A )     . ( 38
)
To ensure that the system remains stable even when the human forces are not zero, i.e., for u H = 0, we shall prove that the system is output-strictly passive w.r.t. the input-output pair (u 3 , y 3 ) = (u H , v H ). Proof. In the proof of Theorem 1 we already showed that ( 17) is a continuously differentiable positive definite function. Considering [START_REF] Hauser | Maneuver regulation from trajectory tracking: Feedback linearizable systems[END_REF] and differentiating [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF] we obtain:

V (x) = -v H B H v H -v R B A v R + u H v H (39) ≤ -v H B H v H + u H v H = u 3 y 3 -y 3 φ 2 (y 3 ), ( 40
)
with φ 2 (y 3 ) = B H v H . Since y 3 φ 2 (y 3 ) ≥ 0 for every y 3 = 0, we can conclude that the system ( 38) is output-strictly passive [START_REF] Khalil | Nonlinear Systems[END_REF].

Conclusions similar to the ones in Sect. V-A can be drawn. In particular, we have that for a bounded input u H , i.e., bounded human forces, the overall energy of the system remains bounded too. Furthermore, the system stabilizes to a constant value as soon as u H goes back to zero or balance the cable force. This means that while the human moves subjected to a non-zero (but bounded) human force u H = 0, the overall state of the system will remain bounded. It will then converge to another specific equilibrium configuration as soon as u H goes back to zero or balance the cable force.

In particular, considering the case in which the human wants to stop in a certain position p H , it has to be that v H = vH = 0. This means that u H = f c . It is then easy to verify that the robot's position at the new equilibrium will be such that

l c = β + lc k c   k c + k H 0 0 0 k c + k H 0 0 0 k c   -1 β β , (41)
where

β = K H (p r R -p H ) + f r c
. Using a Lyapunov-based argument similar to the one in the proof of Theorem 1, it is easy to show that such zero-velocity equilibrium state is asymptotically stable. This provides a high level of robustness to different human behaviors.

Following the same reasoning of Theorem 2, Theorem 3, and Theorem 4, we can ensure that the system remains stable even when the human position reference is time-varying, there are non-zero model errors, and the human forces are not zero.

Theorem 5. System (38) under the control law u A (t) = K H e RT (t) + f r T + u A (t) + M A δ is output-strictly passive w.r.t. the storage function [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF] and the input-output pair

(u, y) = ([u 1 u 2 u 3 ] , [y 1 y 2 y 3 ] ).
Proof. The proof is trivial and follows the same steps of the proof of Theorem 2, Theorem 3, and Theorem 4.

VI. PATH-FOLLOWING PROBLEM

In Sect. V-A, we shown that even if the human position and cable force of reference are time-varying, the system remains stable. This can be exploited for the problem of following a desired path. In the following, we shall define a strategy to lead the human along the desired path, allowing him/her to stop or slightly deviate from the path without causing instability.

Given a desired human path1 defined by its parametric representation p d H (s) ∈ R 3 , where s ∈ [0, 1] is called parameter, we firstly define the desired cable force along the path f d (s) ∈ R 3 . This is the force that we would like the human feels all along the path. f d (s) works as a feedforward term, i.e., it is the nominal pulling force that drags the human along the path. In standard path-following problems [START_REF] Del Río | A path following control for unicycle robots[END_REF], this is similar to defining the desired velocity along the path. Inspired by most of the related works, we define:

f d (s) = f d xy (s) + f d z , (42)
where

f d xy (s) = f d xy (s) ∇ s p d H (s) ∇ s p d H (s) , f d z = f z z W , (43)
and

f d xy (s) is such that f d xy (1) = 0.
In particular, here we chose a trapezoidal profile that starts with a non-zero value. Notice that the planar pulling force, f d xy (s), is tangential to the path and correctly drive the human along the path. Furthermore, at the end of the path, i.e., for s = 1,

f d (1) = f d z , which guarantees p d H (1) being asymptotically stable if p r H = p d H (1) and f r c = f d (1)
, as it is shown in Sect. IV. Now that we have the desired

maneuver η = {(p d H (s), f d (s)) ∈ R 3 × R 3 | s ∈ [0, 1]
}, considering the control strategy described so far (equations ( 6) and ( 8)), we seek a method to choose p r H (t) and f r c (t) such that p H (t) asymptotically converges to the path p d H (s), i.e.,

lim t→∞ p H (t) -p d H (s (t)) , (44)
for any smooth function s (t) : R → [0, 1] [50]. We first set:

p r H (t) = p d H (s (t)), f r c (t) = f d (s (t)). (45)
We only have to define s (t). A very simple way would be to define a time law such that s (0) = 0 and s (T ) = 1 for a given time T ∈ R >0 , e.g., using a spline or a linear function. However, this would fall into the trajectory tracking category which suffers from some robustness issues. Imagine that the human decides to stop or to slightly deviate from the path. At the same time, the desired reference advances increasing the position error. If not tackled accurately, e.g., with saturations, the increase of the position error might saturate the robot inputs making the system unstable. Even if inputs saturation are properly addressed, once the human decides to follow again the robot, the latter will try to quickly catch up with the moving desired state. The desired path will not be followed anymore, and possible collisions with obstacles may occur.

To avoid this issue, we implement a maneuver regulation approach similar to the one in [START_REF] Spedicato | Aggressive maneuver regulation of a quadrotor UAV[END_REF], [START_REF] Hauser | Maneuver regulation from trajectory tracking: Feedback linearizable systems[END_REF], defining:

s (t) = arg min s∈[0,1] p d H (s) -p H (t) . (46)
At every time t, the reference human position is the one belonging to p d H and closer to the actual human position. f c is computed accordingly. It is clear that employing this solution, even if the human stops, the position error will not grow. In fact, p r H will remain the same. Notice that to have (46) well defined, p d H must be non-intersecting. Furthermore, it must be that f d xy (s) = 0 for all s = 1. This is necessary to make the human always advance along the path.

Replacing the maneuver regulation policy (45) into the original controller [START_REF] Gioioso | A forcebased bilateral teleoperation framework for aerial robots in contact with the environment[END_REF], we obtain:

u A (t) = K H (p d R (s (t) -p R (t))) + f d xy (s (t)) + f d z , (47)
where p d R is computed as in [START_REF] Yeh | Exploring proxemics for human-drone interaction[END_REF] with p d R and f d instead of p r R and f r c , respectively. The overall control action is divided into three parts:

1) the first is a proportional term that tends to bring the robot (and thus the human) to the desired position; 2) the second is a feedforward force that tends to pull the human along the desired path; 3) the third is a forcing vertical force that prevents the cable to become slack. Notice that as soon as s (t) = 1, (47) is equal to [START_REF] Gioioso | A forcebased bilateral teleoperation framework for aerial robots in contact with the environment[END_REF] where p r H = p d H [START_REF] Ruggiero | Aerial manipulation: A literature review[END_REF]. Thus the asymptotical stability of p d H (1) is ensured. The stability of the system for s = 1 is also guaranteed thanks to the passivity of the system. Remark 6. Notice that to implement [START_REF] Haddadin | Physical human-robot interaction[END_REF], the actual position of the human is required. This can be easily retrieved either with some embedded sensors in the handle (like GPS), or inverting (10) using the actual robot position and cable force.

VII. EXPERIMENTAL VALIDATION

A. Testbed setup

The system employed in these indoor experiments is shown in Fig. 1. It consists of a quadrotor aerial vehicle connected by a cable to a handle hold by a human being. The latter is wearing all safety protections including glasses, gloves, and a helmet. The holder, simply built with a 3D printer, is equipped with motion tracking markers to obtain a measure of p H . The experiments are performed indoor employing a Motion Capture System (MOCAP) to allow the tracking of the human hand. Here, the use of a reliable positioning system has also the advantage to enhance human safety. Notice that the proposed experiments still encapsulates the main challenges addressed in the work, i.e., the stability of physical humanrobot interaction. The integration of a redundant sensory suit for safe flights outdoors is not the focus of this manuscript and is left as future work.

The cable connecting the human-handle has a rest-length lc = 1 [m], and a negligible mass (less than 10 [g]). The aerial vehicle is a 1.3 [Kg] custom quadrotor platform equipped with a custom Paparazzi-based flight-controller, four brushless motor controllers (ESCs) running an in-house developped firmware that allows regulating the propeller speed in closedloop [START_REF] Franchi | Adaptive closed-loop speed control of BLDC motors with applications to multi-rotor aerial vehicles[END_REF], and an onboard PC that runs a UKF state estimator, a geometric position controller [START_REF] Lee | Geometric tracking control of a quadrotor UAV on SE(3)[END_REF], and the admittance filter. The state estimator fuses the data coming from the onboard IMU (accelerometer and gyroscope) at 1 [kHz] and from the MOCAP (position and attitude) at 60 [Hz]. The measure of the cable force required in ( 6) is obtained with a modelbased wrench observer [START_REF] Ryll | 6D physical interaction with a fully actuated aerial robot[END_REF]. These components run onboard at 1 [kHz] using the open-source software framework TeleKyb32 . The admittance filter parameters have been set to M A = 0.8I 3 and B A = 2.4I 3 , where I 3 = diag (1, 1, 1). Notice that to implement the controller, the virtual admittance parameters of the human are not needed.

The controller [START_REF] Gioioso | A forcebased bilateral teleoperation framework for aerial robots in contact with the environment[END_REF] and the path-following strategy described in Sect. VI are instead implemented in Matlab-Simulink. The 4). The trajectory has been computed using a simple piece-wise polynomial function.

The experiment is divided into three phases (see Fig. 4): Ph.1: The human stays in a safe place. The robot takes off and goes into an initial point in position-mode. After, the admittance filter is activated with u A = 0. This allows the human safely grabbing the handle; Ph.2 Once the human holds the handle, the proposed guiding control and path-following strategy are activated and the robot pulls the human along the desired path p d H (s); Ph.3 Once the human reaches the final position, the robot will be vertically placed on top of the handle. In this condition, the human feels no horizontal pulling force and can release the handle. The human guiding controller and the admittance filter are disabled. Finally, the robot lands in position-mode.

During the experiment, the human is not blinded for safety reasons. Nevertheless, he/she is asked to look at the ground and not at the robot, focusing on the forces felt on the hand. This is done to encourage the human to be led feeling the forces that the robot is applying through the cable, instead of using it as a visual reference. Furthermore, the human is firstly "trained" on a different path to make him/her comfortable with the experiment, and confident with the reliability of the robot.

B. Experimental results

In Fig. 5, we report the detailed results obtained with one participant of the experimental campaign. In Fig. 5a, we show the desired and actual human path from the top, during Phase 2. We also schematically depict the system configuration at some instants to show how the robot pulls the human along the desired path. In Fig. 5b, detailed quantities of the system along the full experiments are shown. During Phase 1, the robot takes off and goes in the initial desired position. The non-zero estimated force along z W corresponds to the weight of the handle that in this phase is suspended below the robot. The initial oscillations of p Hx and p Hy are also caused by the oscillations of the handle.

Passing to Phase 2, the robot starts moving to stretch the cable and pull the human. This can be seen from the estimated force along the cable. Looking at the human position error e H (t) = p r H (t) -p H (t), we remark that its norm is always below 5 [cm]. Although the error is quite small, what is more important is that the stability of the interaction is always guaranteed, even though the human applies non-zero forces. The experiment additionally validates the passivity of the system which guarantees stability even in non-ideal conditions. One can notice that, during Phase 2, there is an error between the desired horizontal force and the actual one. This is because the robot is moving and the damping plays against the generation of the desired force. Passivity-based methods are known to be very conservative. Practically, this is not a problem. If the human is moving toward the good direction, there is no need to increase the guiding force. On the other hand, if the human stops, the robot will apply the desired force, clearly showing the correct direction to the human. Fig. 6: Experimental results of eight tests with four subjects. On the top: the desired human path, the actual human path for the eight experiments, and the mean human path showed in the x-y plane. On the bottom: the path-following error e P H j for every experiment, and the mean path-following error.

Phase 3 starts when the human does not feel horizontal forces. He/she understands to be arrived at the desired position and so releases the handle. After, the robot lands.

To further validate the proposed system, we tested it with a group of four different subjects, performing the experiment twice. To evaluate the path-following performance, we define the path-following error with respect to the desired path as:

e P H (s) = min t∈[0,T ] (p H (t) -p d H (s)) p d H ⊥ (s) , (48)
where p d H ⊥ (s) ∈ R3 is the normal vector to the path in the point p d H (s), and T ∈ >0 is the experiment final time. Given a point the desired path d H parameterized by e P (s) is the distance between p d H (s) and the human position intersecting the normal to the path in H (s). Notice that this definition is a geometric error; it does not depend on the velocity of the human, neither on his/her trajectory, but just on the geometric path 3 . For each s ∈ [0, 1], we then define the mean path-following error, ēP H (s), and the corresponding standard deviation (std), σ H (s):

) 49
where N is the number of performed tests, and the subscript j indicates the j-th test. Figure 6 shows the global performance. One can notice that ēP H (s) is always below 5 [cm], while the maximum standard deviation is below 10 [cm]. This means that, according to the performed experiments, the human will always pass in a tube with diameter 60 [cm] around the desired path. This information could be used at the motion planning level to compute paths that stay sufficiently away from obstacles. We also remark that every subject behaves in a slightly different way (e.g., following the robot with a higher of lower speed, making stops, applying extra forces, etc.). Nevertheless, the system always maintains stability and provides good tracking performance. We remark that these results do not pretend to be statistically significant. The goal is to provide a global evaluation of the proposed method, which further validates its stability and robustness.

Notice that the human tends to stay always 'outward' the desired curve. This is the effect of the human inertia that the robot controller does not take into account. Furthermore, we noticed that the human might have some difficulties understanding that he/she is arrived, resulting in useless small corrections. These marginal aspects are left to future works.

VIII. CONCLUSIONS

In this manuscript, a system composed by a human physically connected to an aerial vehicle as been studied for the fist time. The human holds a handle that is connected by a cable to an aerial platform. The objective is to make the robot capable of safely guiding the human to a desired position, or along a desired path, by pulling him/her exploiting the tether.

We proposed a control strategy that makes the robot compliant to the human-interaction and allows to exert forces to the human. These forces have been properly designed to drive the human to a desired position. Using Lyapunov theory, we formally proved that the desired human position is asymptotically stable for the considered closed-loop system. Furthermore, we showed that the latter is output-strictly passive which guarantees stability and robustness of the method even for time-varying inputs, non-zero model errors, and for nonzero extra human forces. An additional maneuver regulation strategy has been designed to allow the robot guiding the human along a desired path.

The overall method has been experimentally validated with a group of four subjects. The results show that the system is able to keep the path-following error bounded. The robot correctly drives the human along the desired path, compensating for errors, and guaranteeing the stability.

From a practical point of view, we foresee the integration of onboard sensors for outdoor operations and for the onboard detection and avoidance of obstacles.

From a more scientific point of view, the human model could be enhanced to include the main body and arm elements, together with the heading. This should be considered in the controller to improve the control of the human configuration, i.e., position of the main body, heading, and relative position of the arm. A more precise hybrid position/force controller (like the one in [START_REF] Tognon | Observer-based control of position and tension for an aerial robot tethered to a moving platform[END_REF]) could be applied, in order to finely control the intensity and direction of the force felt by the human, regardless to his/her velocity.

HRI aspects, like the acceptability of the system from the human, were not the focus of this work. However they will be analyzed in the future through a complete user study. A human-aware path planner could be designed to explicitly consider at the planning level a path acceptable by the human taking into account human-preferences in terms of distance to obstacles and dynamics of the motion [START_REF] Khambhaita | Viewing robot navigation in human environment as a cooperative activity[END_REF], as well as adaptation to human pace [START_REF] Fiore | An Adaptive and Proactive Human-Aware Robot Guide[END_REF]. The improvement of the indirect force-based communication between human and robot using additional stimulus should be investigated as well. Impulsive forces or wearable haptic devices could be used to enhance the human-robot interface. From one hand the robot should be able to estimate and then react to human intentions (e.g., start/stop the mission). From the other hand the human awareness should be augmented providing additional information (e.g., distance to obstacles or the desired path).

Fig. 1 :

 1 Fig. 1: Representation of the aerial human-tethered guiding system.

Fig. 2 :

 2 Fig.2: Working principle of the control method. On the left the current state, on the right in opaque, the final desired state.

Fig. 3 :

 3 Fig. 3: Block diagram of the overall control method.

Theorem 4 .

 4 System (38) under the control law (8) is outputstrictly passive w.r.t. the storage function[START_REF] Kruse | Human-aware robot navigation: A survey[END_REF] and the inputoutput pair (u 3 , y 3 ) = (u H , v H ).

Fig. 4 :

 4 Fig. 4: Representation of the experimental phases. On each snapshot the robot is highlighted with a purple circle. Robot and human trajectories are marked with purple and red dashed lines, respectively.

  The desired and actual human path showed in the x-y plane. Four schematic representation of the system along the path are shown. The green and blue circles represent the human hand and the robot, respectively. The black line represents the cable. The red and yellow arrows represent the x-y components of u A and fc, respectively.

  Plots of the human position error (zoomed during the guiding phase), desired and actual human hand position, robot position, and cable force.

Fig. 5 :

 5 Fig. 5: Detailed experimental results of one subject.

The path is considered given and is such that no obstacles are present in its proximity. The minimum distance to the obstacle will be given by the precision of the control method.

TeleKyb3 is freely available at https://git.openrobots.org/projects/telekyb3

Other definitions of the distance between paths can be used. We believe that (48) allows to clearly show the envelope around the desired path in which the human will probably pass.

ACKNOWLEDGMENTS

We thank Gabriele Minini for his contribution on the early stage of this work. We are also grateful to Anthony Mallet for his work on the software framework.


Maud Guglielmino

Pierre Bernhardt

Claire Trocquet

Christophe A Serra

Stéphane Le Calvé
email: slecalve@unistra.fr.

On-line

Keywords: Formaldehyde, microfluidics, annular flow, microdevice, indoor air 35 36

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Formaldehyde is a major pollutant of indoor air due to its multiple sources (materials, combustion, painting, etc.). Several studies [1][2][3] have shown that indoor formaldehyde concentrations are 2 to 15 times higher than those measured outdoor, and may vary typically between 10 to 100 μg m -3 . Casset et al. (2006) have shown that formaldehyde is implicated in allergic diseases, particularly for asthmatic people [4]. Starting from 2004, formaldehyde is considered as a carcinogenic compound for human by the International Agency for Research on Cancer [5]. French recommendations aim at limiting formaldehyde concentrations in public buildings to 30 μg m -3 by 2018. Therefore, sensitive methods and potentially transportable or highly portable instruments capable of monitoring airborne formaldehyde concentration are highly required.

The main analytical methods used for quantification of airborne formaldehyde concentrations are summarized in Table 1. The conventional and reference method [6][7][8][9] uses an active or passive sampling on 2,4-Dinitrophenylhydrazine (DNPH) cartridge for several hours or several days respectively, followed by an HPLC/UV visible analysis which is time-consuming and involves a laboratory treatment with bulky instruments. In addition, this approach does not provide real-time results and only gives an average value of concentration.

Many alternative methods oriented towards real-time and in-field detection, have been developed such as Hantzsch monitor [10][11], Proton-Transfer-Reaction Mass Spectrometry (PTR-MS) [12][13], GC-MS [14], infrared diode laser spectroscopy [9]. However most of these methods rely on bulky apparatus and then are not adapted for indoor air monitoring where portability is a key point (see Table 1). Among the methods which were developed, some used specific reagents that can react with formaldehyde to produce a compound which can be detected either by colorimetry or fluorescence [10,[START_REF] Zheng | Développement d'un analyseur rapide et transportable du formaldéhyde dans l'air[END_REF][START_REF] Sakai | [END_REF][17][18]. Miniature and portable systems based on color change in solid materials, which is conveniently used for the detection, have been developed [19][20]. Nevertheless, autonomic and fully continuous monitoring was not possible since with time the solid detection element got saturated and could not be renewed. In addition, the resolution of such sensors was usually not as good as needed to detect the very low formaldehyde concentrations recommended by the state legislation.

Another type of transportable instrument, based on the uptake and concentration of formaldehyde into an aqueous solution followed by the derivatization reaction at 60-80°C and the fluorescence detection, has also been developed in our laboratory [20][START_REF] Le Calvé | Device and method for determining the concentration of a compound in an aqueous or gaseous phase[END_REF]. Such very sensitive analytical instrument with detection limit lower than 0.5 µg m -3 is considered to be transportable and real-time due to its low weight (8 kg) and its time resolution (10 minutes). However, its high liquid reagent consumption of 1.1 mL min -1 does not provide sufficient autonomy since 100 mL can permit only 1h40 operation time and 10 measurements. In addition, more miniaturization is needed for in-field monitoring.

Gas/liquid flows in large channels have been studied intensively owing to the numerous applications such as mass bubble column reactors [22] or mass transport in pipes [23]. Different flow patterns (e.g. bubbly, stratified, slug, annular, churn etc.) were reported as a function of fluid properties (mainly viscosity and density), liquid and gas phase velocity or pipes dimensions and tilt. In large channel gravity and inertial forces are predominant while in microchannels, like capillary tubes, surface tension and viscous force play a dominant role; thus transitions between the different flow patterns are no longer the same as in case of large channels. Many authors [24][25][26][27][START_REF] Ishii | Thermo-fluid dynamics of two-phase flow[END_REF][START_REF] Rebrov | [END_REF] have reported that in such tiny environment stratified flows were never observed. Triplett et al. (1999) studied the air-water two-phase flow at 25°C in a 1.1 mm ID capillary and observed that the bubbly pattern is obtained when the liquid flow rate is much greater that the air flow rate. When the former is of same order but still greater that the latter, the slug pattern is observed [30]. Then the churn, slug-annular and annular patterns are obtained when the gas flow rate is increased, when the gas to air flow rate ratio reaches 100 and for further higher gas flow rates respectively. Chung and Kuwaji (2004) studied the two-phase flow of nitrogen and water at a temperature no higher than 30°C in 250 and 530 µm ID capillaries [31]. They found that all flow patterns seen by Triplett et al. (1999) [30] were also present in both capillaries [31].

The overall goal of our work is then to develop a formaldehyde analytical method which combines precision, selectivity and fast analysis in a single portable miniaturized instrument which will be more suitable for field measurement with reagent autonomy sufficient for a two weeks continuous monitoring. The work reported in this paper aimed at improving the existing microfluidic system described in a previous publication [32] and which was based on a microfluidic device operating in 3 combined steps, i.e. gaseous formaldehyde uptake into the aqueous solution using an annular gas/liquid flow at room temperature, the derivatization reaction with acetylacetone at 65°C and finally 3,5-Diacetyl-1,4-dihydrolutidine (DDL) detection by colorimetry by means of a Liquid Wave Core (LCW) of 44 cm length and 207 µm ID [32]. More precisely, since the configuration of the annular flow allows a residence time potentially long enough to convert totally formaldehyde into DDL in liquid phase, the first objective was to couple the uptake and reaction steps at 65°C in a single step to reduce the response time of the analytical instrument. As such we will investigate the liquid/gas phase diagram in capillary of different dimensions at high temperature which was to the best of our knowledge scarcely reported in the literature. The second objective was to reduce the reactant consumption of 35 µL min - 1 previously reported [32] and therefore increase the microdevice autonomy.

Materials and method

Chemicals

The reaction between acetylacetone solution (Fluoral-P) and formaldehyde producing DDL was first investigated via a Hantszch reaction [33] in details. Acetylacetone solution (0.01 M) was prepared by mixing 0.3 mL of acetic acid (100%, Merck), 0.2 mL of acetylacetone (99%, Merck), 15.4 g of ammonium acetate (98%, Sigma-Aldrich) in 200 mL Milli-Q water (18.2 MΩ.cm at 25°C, Millipore). The reagent bottle was equipped with a DNPH cartridge to purify the air entering the vial so as to avoid any contamination of the reagent with formaldehyde present in the ambient air.

Gaseous formaldehyde mixtures, whose concentration was varied from 0 to 154.4 µg m -3 (0 to 5.1x10 - 6 M once quantitatively transferred into the liquid phase), were prepared by means of a homemade gaseous formaldehyde source whose development and optimization was reported in a previous work [START_REF] Le Calvé | Device and method for determining the concentration of a compound in an aqueous or gaseous phase[END_REF]. Briefly, this source generates gaseous formaldehyde by applying a controlled gas flow (5-20 NmL min -1 ) in a microporous tube immersed in a concentrated formaldehyde solution of 0.0925% regulated at 10°C by a Peltier module. This enriched gaseous formaldehyde stream is then diluted with a controlled flow (500-2000 NmL min -1 , Messer) of pure air (> 99.999%, total VOCs < 50 ppb, HCHO < 1 ppb). The concentrated formaldehyde solution of 0.0925% was prepared by mixing a commercial formaldehyde solution (37% in water, Sigma-Aldrich) with Milli-Q water (18.2 MΩ.cm at 25°C, Millipore). Finally, the generated gaseous formaldehyde concentrations were precisely quantified using the conventional sampling method on DNPH cartridges followed by HPLC/UV analysis already described elsewhere [6,34]. The overall relative uncertainties on the measured gaseous concentrations were in the range 8-12% and were calculated from uncertainties of flow meters of the formaldehyde source, the gas sampling and HPLC analysis of DNPH cartridges.

For liquid calibration, formaldehyde solutions in the range 0 -207.6 μg L -1 (0 -10.2 10 -6 M) were prepared by mixing a commercial formaldehyde solution (37% in water, Sigma-Aldrich) with Milli-Q water (18.2 MΩ.cm at 25°C, Millipore). They were subsequently diluted by half in a solution of acetylacetone 0.02 M to directly obtain formaldehyde concentrations ranging from 0 to 153.8 μg L -1 (0 to 5.1 10 -6 M) in a solution of acetylacetone 0.01 M.

Setups and experimental conditions

Overall setup for detection of gaseous formaldehyde

The setup used for the quantification of gaseous formaldehyde is represented in Figure 1. It consisted in three sequential steps: 1) fluid feeding lines, 2) gaseous formaldehyde uptake into the aqueous solution using an annular gas/liquid flow at 65°C and simultaneously derivatization reaction with acetylacetone at 65°C and finally 3) DDL detection by colorimetry.

The aqueous solution of acetylacetone (0.01 M) was pumped at a flowrate of 5 µL min -1 thanks to a peristaltic pump (Ismatec, Reglo Lab) while a gas pump (NMP 850 KTDC, KNF) and a gas flow controller (low P flow, 0-100 NmL.min -1 , Bronkhorst) ensured the constant flow rate (5 NmL min -1 ) of the gaseous formaldehyde mixture. The uptake arrangement was composed of a 1/16" PEEK T-junction (P-728-01; Upchurch) paired with two hydrophilic fused co-axial silica capillaries (Polymicro). The gas mixture flowed through the inner capillary (100 µm ID x 150 µm OD) while the gas/liquid flow exited the T-junction through the outer capillary whose size was varied (320, 430 or 530 µm ID x 420, 550 or 680 µm OD). The gas flow rate was thus 3 orders of magnitudes higher that the liquid flow rate which under a specific flow rates ratio gave rise (cf. §3.1) to the formation of an annular flow as illustrated in the inset of Figure 1. Due to a very high Henry's law constant of 5020 M atm -1 [35] and a large interface area between gas and liquid phases promoted by the annular flow, the formaldehyde was transferred from the gas to the liquid phase. Both uptake of gaseous formaldehyde and derivative steps occurred simultaneously at 65°C in order to reduce the response time compared to two sequential steps as reported in our previous work [32]. Furthermore, it also allows increasing the compactness of the setup. Finally, operating at 65°C speeds up the conversion of formaldehyde to DDL for reproducible quantification of formaldehyde.

The gas was removed in flow by means of a microporous tube (µPT) and the resulting gas-free liquid phase containing the DDL was transferred to the colorimetric cell through a 250 µm ID x 1/16" OD PEEK capillary. This cell was composed of 4 different parts (cf. Figure 1): i) a Liquid-Core-Waveguide (LCW, BioGeneral), 400 µm ID and 5 cm length, with an inner coating of Teflon® AF 2400 allowing the light to travel through the liquid solution by total reflection, the LCW was wrapped in an aluminum foil to avoid any external parasite light; ii) 2 homemade one-inlet/two-outlet connectors with low dead volume and made of black polymeric material to avoid any parasite light; one port of both T-connector was used to connect the LCW, another port was used for the inlet and outlet (waste) of the DDL solution, finally the last port was used for a SMA adapter to connect in a straight manner the 1500 µm optical fibers (IDIL Fibres Optiques); iii) a Deuterium-halogen light source (DH-2000, Ocean Optics) and iv) a minispectrometer (C10082CAH, Hamamatsu).

Setup for gas/liquid phase diagram determination

To determine the flow pattern of the gas/liquid flow inside the outer capillary at 65°C, the setup represented in Figure 2 was used. A syringe pump (Phd 2000, Harvard Apparatus) was used to pump a fresh acetylacetone solution (0.01 M) through a TEFLON tubing (300 µm ID x 1/16" OD). Another syringe pump was used to inject air through another TEFLON tubing (810 µm ID x 1/16" OD). Then the two fluids entered a temperature controlled oven, the lengths of both tubings were set to 25 cm and 110 cm respectively to insure a residence time long enough so that the fluids reached 65°C at the oven's exit. Then they flowed through the same aforementioned uptake arrangement detailed in the inset of Figure 1 for which the same three capillaries were investigated (320, 430 or 530 µm ID x 420, 550 or 680 µm OD). 10 cm downstream to the T-junction, an optical microscope (Eclipse TS 100, Nikon) was placed above the outer capillary to take pictures and record the flow pattern thanks to a CCD camera (PIKE F-032C, Allied Technology) and an imaging software (Hiris, R&D Vision). Thus to determine the flow pattern inside the outer capillary, the liquid flow rate was varied from 5 to 35 µL min -1 while the gas flow rate was varied from 5 to 35 mL min -1 . Since the gas flow rates are quite low, the compressible effect can be neglected, the Mach number at 65°C was calculated to be equal to 6%. Thus all the gas flow rates reported for the phase diagrams are expressed as the flow rate induced by the syringe pump at 20°C (room temperature). Upon modification of an operating parameter, at least 10 min were elapsed before to record the flow pattern such as the new steady state was reached.

Setup and experimental conditions for the gaseous or aqueous formaldehyde calibration

The length of the uptake cell was 150 cm whereas the internal diameter and length of the colorimetric detection cell were 400 µm and 5 cm, respectively. The liquid and gas flow rate were fixed to 5 μL min - 1 and 5 NmL min -1 , respectively. In order to evaluate the uptake yield, both gas and liquid calibrations were then performed using the setup detailed in Figure 1.

In fact, the experimental apparatus for carrying out liquid phase formaldehyde calibration is similar to the one used for the calibration of the concentrations of gaseous formaldehyde. The only two differences are as follows: 1) synthetic air free of formaldehyde replaces the gaseous formaldehyde mixtures to keep exactly the same residence time in the uptake cell and 2) acetylacetone solution 0.01 M is replaced by a mixture of acetylacetone doped with a known concentration of formaldehyde prepared as detailed in section 2.1.

Results and discussion

Gas/Liquid phase diagram

To determine the combination of gas and liquid flow rates that provide an annular flow, the acetylacetone/air phase diagram was determined experimentally following the procedure described in section 2.2.2. Depending of the flow rates of both liquid and gas phases, up to 4 different flow patterns were observed, namely slug, slug-annular, annular and chaotic flows as illustrated in Figure S1. The phase diagram for the 530 µm ID outer capillary at 65°C is presented in Figure 3 while results for the two other capillaries (430 and 320 µm ID) are reported in Figure S2A&B respectively. Whatever the capillary ID, the slug patterns were found in the top left part of the phase diagram, i.e. when the gas flow rate was low and liquid flow rate was high. When the gas flow rate was increased, an intermediate region appeared and the flow adopted a slug-annular pattern. Finally when the gas to liquid flow rate ratio was around or greater than 1,000, a large annular region covered the whole right hand part of the phase diagram. However for the highest capillary ID (530 µm) a chaotic pattern was observed in the bottom right corner of the phase diagram, i.e. for the lowest liquid flow rates and the highest gas flow rates conversely to the two other capillary dimensions (Figure S2A&B). Compared to the work of Chung and Kuwaji (2004) conducted with nitrogen and air at temperatures ranging from 22.9 to 29.9 °C in a 530 µm ID capillary [36], we did not observed the bubbly and churn pattern just because we kept the liquid flow rate quite low in order to increase the autonomy of our analytical microdevice, the maximum gas and liquid flow rates were set to 35 mL min -1 and 35 µL min -1 respectively.

In a previous paper [32], we investigated the flow pattern for an outer capillary of 530 µm ID x 680 µm OD at 23°C. Effect of temperature (23°C vs. 65°C) on the transition lines between the different flow patterns is presented in Figure S3. It is observed that when the temperature increases, the transition between slug-annular and annular patterns is shifted toward lower gas and higher liquid flow rates. Same observation can also been drawn for the transition between slug and slug-annular patterns. Finally, the phase diagram at 65°C exhibits a chaotic pattern which is not present in the phase diagram at 23°C. These observations can be explained by the change in viscosity of the liquid and gas when the temperature increases from 23°C to 65 °C (Table S1). While the viscosity of liquid decreases by 57%, the viscosity of the gas is increasing by 13%. As a consequence, the slug-annular flow is shifted to the left hand part of the diagram as the gas is exerting on the liquid a higher shear force resulting in the draining of the liquid film along the wall of the capillary. Thus, the annular region is expanding. For the same reason, the transition between slug and slug-annular is also shifted toward the left hand part of the diagram, therefore the slug region is reducing. Appearance of the chaotic region in the bottom right region of the phase diagram at 65°C may result from the combined effect of high gas flow rates and increased gas viscosity which at some point disturbs the liquid film flowing along the wall of the capillary (see figure S1).

Thus, operating at 65°C still allows working in an annular flow regime for the same range of liquid and gas flow rates as for 23°C but gives the opportunity to promote a faster derivatization reaction and to reduce the analytical response time.

Influence of the capillary ID on the uptake efficiency

The effect of the capillary ID used in the uptake cell was studied using successively capillaries of different internal diameters (320, 430 and 530 µm) using the same experimental conditions defined in section 2.2.3. For each capillary, both gas and liquid calibrations were conducted in order to determine the uptake yield of gaseous formaldehyde into the solution (see section 3.5).

Gaseous formaldehyde calibration

The calibration of the absorbance as a function of the concentration signal was performed for each uptake capillary cell, i.e. 320, 430 and 530 µm ID. Figure 4 shows that the absorbance signal increases linearly (0.995 <R² <0.998) when the concentration of gaseous formaldehyde increases from 0 to 154.4 μg m -3 (0 to 5.1×10 -6 M) whatever the capillary ID. The absorbance uncertainties were estimated to be the standard deviation calculated from 3 or 4 consecutive measurements and varied between 3 and 6.5 %. They correspond to the overall analytical system including the gaseous formaldehyde generation and the analytical instrument itself. In addition, Figure 4 shows that the higher the capillary ID, the higher is the slope, from 2.47×10 4 M -1 for 320 µm ID to 3.60×10 4 M -1 for 530 µm ID. These results may suggest that uptake yield could increase with the capillary ID. In order to subsequently determine the effective uptake efficiency, the main abscissa of the figure 4 corresponds to the theoretical aqueous formaldehyde concentration trapped (in units of mol L -1 ) in the case of a full uptake from gas phase.

Liquid formaldehyde calibration

As observed for the gaseous formaldehyde calibration, Figure 5 shows that the absorbance increases linearly (0.994 <R² <0.996) when the liquid formaldehyde concentration increases from 0 to 153.8 μg L -1 (0 to 5.1×10 -6 to M). The standard deviation calculated from 3 or 4 consecutive measurements ranged between 2 and 5 %.

Similarly to the gaseous formaldehyde calibration curves shown in Figure 4, the higher slope is obtained for the highest capillary ID (see Figure 5). According to the Beer Lambert law and considering a 5 cm pathlength, the slope can be used to extract the molar extinction coefficient  of DDL. In our case, the  values are the following: 4176 ± 406 M -1 cm -1 for 320 µm ID, 5120 ± 430 M -1 cm -1 for 430 µm ID and 7216 ± 611 M -1 cm -1 for 530 µm ID. The latter value is consistent with that determined from our previous work using a colorimetric cell LCW of 207 µm ID and 44 cm length (ε = 7320 ± 610 M -1 cm - 1 [32] and in good agreement with the literature data (ε = 7500 M -1 cm -1 [11]). These results suggest therefore that the residence time and consequently the reaction time inside the capillary uptake cell permits a full conversion of formaldehyde into DDL for a 530 µm capillary ID but it is not sufficient for both 320 µm and 430 µm capillary IDs. Indeed, in an annular flow, the thickness of the liquid film decreases with the decrease of the internal diameter of the capillary [37]. For a fixed liquid flow rate (here 5 μL min -1 ) and a given cell length (here 150 cm), this implies an increase in the liquid linear velocity and thus a shorter residence time of the liquid. Therefore, it is more relevant to use the biggest capillary.

Uptake yield

The joint gaseous and liquid formaldehyde calibrations allow the determination of the uptake yield for the 3 investigated uptake cells characterized by their capillary ID. The uptake yield is defined as the ratio between the gaseous calibration slope multiplied by 100 and the liquid calibration slope obtained in the same experimental conditions, i.e. the same liquid and gas flow rates. The formaldehyde uptake yields obtained are the following: 118 ± 22 % (320 µm ID), 107 ± 18 % (430 µm ID) and 100 ± 16 % (530 µm ID) where the uncertainties are calculated from the uncertainties slopes (2). This shows that whatever the capillary ID used, the uptake efficiency should be total with an annular flow, or at least almost total if we consider the errors bars.

Analytical performances

As demonstrated above with the determination of , it appears that the derivatization reaction of formaldehyde to DDL is only complete for a capillary with an inner diameter of 530 µm. Consequently, the analytical performances of the device described in section 2.2.1 were evaluated for an uptake capillary of 530 µm ID and 150 cm long, a colorimetric cell made of 5 cm LCW using liquid and gas flows of 5 µL min -1 and 5 NmL min -1 , respectively.

Linearity range, detection and quantification limits

The gas calibration curve, shown in Figure 4, demonstrates that the response of the new analytical method is linear (R² = 0.998) for gaseous formaldehyde concentrations ranging from 0 to 154 μg m -3 (0 to 5.1 10 -6 M) which corresponds to typical concentrations found in indoor air.

The gas calibration curve was used to determine both limit of detection (LOD) and limit of quantification (LOQ) of the device in the gas phase. From the lowest formaldehyde concentration investigated, i.e. about 20 µg m -3 , the detection and quantification limits were estimated to be 1.8 μg m -3 (S / N = 3) and 6.1 μg m -3 (S / N = 10), respectively.

Repeatability and reproducibility

For both repeatability and reproducibility measurements, the zero of the measurement was performed every 6 hours to prevent any slight baseline drift due to the possible reagent contamination or slight drift of the UV lamp energy.

A series of 10 measurements was performed using the analytical method developed above and under the optimum conditions determined previously, i.e. 5 µL min -1 and 5 NmL min -1 for liquid and gas flow rates respectively, with the same operator using the same sample. The formaldehyde concentration generated by the source was 51.7 μg m -3 . The corresponding results are shown in Figure 6. From these 10 measurements, the Relative Standard Deviation (RSD) is determined equal to 2.7%. This value, less than 5%, means that the repeatability of this novel analytical method is extremely satisfactory.

3 series of 5 measurements each covering one full week were then conducted, with the same operator using the same gas formaldehyde concentration, on three different days. The results are shown in Figure S4. The RSD determined from all of these measurements is equal to 3.6%. As for repeatability, this RSD value being less than 5% demonstrates the very good reproducibility of the new microfluidic analytical method.

Comparison with available techniques

The analytical performances of our prototype have been compared in Table 1 with those reported in the literature [9-10, 12, 14-16, 18, 38-39] for other methods and apparatus, in terms of limit of detection, time resolution, possible interferences, liquid reagent consumptions when it was relevant and weight as a portability indicator. All the chromatographic and spectroscopic methods have been already discussed in the introduction. In addition, they cannot be considered as neither transportable nor portable so that no more comparison is given here.

Among the chemical methods which are potentially portable or at least transportable, the colorimetric microfluidic device developed in this work appears to be the most portable regarding its weight and the most autonomous considering its very low liquid reagent consumption of 5 µL min -1 . In addition, the use of only one reagent makes it simpler for operation. Its time resolution of 1-2 minutes and its sensitivity of 1.8 µg.m -3 permit to rapidly monitor changes in formaldehyde concentration in air and allow accurate measurements below the threshold value of 30 μg m -3 .

Conclusions

Gas/liquid phase diagrams at 65°C for liquid and gas flow rates ranging from 5 to 35 μL min -1 and from 5 to 35 mL min -1 respectively were experimentally drawn for different capillary IDs (320, 430 and 530 µm) and revealed up to four different flow patterns: slug, slug-annular, annular and chaotic. This latter pattern was only found for the highest gas flow rates investigated in the 530 µm ID capillary. Compare to similar work carried out on nitrogen-water two-phase flow in a 530 µm ID capillary, absence of bubbly and churn patterns results from the extremely low liquid flow rate we used so as to maximize our microdevice autonomy. In our system, the annular flow pattern was obtained for gas to liquid flow rate ratio around or greater than 1,000. As such we have identify the lowest eligible flow rates to obtain an annular flow and reduce the reagent consumption, i.e. liquid and gas flow rates of 5 μL min -1 and 5 NmL min -1 respectively.

In addition, both liquid and gaseous calibration curves showed a linear response of the analytical method regardless of the capillary ID for formaldehyde concentrations representative of those found in indoor air. However, the derivatization reaction was complete only for a 530 µm capillary ID, the capillary length being not long enough for the other tested capillaries.

Using the 530 µm capillary ID, the detection and quantification limits were then estimated to 1.8 and 6.1 µg m -3 respectively for a LCW pathlength of 5 cm. From a purely analytical point of view, a longer acquisition time of the spectrometer could permit to reduce the noise and consequently to improve the LOD and LOQ. Another possibility could consist in increasing the pathlength for example from 5 to 20 cm. However, the current LOD of this novel microdevice is sufficient enough to achieve quantitative measurements in indoor air to meet the regulation which fixes a threshold value of 30 μg m -3 by 2018. These results are very promising and could still be improved in the near future by using the fluorescence detection of DDL as previously reported by Le Calvé et al. ( 2012) [START_REF] Le Calvé | Device and method for determining the concentration of a compound in an aqueous or gaseous phase[END_REF].

The use of this new device achieves both objectives, namely a shorter response time and greater autonomy. The acetylacetone consumption was reduced by a factor of 10 compared to the previous device since the liquid flow rate decreased from 35 to 5 µL min -1 . For a given flow rate of 5 µL min -1 , the gain in response time was estimated about 4 minutes, i.e. the time required for ensuring full conversion of formaldehyde to DDL in the oven (3 minutes) and for passing through an additional microporous tube placed after the oven (1 minute), the new response time being estimated to 6 minutes. The corresponding autonomy with 100 mL of acetylacetone solutions is estimated to 13.9 days which is significantly longer than those obtained in previous works, i.e. 2.0 days [32] or 1.7 hours [START_REF] Le Calvé | Device and method for determining the concentration of a compound in an aqueous or gaseous phase[END_REF].

Finally, the development performed in this work has contributed to obtain a highly portable formaldehyde microanalyser weighting less than 4 kg and having an autonomy of two weeks with an acetylacetone reservoir of 100 mL for a continuous monitoring.

Acknowledgement

This work was supported by ADEME and Region Alsace, REALISE, the program PRIMEQUAL and ANR (ANR-11-ECOT-0013).