Vincent Lebastard
email: vincent.lebastard@emn.fr.

Frédéric Boyer

Sylvain Lanneau

Reactive underwater object inspection based on artificial electric sense

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In spite of many recent successes in underwater robotics, there is still no robot actually capable of operating in highly confined spaces in turbid waters [Lane, 2012].

These environments require manoeuvrability and sensing abilities that exceed the capacities of today's technologies. Though many efforts have been made in recent years to design new solutions for manoeuvrable underwater navigation from reconfigurable AUVs [Lane, 2012] to bio-mimetic fish like robots [START_REF] Triantafyllou | Hydrodynamics of fishlike swimming[END_REF], the issue of sensing in these difficult conditions has had less attention from a research community that has traditionally been more oriented toward vision and sonar [Lane, 2012]. One promising approach takes inspiration from a group of fish species who posses "active electric sense". As Lissman and al. [START_REF] Lissmann | The mechanism of object location in gymnarchus niloticus and similar fish[END_REF] discovered in the 50s, active electric fish perceive their immediate environment by measuring with a dense array of transcutaneous receptors, the perturbations of an electric field generated by an Electric Organ (EO) in the tail, that is polarized with respect to the rest of their body [START_REF] Der | Non-visual orientation and communication by fishes using electrical fields: A model system for underwater robotics[END_REF]. To use a visual metaphor, this field "lights" the close objects, projecting a dense electrical image of them on the skin which acts as an electric "retina" [von der [START_REF] Der | Electric fish measure distance in the dark[END_REF]]. However, unlike vision, electric sense has no pre-reception mechanism comparable to the lens of the eye [Caputi, 2004]. The electric images projected onto the fish skin are fundamentally blurry and difficult to interpret [START_REF] Der | Non-visual orientation and communication by fishes using electrical fields: A model system for underwater robotics[END_REF]. Beyond these biophysical difficulties, a simple physical analysis of the electric responses of the polarized objects shows that their position and intrinsic geometry (shape, size) are intricately related in the measurement of the transcutaneous currents projected onto the skin [Rasnow, 1996]. Despite these difficulties, electric fish are able to perform complex cognitive tasks such as shape recognition [START_REF] Der Emde ; Von Der Emde | Active electrolocation of objects in weakly electric fish[END_REF]. One reason why the fish can do this is that they intensively move their bodies to extract useful information from these blurry images [START_REF] Hofmann | From static electric images to electric flow: towards dynamic perceptual cues in active electroreception[END_REF]. Electric fish have developed stereotyped locomotion patterns named probing motor acts (PMA) that they perform during object inspection [Toerring andBelbenoit, 1979, Toerring andMoller, 1984]. Three of these patterns are illustrated in figure 1. The first pattern (1) is named "stationary probing". In this behavior the fish approaches the object in straight line then stops suddenly when its head has reached a certain distance from it. The second behavior is named "lateral va et vient". This consists of forward and backward swimming movements at a constant distance from the object. The third behavior is termed "circling probing", which consists of a forward circling movement, whereby the fish swims at a constant distance around the object. As remarked in [START_REF] Toerring | Locomotor and electric displays associated with electrolocation during exploratory behavior in mormyrid fish[END_REF], the characteristic distances observed in each of these three behaviors are independent of the object's intrinsic properties (shape, size and material). [Toerring andBelbenoit, 1979, Toerring andMoller, 1984])

Since the 2000s a few roboticists have started to develop sensor technologies and algorithms inspired by electric fish with the aim of equipping a new generation of AUV capable of navigation and perception in constrained spaces and turbid waters [MacIver and Solberg, 2001, MacIver et al., 2004, Silverman et al., 2012, Porez et al., 2011, Lebastard et al., 2010, Boyer et al., 2012, Chevallereau et al., 2014]. Using various measurement principles [START_REF] Silverman | Location and orientation estimation with an electrosense robot[END_REF], Servagent et al., 2013], several robotics issues have been tackled from object localisation and recognition [START_REF] Lebastard | Underwater robot navigation around a sphere using electrolocation sense and kalman filter[END_REF] to reactive navigation [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF], Boyer et al., 2015]. Though these first results are encouraging we are still far from equalling the fishes' performance. One reason for this is that compared to the fish, which can build dense electric images, our sensors can only provide a few measurements collected by a sparse array of electrodes [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF]. To overcome these limitations, one promising approach consists of coupling action and perception into low-level sensory loops able to reproduce the fish PMA with the aim of enriching the useful information available from a sparse receptor array.

This article describes attempts to reproduce the basic PMA illustrated in figure 1 using a motion-controlled sensor that mimics active electric sense. These PMA make use of a global fully autonomous strategy devoted to the inspection of electrically contrasted objects [Boyer and Lebastard, 2012]. The strategy begins with a primary behavior that consists of seeking any electrically contrasted object in an unknown scene, as introduced in [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF]. A stopping phase, inspired by the stationary probing act (1), a clearance phase inspired by the lateral "va et vient" probing (2) and a circling probing phase (3) are then described. The whole strategy is suited to the inspection of any electrically contrasted object in an unknown environment. Furthermore, it works in real time and does not require any prior model of the environment. This is essential since modelling the electric response of arbitrary shaped objects can only be performed with heavy finite elements numerical solvers which are fundamentally unsuited to the computational capabilities requirements of underwater robotics. We have presented the approach in [Boyer and Lebastard, 2012], but in a very empirical way, and with no reference to the PMA nor detailed analysis of the different behaviors. In the present paper, we further analyse each phase of the entire strategy using models, simulations and experiments. This exhaustive approach has produced further insights into each of the phases, and open new perspectives and issues for the application of artificial electric sense to underwater robotics.

The article is structured as follows. In section 2 we present our sensor technology. In section 3 the experimental set up on which our control strategy has been developed and assessed is described. In section 4, the electric model of the slender sensors in presence of small ellipsoidal objects is presented and assessed in section 5. In section 6 a control strategy optimized for object inspection is presented. Experiments based on this control law are detailed in section 7. Section 8 offers a conclusion and perspectives for further research.

Sensor

The first sensing system inspired by electric sense was designed in [START_REF] Solberg | Active electrolocation for underwater target localization[END_REF]. It was composed of four point electrodes, two of them being set under voltage, while the voltage between the other two was measured, supplying data to a particle filter algorithm designed to estimate the location of spherical objects. Since then, other sensor designs have been developed using either a U -U [START_REF] Bai | Sensing capacitance of underwater objects in bio-inspired electrosense[END_REF] or U -I [Boyer et al., 2012, Servagent et al., 2013] strategy. The first letter designates the emission (here a voltage U), the second, the measurement variable. In all cases, the sensor is an insulating axisymmetric shell on which a set of metal electrodes are arrayed with bilateral (left-right) symmetry. In both cases, the electric field is generated by setting a voltage on at least two electrodes in contact with the water. Though both techniques (U -U and U -I) share this common emission principle, in the U -U mode, the other electrodes are paired floating potential electrodes between which the voltage is measured. While in the U -I mode, all of the electrodes except the emitter are grounded, and the currents that flow across each of them are measured. A set of sensors based on the U -I measurement principle, named "slender probes" from their high aspect ratio (length/thickness), have been presented in [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF] and are currently being used to study electric sense. The probes' macro-electrodes are rings or hemispheres numbered E i (i = 1, 2...n) from the tail to the head. Figure 2 shows one of these probes with 3 macro-electrodes E i=1,2,3 . Except for the emitter E 1 , located in the tail, each of E i is divided into a pair of two identical left-right measurement electrodes electrically isolated and named e il and e ir (l meaning "left" and r meaning "right"), such that E i = e il ∪ e ir , for i = 1, 2, 3. Electrode E 3 is termed the "headmacro-electrode" or more concisely the "head-electrode", E 2 , the "neck-electrode", and E 1 , the "tail-electrode", or more simply "the emitter". The probe's overall length is 22cm, and its diameter is 2 cm, yielding an aspect ratio ≃ 10. Using the standard electronics described in [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF], this sensor obtains a sensitivity of about ±0.2% of the emitting current without object. In all the subsequent developments we used the sensor of figure 2 exploiting the head and neck electrodes. The vertical beam attached to the probe is used to control the motion of the probe as detailed in the next section. Once set under a voltage vector U = (U, 0, 0) T applied to (E 1 , E 2 , E 3 ), the response of the sensor with no object in the scene is given by the simple matrix model:

I (0) = C (0) U, (1)
where

I (0) = (I (0) 1 , I (0) 2 , I (0)
3 ) T denotes the vector of the current flowing across E 1 , E 2 , and E 3 respectively, while C (0) is the conductance matrix of the scene with no object, which depends on the sensor's geometry and the surrounding water's conductivity, which is assumed to be homogeneous and equal to γ o . Note that this matrix, which plays a crucial role in the subsequent inspection strategy, can be measured initially during a preliminary calibration or alternatively by using a (boundary) finite element (BEM) simulation. For instance, a BEM simulation for the 3-electrode probe in figure 2 yields the following expression:

C (0) = γ o 100    5.8162 -2.9179 -2.
where γ o is the conductivity of the surrounding water, which we assume to be homogeneous, isotropic and constant over time. Note that, due to the sensor's symmetries, the electric currents flowing across the sub-electrodes e il,r are simply given by :

I (0) 2 = 2I (0) 2r = 2I (0)
2l and I

(0) 3 = 2I (0) 3r = 2I (0)
3l when there is no object in the scene and (1) contains all the information that the sensor can give in this case. Not also that we do not measure the current flowing across the emitter E 1 , but reconstruct it from the conservation law of currents I 1 = -I 2 -I 3 . Finally, with our sensitivity, to recover the fish range which is about one and a half times the total body length [von der Emde, 1999], we need to set U = 5V , as this is done in all the following tests.

Experimental setup

An automated test bench has been built to test and validate our sensors and algorithms (see figure 3). This test bench comprises a tank of one cubic meter side with insulating walls and a three-axis cartesian robot (gantry). The tank is filled with tap water whose conductivity varies from 350 to 420µS/cm. The gantry is fixed on top of the tank and allows controlled translation along the x and y axes. The orientation in the (x, y) plane can be controlled with a yaw-rotation stage. The three axes are motioncontrolled with the dSpace system using Matlab/Simulink software. The maximum speed is 0.3m/s for both x and y translations and 13.5rpm for rotation. The height of the sensor in the tank can be adjusted by using a rigid glass epoxy fiber tube. This vertical tube provides a conduit for the connecting wire that passes the signals coming from the electrodes to the electronic board. The signal is first amplified and filtered in the board's analog chain before passing to a 16 bit ADC Dspace card (DS2004) for digitization. The Dspace card (DS2004) can convert 16 channels simultaneously with a resolution of 0.3mV/bit (LSB) and a maximum conversion speed of 1.25 MHz. Our test bench is described in more detail in [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF].

Model

We present a model for the control of a probe in this section. This includes a model of locomotion (section 4.1) and a model of electric perception (section 4.2). We will consider scenes in which a single small ellipsoidal object has been placed, and then more complex scenarios from experiments.

Probe kinematic model and parameterization of a scene

Throughout the article, we consider the probe described above, with its pose parameterized as indicated in figure 4. The probe undergoes 2D motion in the horizontal plane. These motions are generated by a linear velocity V parallel to the probe length and a yaw angular velocity Ω along the vertical axis. With these restrictions, the locomotion model of the probe can be modeled by the unicycle kinematics:

   ẋ ẏ β    =    cos β -l sin β 2 sin β l cos β 2 0 1    V Ω , (3)
where (x, y, β) T is the vector of Cartesian pose of a frame attached to the sensor head with respect to an inertial frame (O, e x , e y ) attached to the symmetry plane of the scene in which the probe moves (see figure 4(a)). (V, Ω) define the control inputs of the sensor, whose motions are modeled through (3). Note that in the case of an autonomous underwater vehicle (AUV), the above kinematic model can be used as a high control stage feeding a lower stage in charge of controlling the Cartesian velocities (V, Ω) through servo-propellers. The scene surrounding the probe will be assumed to be composed of a single small ellipsoidal object, whose pose is parameterized in the sensor mobile frame as indicated on figure 4(b) for our 3-electrode probe. Since the scene is at rest in the inertial frame, this (ego-centered) parameterization enables us to easily derive the following set of relations:

O e x e y e e ⊥ β x 3 = (x, y) T E 3 E 1 x 1 E 2 x 2 V Ω (a) e e ⊥ e ρ e α u v x 0 α 3 = α α θ ρ 3 = ρ ρ 1 ρ 2 α 2 α 1 l 2 = ∆l l 1 = l 2R (b)
   ρ α θ    =    -cos α -l sin α 2 sin α ρ -l cos α 2ρ -1 -sin α ρ l cos α 2ρ    V Ω , (4)
defining the kinematic model of the scene with respect to the sensor frame.

Model of electric measurement

The model of the electric measurement of a U -I sensor was introduced in [Boyer et al., 2012] for an arbitrary n-electrode slender probe. The key idea of [Boyer et al., 2012] consists in remarking first that the currents measured by the sensor, here our 3-electrode probe, are physically due to the electric equilibrium recovery of the sensor when it is unbalanced by an applied field of potential φ a . To derive the model of the electric sensor response, it is convenient to shift from the measured currents I = (I l1 , I r1 , I l2 , I r2 , I l3 , I r3 ) T to the following set (i=1,2,3):

δI axi = (I il + I ir ) -I (0) i , δI lati = I lati = I il -I ir , (5)
where δI axi and δI lati measured on E 1 , E 2 and E 3 , are named the lateral and axial currents respectively.

Model of the electric response of the sensor under an applied potential φ a

Our probe's high aspect ratio allows the electric reaction to be decomposed into two components. As the electrokinetics are linear, they can be superimposed by addition to construct the total response (i.e. the measured currents crossing the receiver electrodes) of the sensor. As detailed in [Boyer et al., 2012], the first component is due to the polarization by φ a of the sensor along its length and is termed "axial polarization". It is produced by the vector of potentials applied to all the macro-electrodes of the probe (E 1,2,3 ):

Φ a = (φ a (x 1 ), φ a (x 2 ), φ a (x 3 )) T , (6)
where x i generically denotes the position of the E i -electrode center (see figure 4). The second is due to the polarization by φ a of the sensor along the direction perpendicular to its length and is termed "lateral polarization". This polarization is primarily generated by the following vector of axial excitation applied to the receiver electrodes (E 1,2,3 ):

E a⊥ = -(∇φ a (x 1 ).e ⊥ , ∇φ a (x 2 ).e ⊥ , ∇φ a (x 3 ).e ⊥ ) T . ( 7
)
where e ⊥ is a unit vector perpendicular to the rostro-caudal axis of the sensor (figure 4(a)). The axial excitation ( 6) is comparable to the vector of voltages U imposed by the internal electronics, when all potentials U i=1,2,3 are controlled independently as discussed in section 2. Thus, the currents produced by the axial polarization, gathered in the vector δI ax = (δI ax1 , δI ax2 , δI ax3 ) T , are simply given by substituting U by -Φ a in the model of the sensor with no object (1):

δI ax = -C (0) Φ a , (8)
where the minus sign indicates that the sensor has to oppose Φ a in order to recover its electric balance. The currents produced by the lateral polarization are gathered in the vector δI lat = (δI lat1 , δI lat2 , δI lat3 ) T , modeled by the following relation:

δI lat = P ⊥ .E a⊥ , (9)
where P ⊥ is a tensor whose expression in the sensor frame is given by the diagonal matrix P ⊥ = diag k=1,2,3 (p ⊥i ), where the p ⊥i are some "lateral polarizability factors" modeling the lateral distribution of the currents on the ring electrodes E i when they are submitted to a (locally uniform) perpendicular electric field (here -∇φ a .e ⊥ ).

In the following subsection we give the model of the applied potential φ a when the sensor is positioned near to a small ellipsoidal object.

Model of the potential φ a generated by a small ellipsoidal object

The purpose of this subsection is to present the model of the ambient field φ a permitting the calculation of the measured currents δI ax and δI lat when the probe is in presence of a small ellipsoidal object whose center is located in x o , using (6-9). Because the object is small, the potential produced by its polarization φ a , can be approximated by its leading order dipolar response in any point x of the scene as:

φ a (x) = - (x -x o ).P.∇φ (0) (x o ) x -x o 3 , ( 10
)
where p = P.∇φ (0) (x o ) is the electric dipole generated by the polarization of the object by -∇φ (0) (x o ) E 0 , i.e., the electric field in x o that would be produced by the sensor should there be no object nearby. For any orientation of the incident field, this dipole is modeled by P which represents the dipolar tensor of the object. This is a linear operator intrinsically related to the object. Such a tensor has been calculated analytically for the case of an ellipsoid in [START_REF] Ammari | Polarization and moment tensors: with applications to inverse problems and effective medium theory[END_REF]. In an orthogonal frame (O o , u, v, w) whose origin O o is located at the geometric center x o of the ellipsoid, and with unit vectors u, v, w supporting the ellipsoid axes of half-length a, b and c respectively, the expression of the matrix of the components P of P is:

P =    λ u 0 0 0 λ v 0 0 0 λ w    = (γ -γ o ) v    1 γo+A(γ-γo) 0 0 0 1 γo+B(γ-γo) 0 0 0 1 γo+C(γ-γo)    (11)
with v the volume of the object, γ the conductivity of its constitutive material, and where A, B and C are three geometric constants given by the following elliptic integrals:

A = bc a 2 +∞ 1 1 ξ 2 ξ 2 -1+( b a ) 2 ξ 2 -1+( c a ) 2 dξ, B = bc a 2 +∞ 1 1 ξ 2 -1+( b a ) 2 3 2 ξ 2 -1+( c a ) 2 dξ, C = bc a 2 +∞ 1 1 ξ 2 -1+( b a ) 2 ξ 2 -1+( c a ) 2 3 2 dξ. (12)
These expression are valid for any small ellipsoidal object. In particular, for elongated (prolate) ellipsoid for which a ≥ b = c, A, B and C take the simpler forms:

A = b a 2 +∞ 1 1 ξ 2 ξ 2 -1+( b a ) 2 dξ, B = b a 2 +∞ 1 1 ξ 2 -1+( b a ) 2 2 dξ, (13)
and C = B. When the object is made of an ideal conductive material such as metal γ ≫ γ o , and it is easy to show that λ u , λ v , λ w > 0. As a result, the object polarization tends to reinforce the ambient field φ (0) while the electric current lines of the total field φ a are funneled locally by the object. Meanwhile, when the object is made of an ideal insulating material (e.g. plastic or glass), γ ≪ γ o , and λ u , λ v , λ w < 0. In this case, the object produces a secondary field that opposes the ambient field, and the electric lines of the total field are repelled locally from the object boundaries. As shown in (10), the influence of the object on the sensor becomes comparable to that of the ambient field when its distance from the object is of the order of max(a, b, c). Furthermore, in the case of a sphere a = b = c, so A = B = C = 1/3, and P then reduces to the well known expression given in [Rasnow, 1996]:

P = χa 3 1 3×3 , (14)
with 1 3×3 the three order identity matrix, a, the sphere radius, and χ = γ-γo 2γo+γ the so-called contrast factor of the sphere. We have chosen a small prolate ellipsoid with a = 2b = 2 cm as a test object in all that follows. Its major axis lies in the equatorial plane of the sensor. In these conditions, it is practical to introduce the following decomposition of P :

P = P + P , (15)
where P and P define two further tensors, the first representing the isotropic response of the object, the second, its anisotropic response, each of them being easily deduced from P through the following expressions:

P = 1 2 (λ u + λ v )1 3×3 1 2 λ1 3×3 , P = 1 2 (λ u -λ v )1 3×3 1 2 λ1 3×3 .
(16)

Model of the measured currents generated by a small ellipsoidal object

We now have all the necessary ingredients to derive an analytical model of the currents measured by the probe in presence of an ellipsoid. By introducing (11), together with (15), into (10) we can compute the potential φ a produced by the polarization of the object. When this is introduced into (6-9), we obtain the model of the six measured currents (5) as (i=1,2,3):

δI lati = δ Īlati + δ Îlati , δI axi = δ Īaxi + δ Îaxi , (17)
where the lateral and axial currents are both decomposed into two contributions, one (over-barred) produced by the isotropic part of P, and the other (over-hatted), by its anisotropic part. After tedious but simple computation, we find the analytical expressions of the four (axial/lateral and iso/anisotropic) currents to be:

δ Îlati δ Īlati = - p ⊥i 4πγ o ρ 3 i 3 k=1 I (0) k ρ 3 k λ fik (α, θ, ρ, α i ) λ fik (α, θ, ρ, α i ) , (18)
with

fik = l k [3 sin(α i ) cos(2(θ + α) -α i ) -sin(2(θ + α))]+ ρ[3 sin(α i ) cos(2θ + α -α i )-sin(2θ + α)], (19)
fik = 3l k sin(α i ) cos(α i ) + ρ(3 sin(α i ) cos(α -α i ) -sin(α)), (20)
and

δ Îaxi δ Īaxi = - 1 4πγ 0 3 k,j=1 C (0) ij I (0) k ρ 3 k ρ 3 j λĝ jk (α, θ, ρ) λḡ jk (α, θ, ρ) , (21)
with

ĝjk = l k l j cos(2(θ + α)) + (l k + l j )ρ cos(2θ + α) + ρ 2 cos(2θ), (22)
ḡjk = l k l j + (l k + l j )ρ cos(α) + ρ 2 . ( 23
)
In these expressions ρ i = ρ i = l i e + ρe ρ , with l 1 = l, l 2 = ∆l and l 3 = 0.

Referring to figure 4, we have ρ i sin(α i ) = ρ sin(α) and ρ i cos(α i ) = ρ cos(α) + l i , all the other notations having been introduced above. As expected, contemplation of the above expressions shows that while the isotropic part of the currents (i.e. δI lat and δI ax ) only depend on the position of the object through the parameters (ρ, α), the anisotropic components (δ Îlat and δ Îax ) depend both on the object's position and its orientation θ. Similarly, removing λ from the expressions above gives the model of the currents measured by the sensor in presence of a single sphere as was first introduced in [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF] for a 2-electrode probe.

Evaluation of the analytical model

Before addressing our control problem, let us assess the previous analytical model of measurements. To this end, we used the 3-electrode sensor illustrated in figure 2, together with two copies of our test ellipsoid, one insulating, the other conductive. Both were placed in the scene as illustrated in figure 5. The sensor is then translated through the scene along a rectilinear path while the currents that penetrate it are computed using both our BEM reference numerical simulator, and the analytical model described in the previous section. The results are displayed in figure 6. The lateral and axial currents of ( 5) are plotted against the probe's position along its path.

A good match between the analytical model and the reference simulator was obtained.

The reference simulator runs in-house code based on the Boundary Elements Method (BEM). It integrates the Laplace equations of the electric potential field without approximation of the boundary geometry (sensor and objects), except those unavoidably introduced by their finite-element meshing. The discrepancies observed in figure 6 are only due to the reduction process developed in [Boyer et al., 2012] which is essentially based on the method of reflections and the perturbation expansion of the Laplace equations with respect to the aspect ratio of the sensor. Finally, note that compared to a similarly sized conductive ellipsoid, the measured currents obtained for the insulating ellipsoid is negative and its magnitude is nearly half lower, which agrees with the findings in [Rasnow, 1996].

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -2 0 2 4
x 10 -7

x(m)

δIax3(A) BEM Model -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -2 0 2 4
x 10 -7

x(m)

δIax2(A) BEM Model
(a) Axial current versus the position of the probe along the path.

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -1 0 1 2 x 10 -7
x(m)

δIlat3(A) BEM Model -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -1 0 1 2 x 10 -7
x(m)

Reactive control strategy

In this section, we introduce a control strategy that allows the sensor to seek and inspect electrically contrasted objects in unknown scene. The strategy is presented in this section. Its physical interpretation then follows. Inspired by the fish PMAs outlined in section 1, by "sense and inspect" we here mean: to seek them and once discovered, to stop in front of them and to follow their boundaries without touching them. This strategy is achieved in a purely autonomous manner through the combination of three basic behaviors, co-ordinated according to the algorithm presented in figure 7(a). Its application to a small object is illustrated in figure 7(b). The sequence of the implemented behaviors is summarized as follows:

• 1) Seek any object that is electrically contrasted with respect to the ambient medium. This first phase (from A to B in figure 7) consists of seeking an electrically non-transparent object by applying an attractive control law with a constant forward velocity V > 0.

• 2) Retreat from the electric influence of the object. This second phase (from B to B ′ in figure 7) corresponds to the initialization of an orbiting motion of the sensor around the object. It is obtained by applying a repulsive control law with V = 0 and Ω = 0.

• 3) Follow the boundaries of the object. This third phase (from B ′ to C in figure 7) stands for the orbiting phase around the object's boundaries with a constant forward velocity V > 0. As we shall see below, all these behaviors can be achieved with our three-electrode probe. In particular, except for the tail emitter, which is required in all circumstances, the first and second phases require only the head electrode measurements (δI ax3 , δI lat3 ), while the third phase is based on the measurements of the neck electrode (δI ax2 , δI lat2 ). Moreover, as indicated by the algorithm in figure 7(a), the transition between the phases are governed by discrete events which depend only on the time variations of the measurements (and not their magnitude). The first transition from phase 1 to phase 2, is activated by the change of sign of the axial current flowing across the neck electrode (δI ax2 ). The second transition (from phase 2 to phase 3) is activated when the axial current flowing across the head-electrode (δI ax3 ) reaches its minimum absolute value. In the subsequent section we physically interpret this strategy.

Physical interpretation of the reactive strategy

The above strategy can be easily explained in terms of interactions between the sensor and the electric field lines emitted by the objects in their immediate environment. In (7), note that the lateral current δI lat3 can be detailed as δI lat3 = -p ⊥3 ∇φ a (x 3 ).e ⊥ with φ a given by (10). Thus, δI lat3 is proportional to the lateral flux of the electric field reflected by the object, and in the first phase, the attractive law which as we shall see later, forces δI lat3 = 0, also ensures the sensor aligns the front part of its body with the electric field lines of φ a , while following them at a constant velocity V . As a result, since all the electric lines of φ a (the integral of the electric field reflected by the objects) emanate from the objects close to the sensor, the sensor is attracted by these objects. At the end of this first phase, as the sensor head gets closer and closer to, say a conductive object, the electric field lines originally (when there is no object around the sensor) captured by the neck electrode are attracted from the neck toward the head electrode. On the other hand, if the object is insulating the electric field lines are repelled backward along the sensor axis from the head to the neck electrode as illustrated in figure 8. In both cases a moment in time arrives at which δI ax2 changes sign. At that time the control law switches from the attractive behavior of phase 1 to the repulsive behavior of phase 2. When this occurs, the sensor starts to avoid the object while V = 0, until | δI ax3 | reaches a minimum, which implies that the tail (E 1 ) and the neck and head electrodes (E 2 , E 3 ) are electrically equidistant from the object. If one of these two electrodes is closer to the object than the other, the electric lines emitted by the tail or the lines received by the head are more perturbed (than in the equidistant case) and the corresponding measurement | δI ax3 | is larger (see figure 8). When this condition is satisfied, δI ax3 is memorized as a reference value that the sensor tries to maintain by steering its body's axis while maintaining a constant forward velocity, thus maintaining an iso current path around the object's boundaries. In the sections, we analyse each of these phases and transitions using the above analytical model, the BEM simulations, and the results from our experimental setup presented in section 3.

First phase: seek any electrically contrasted object

In this subsection, we analyse the behavior "Seek any electrically contrasted object". This basic behavior is introduced in [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF] where it was achieved through the simple feedback law:

V = V d , and: Ω = k δI lat3 |δI ax3 | . ( 24
)
Expression (24) ensures the attraction of the probe toward electrically contrasted objects of any shape and size. This was shown in [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF] where the probe's closed loop dynamics were analysed with an analytical model of the response of spherical objects. Here, we start from this simple case and advance the analysis to the more general case of ellipsoidal objects. In (24), V d is a constant positive value ensuring the forward motion of the probe with a constant axial velocity V d e and k is a steering gain which ensures the probe is attracted or repelled by any electrically contrasted object depending on whether k is positive or negative respectively. Introducing the control law ( 24) in the kinematic model of the scene (4), and using the model of the electric currents (18-21), gives the closed-loop dynamics of the probe in the presence of a single small ellipsoidal object in the form x ′ = f (ξ, x) with x = (ρ, α, θ), the state vector, and ′ the derivation d./ds with s = V t the length travelled by the probe along its path in the scene. In the simple case of spherical objects, the symmetry of the object ensures the closed loop-dynamics to take the simpler form x ′ = f (ξ, x) with x = (ρ, α). Thus, for a given value of the reactivity of the law ξ = k/V , the entire dynamics can be pictured in a single phase portrait in the plane (ρ, α). This is illustrated in figure 9, for a positive k and a moderate (figure 9(a)) and a high (figure 9(b)) reactivity set to ξ = 50 and ξ = 500 respectively. These portraits show the trajectories of the object in the mobile frame of the sensor whose detection range (fixed to one and a half times that of the total sensor length) is drawn with a dashed green line. It may be seem that any sphere initially located within the cone drawn with a dashed green line, termed the attraction cone, is steered to the head. Furthermore, increasing ξ increases the reactivity of the attraction and the aperture of the attraction cone. Note that for high values of reactivity, the spherical object first moves to a straight line in the longitudinal axis of the sensor and then flows along it toward its head. In the inertial frame, this means that the sensor directs its head toward the sphere's center before going forward to reach it. This can be easily explained using the analytical model of lateral currents (18) restricted to the isotropic component of the object response. In fact, (24) steers the probe in order to ensure that δI lat3 = δI lat3 = 0. Then, noting that ρ i sin α i = ρ sin α ∀i in expression (20) of δI lat3 , shows that increasing ξ makes the closed loop dynamics converge more and more rapidly toward the asymptotic branch defined by α = 0. Note that this branch is defined by the root locus of δI lat3 (ρ, α) = 0, whose point singularities due to the ρ-dependency, are systematically crossed by forcing the condition V = V d of (24). Now, in the case of an ellipsoidal object, the closed loop dynamics take the more general form x ′ = f (ξ, x) with x = (ρ, α, θ), since they also depend on the orientation θ of the anisotropic object. In spite of this major difference, the attractive behavior is preserved as illustrated in figure 10(a) which displays a typical trajectory around the test ellipsoid in the probe frame controlled by (24). Nevertheless the asymptotic branch along which the probe converged toward a small sphere now has two branches symmetrically tilted with respect to the probe axis, as shown in figure 11. As in the case of the sphere these two branches are analytically defined as the root locus of δI lat3 = δI lat3 + δ Îlat3 = 0, which now depend on both α and θ through the anisotropic component δ Îlat3 . For sufficiently high values of ξ, the control ensures δI lat3 ≃ 0, and (4) shows that after reaching one of the two branches (between positions A and B in figure 10(a)), the angles α and θ evolve along the branches (between positions B and As expected, this θ-dependency does not affect the overall attractive behavior as it is illustrated in figure 11(a Note that the aperture angle between the two symmetric asymptotic branches only depends on the anisotropic component of the object's response and that this angle does decrease with the magnitude of the anisotropic response. This is shown in figure 12 which plots the dependency of this aperture angle as a function of λ, or equivalently, of the ellipsoid aspect ratio a/b.

First transition: stop while facing the object

Returning to the global strategy illustrated in figure 7, we consider our probe to be in the same condition as the previous phase, i.e. it approaches the object to headfirst. We now consider the transition leading to the probe stopping near the object. Ideally, we would like to find a condition of the measured currents which, when satisfied, would ensure the probe stops at a fixed ρ, independent of the incidence angle α at which the probe discovers the object, and of the α + θ orientation of the object with respect to the probe. Such a condition could be encoded into a function of the measurements capable of imaging depth perception along the probe's length as it approaches the object. To introduce such a function, we make use of the axial currents, which depend only on depth [Boyer et al., 2012], and reconsider the expression (10) of φ a that once inserted into Φ a = (φ a (x 1 ), φ a (x 2 ), φ a (x 3 )) T of (1), can be set in the following matrix form:

δI ax = C (0)    φ 1 (x 1 ) φ 1 (x 2 ) φ 1 (x 3 )    =    C (0) 11 C (0) 12 C (0) 13 C (0) 21 C (0) 22 C (0) 23 C (0) 31 C (0) 32 C (0) 33       ρ 1 .p ρ 1 3 ρ 2 .p ρ 2 3 ρ 3 .p ρ 3 3    , ( 25
)
from which we extract the axial currents flowing across the neck electrode:

δI ax2 = C (0) 21 ρ 1 ρ 3 1 + C (0) 22 ρ 2 ρ 3 2 + C (0) 23 ρ 3 ρ 3 3 .p C 2 .p. ( 26
)
Due to the conservation of currents, the conductivity matrix of the sensor when no object is nearby, satisfies the relation

C (0) 21 + C (0) 22 + C (0) 23 = 0 with three commensurable C (0)
2i (see ( 2)). Thus, in the general case where α = 0, the factors ρ i / ρ i 3 in ( 26), define at each moment in time a basis of three commensurable vectors of R 3 , so one can expect that, when the sensor approaches an object, a reasonable position (ρ, α) of the object such that C 2 = 0, C 2 (ρ, α) denoting a 3 × 1 vector-function defined in (26), always exists. Of course, when C 2 = 0, δI ax2 changes sign by crossing δI ax2 = 0. The idea implemented hereafter consists in stopping the sensor when δI ax2 changes sign, i.e., when the condition δI ax2 = 0 is satisfied. This may happen in two other circumstances (than C 2 = 0) namely when p = 0 and C 2 ⊥ p. The first circumstance is absurd (it means that there is no object in the scene), while the second is purely fortuitous (and has never been observed after numerous simulations and experiments). The condition δI ax2 = 0 can be considered as equivalent to C 2 = 0. Using ( 26), in which we insert the parametrization of figure 4 and the consequence of current conservation

C (0) 21 + C (0) 22 + C (0) 23 = 0, the condition C 2 = 0 can be detailed as:        C (0) 23 ρ ρ 3 - ρ + l cos(α) ρ + l 3 + C (0) 22 ρ + ∆l cos(α) ρ + ∆l 3 - ρ + l cos(α) ρ + l 3 = 0 C (0) 23 l sin(α) ρ + l 3 + C (0) 22 l sin(α) ρ + l 3 - ∆l sin(α) ρ + ∆l 3 = 0, (27)
where l = le and ∆l = ∆le . The two equations ( 27) define a non-linear algebraic system whose roots (ρ, α) are the positions of the object in the sensor frame for which δI ax2 = 0. The root loci of the system (27) are drawn in figure 13(a) for various distances ∆l between the head and the neck electrodes. These plots show that for any ∆l there always exists a set of (ρ, α) defined by a bell shaped curve such that δI ax2 = 0. The different object positions strongly depend on the design variable ∆l (shifting ∆l from 0.09 m to 0.1 m doubles the ρ distance at a fixed α = 0). Let us also note that ρ weakly depends on α for reasonably low values of ∆l, a situation reinforced by the fact that the domain of possible root locus is in reality restricted by the sensor range and the cone aperture angle imposed by the reactive law outlined in section 6. Figure 13(b) displays the root loci of (27) in a more realistic sub-domain for ellipsoids of aspect ratios between 1 and 5, and a high reactivity of the attractive law of phase 1 (note that in this case, the incident angle at which the probe discovers the object is dependent on the ellipsoid aspect ratio as in the plot in figure 12).

Second phase and second transition: object clearance

We now consider the second phase and the transition that inhibits it (see algorithm in figure 7). They both define a single behavior in which the sensor clears the object after having found it (phase 1) and has stopped near it (transition 1). This behavior is obtained by applying the reactive control law (24) but in its repulsive variant, i.e. with:

V = 0 , and:

Ω = -k δI lat3 |δI ax3 | . ( 28
)
Compared to (24), we reverse the sign of the steering gain (k > 0) to ensure the sensor turns its head away from the object. This is performed with V = V d = 0, maintaining the distance between the probe and object centers constant while manoeuvering. While performing this probe rotation, the head axial current I ax3 = I (0)

3 + δI ax3 undergoes a stereotyped time-evolution which we will now analyse further in order to define a condition in which the probe stops its rotation while remaining clear of the object. Note that, from (1) and ( 21), we can rewrite the model of axial currents in the matrix form:

I ax = C (0) -C (0) GP G T 4πγ o C (0)    1 0 0    U C    1 0 0    U, (29)
where C defines the conductivity offered by the scene, while G is a 3 × 3 matrix defined as:

G =     cos(α 1 ) ρ 2 1 sin(α 1 ) ρ 2 1 0 cos(α 2 ) ρ 2 2 sin(α 2 ) ρ 2 2 0 cos(α 3 ) ρ 2 3 sin(α 3 ) ρ 2 3 0     . (30)
In ( 30), the third column is zero because the scene is symmetrical in the horizontal plane. Note also that ρ 3 = ρ and ρ i sin(α i ) = ρ sin(α) for any i. Moreover, as described in [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF], we have the following relation at leading order (with respect to the object volume):

C = C (0) -C (0) GP G T 4πγ o C (0) ≃ R (0) + GP G T 4πγ o -1 , (31)
where R (0) is the resistance matrix of the sensor immersed in a scene with no object, while the matrix (4πγ o ) -1 GP G T represents the perturbation of the resistivity of the scene offered to the probe due to the presence of the object. Based on this insight, the axial current I ax3 = (0, 0, 1)I ax is an image of the resistance of the scene between the tail-emitter E 1 and the head-electrode E 3 . As such, I ax3 encodes not only the intrinsic features of the object (material, shape, size) and its tilt angle through P , but also its position in the sensor frame through the G matrix of (29). In particular, due to the factors ρ -2 i in (30), this current increases (respectively decreases) when one of the two tip electrodes approaches a conductive (respectively insulating) object. Reciprocally, I ax3 decreases (respectively increases) when one of the two tip electrodes moves away from a conductive (respectively insulating) object. All of these trends reflecting the variations in the scene resistance between the two tip electrodes E 1 and E 3 . This is illustrated in figure 14(b) which shows the simulated evolution of I ax3 as a function of the tilt angle of the probe (denoted µ) for our conductive ellipsoidal test object with a fixed center, but different orientations of its axis in the horizontal plane (of the motion). The evolution starts from a probe pose in which the head points toward the object center (see figure 14(a)). As expected, as rotation is initialised, the sensor head is at its closest to the conductive object, so the resistance is minimal and the axial current is maximal. Furthermore, while rotating, the same situation obtains each time one of the two tips (head/tail) of the probe is closest to the object, i.e., at every multiple of 180 • . The odd multiples correspond to the case where the tail faces the object, which generates a local maximum of I ax3 . The even multiples repeat the starting pose in which the head faces the object while a global maximum of I ax3 is reached. Furthermore, between each of its maximums, I ax3 reaches a minimum value which corresponds to a pose in which the sensor is electrically as far as possible from the object (V being fixed to zero). These observations hold perfectly when the test ellipsoid has one of its axe of symmetry aligned with the probe axis at the initiation of the phase. This is illustrated in figure 14 where the red and purple plots correspond to an object initially tilted at angles of θ 0 =0 • and θ 0 =90 • respectively, and show that I ax3 does exhibit a maximum at µ = 0 • , and reaches a minimum when µ = 90 • . However, rotating the ellipsoid to an arbitrary initial angle θ 0 introduces an asymmetry in the initial scene that shifts the maxima and minima of a tilt offset which does not exceed ±15 • (see the other plots obtained when the ellipsoid is rotated of 30 • and 60 • ). For a resistive object, the evolution of I ax3 as a function of the tilt angle of the probe µ, is similar but reversed with respect to the I ax3 = 0 axis. Hence, in both cases (conductive or resistive) |I ax3 | cyclically and alternately reaches local maxima and minima in the same poses of the probe. In the context of our global strategy, when |I ax3 | reaches its first minimum (after the activation of the law (28)), we switch off the repulsive law while memorizing the axial current I ax3 and activate the subsequent and final phase (numbered 3 in the algorithm in figure 7).

Third phase: following the object boundaries

In this subsection, we analyse the behavior "Following the boundaries of an electrically contrasted object". This basic behavior is achieved through the simple sensor-based feedback law:

V = V d , and Ω = k ′ (I ax3 -I d ax3 ), (32)
where V d is a constant positive value ensuring the probe moves forward with a constant axial velocity V d e and k ′ is a second steering gain (after that of (24,28)) which makes the probe track an iso-value I ax3 = I d ax3 , where I d ax3 is the value of I ax3 memorized at the end of the previous phase, i.e. when |I ax3 | reaches its first minimum after activation of the repulsion. Using the model developed in the previous section, we know that I ax3 images the position of the object in the sensor frame. Moreover, figure 14 shows that the minimum values of I ax3 are weakly affected by the tilt angle of the ellipsoid.

As a result, imposing I ax3 = I d ax3 through the control law (32) with I d ax3 being the minimum values measured in similar conditions to those of figure 14, should maintain the object in the same position in the mobile sensor frame. In other words, the probe moves forward at a constant linear velocity V d , while tilting its body to maintain I ax3 = I d ax3 . It thus revolves around the object at a nearly constant distance from it. This basic idea is confirmed by simulation of the closed loop dynamics x ′ = f (ξ, x) obtained by inserting (32) into the kinematic model of the scene (4). In figure 15(a) we plotted the paths of the probe starting in a pose perpendicular to the line linking the center of the probe with a spherical object, i.e. µ(t = 0) = 90 • for different values of reactivity ξ ′ = k ′ /V . In figure 15(b), the same numerical simulations were performed but with an initial tilt offset of ±15 • with respect to the perpendicular configuration. Such an offset takes into account the slight variations of the minimum detection occurring at the end of the previous (clearance) phase. This control law has been tested in many experiments in the test bed described in section 3 with objects of different shape, size and materials. We report two representative tests in which the sensor starts from a given initial pose A, then moves in straight line while at a given instant (illustrated by pose B on these plots), the axial current I ax3 is memorized and imposed as I d ax3 in the reactive law (32) which is applied at all further instants. This simple test is illustrated for a conductive ellipsoid and cube in figures 16 and 17 respectively. In both cases, the plotted probe trajectory shows that after switching from open loop to closed loop control (32), the sensor revolves around the object in order to maintain the prescribed memorized value of I ax3 .

Experimental tests of the overall strategy

We report here some of the experimental results obtained by applying the above object inspection control strategy. The experimental conditions are those described in section 3. Two types of objects are inspected. The first is an insulating cube, the second is a combination of large flat insulating objects (finite walls). The electric feedback loops ( 24) and ( 32) are tuned once for all with k = 50V and k ′ = 2.5V respectively, and then applied to all the tests. Figures 18 (top) and 19 (top) display the motion path of the sensor for the inspection strategy. As expected, for each experiment the sensor first starts by seeking the electrically contrasted objects. After having found one, it stops while facing it (in pose B), retreats from its electric influence through a clearance rotating phase, and then follows the boundaries of the object starting from pose B ′ . For the smooth objects, the path around them nearly follows their boundaries (a circle for the sphere, an ellipse for the ellipsoidal object). Figure 18 shows a nearly circular sensor path centered on the insulating cube located in the middle of the tank, with two orientations 0 • and 45 • with respect to the initial position of the sensor's axis. This can be explained by the fact that the electric responses of a sphere and that of a cube are very similar, and in fact indistinguishable at the leading order. However, this basic property is only true for small objects. Thus, due to the largeness of our test-cube, the sensor trajectory is not a regular circle. These distortions of the circular path map the sharp edges of the cube. The time-evolution of the currents δI ax2 and δI ax3 is plotted in figure 18 (bottom) for the cube (similar plots are observed for the sphere and the ellipsoid). As expected, phase 2 (clearance from pose B to pose B ′ ) is activated by the change of sign of δI ax2 while phase 3 (revolution after pose B ′ ) is activated when | I ax3 |=| I (0)

ax3 + δI ax3 | reaches its minimum value. In this circumstance the value of I ax3 , indicated by the horizontal grey dashed line, is memorized to inform the reactive orbiting law (32). Finally, the case of the inspection of large objects is illustrated in the two last examples in figure 19. In the first example, the object is an insulating wall placed in the middle of the tank. In the second example, there is no object in the tank except its 4 insulating vertical walls and a removable small insulating plane initially placed at the bottom left corner of the tank. In the first (respectively the second) case, after having stopped in front of the center wall (respectively the right wall), the sensor orbits around the center wall (respectively the tank's boundaries). In the first case, we plotted several paths obtained using the overall strategy with different initial poses. In the second case, the strategy is initiated with the small removable wall at the left-bottom corner, and after one revolution around the tank, this small panel is suddenly removed. When moving past this corner, the sensor re-adjusts its path (indicated by the dashed line with the removable panel in the tank and in continuous line, after it was removed) to follow the two original fixed walls of the tank.

Conclusion and perspectives

In this article we have addressed the problem of object inspection using a sensor inspired by electric fish. The solution enables to seek any electrically contrasted object and then to orbit around its boundaries. Starting from the memory-less reactive steering strategy dedicated to object seeking and/or obstacle avoidance initially presented in [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF] for a "binocular" sensor with two left-right head receptors, we progressively increased the number of electrodes and provided our controller with memory in order to perform object inspection. This minimal approach enabled clear identification of the key physical principles required for these control objectives. In particular, compared to our previous work on reactive navigation, the object inspection requires that our sensor performs a depth measurement while facing the object. This additional capability requires an additional neck electrode whose axial (left+right) perturbing current δI ax2 = I ax2 -I (0) 2 enables this perceptual feature. In the context of underwater robotics, this control approach could be used for the autonomous navigation of an AUV in turbid waters and muddy environments, for instance to collect data that could subsequently be processed off-line to build a model of objects. Remarkably, the approach does not require any model of the electric interactions between the sensor and the surrounding objects. According to the concept of embodied intelligence [ [START_REF] Pfeifer | Self-organization, embodiment, and biologically inspired robotics[END_REF], the model is in fact totally embodied into the morphology of the sensor and the physics (electrokinetics of perception and kinematics of locomotion) of its interactions with its environment (see section 6.2). In this respect, note that the bi-lateral (left-right) symmetry commonly observed in most of animals, and electric fish in particular, is a key feature for steering the robot with the reactive law (24). The additional morphological parameter ∆l (the distance between the head and neck electrodes) is another key parameter which totally encodes the stopping pose illustrated in figure 13(a). This approach allows the same behavioral inspection strategy to be obtained in all circumstances, particularly when there is no prior model of the object. This situation is often encountered, and represents remarkable advantages since, except for small ellipsoidal objects or infinite walls and corners, we only have numerical finite element models which are unusable in real time robotics applications. Beyond the empirical scenarios reported in the article, the insights developed here open promising perspectives of accurate sensor positioning with respect to an object. For example in the stopping transition illustrated in section 6.3, at the end of phase 1, the sensor is aligned with one of the object's axis of symmetry, and so therefore it is pointing directly at its geometric center, the incident angle α being zero. In this simplified case, the condition δI ax2 = 0 defines an invariant distance to the object, i.e. a condition on the measurements only, and whose satisfaction ensures a fixed distance between the probe and the object, this distance being independent of the intrinsic object's properties. As a result, replacing phase 1 by active self-alignment of the probe along one axis of symmetry of the object (through feedback control), would allow the sensor to be positioned at a fixed distance from the object's center at transition 1, as is observed in fish when they perform the PMA 1 illustrated in figure 1. This strategy is currently being studied and will be presented in a future article.

Figure 1 .

 1 Figure 1. Probing Motor Acts (PMA) that fish use during object inspection: (1) Stationary probing. (2) Lateral "va et vient" probing. (3) Circling probing. (after Moller and Toerring[Toerring andBelbenoit, 1979, Toerring andMoller, 1984])

Figure 2 .

 2 Figure 2. Photograph (left) and schematic view (right) of a 3-electrode sensor organized in 3 polarizable rings, 2 of them (E 2 and E 3 ) being divided in two half rings (e il , e ir ) allowing two lateral (left and right) current measurements.

Figure 3 .

 3 Figure 3. (left) Electrolocation test bench. (right) Example of a set of simple shaped objects (ellipsoids) of various sizes and materials (plastic, metal).

Figure 4 .

 4 Figure 4. (a) Parametrization of a sensor pose , (b) and of a scene composed of an ellipsoidal small object.

Figure 5 .

 5 Figure 5. The 3-electrode probe is forward translated in the presence of one insulating and one conductive ellipsoid of same geometry (a = 0.012, b = 0.006). Dimensions in meters.

  Lateral current versus the position of the probe along the path.

Figure 6 .

 6 Figure 6. (a) Axial and (b) lateral currents predicted by the analytical model and the BEM simulation with a 3-electrode sensor translated in the scene as pictured in figure 5.

Figure 7 .

 7 Figure 7. (a) Inspection task algorithm. (b) Typical object inspection scenario.

Figure 8 .

 8 Figure 8. Physical interpretation of the reactive strategy in terms of electric field lines: (top) The electric lines are pushed forward or backward depending upon whether if the object facing the sensor is a conductor or an insulator. (bottom) When the head-neck and tail electrodes are electrically equidistant to the object, the head current reaches a minimum.

Figure 9 .Figure 10 .

 910 Figure9. Portrait of the trajectories of a sphere in the reference frame of the sensor when it is controlled by the attractive version of (24), (a) for ξ = 50, and (b) ξ = 500.

  ) and 11(b), which respectively show the portrait of trajectories at detection time: θ + α = 0 • , and θ + α = 30 • .

Figure 11 .

 11 Figure 11. Portrait of trajectories of an ellipsoid in the sensor frame with ξ = 500 and (a) a zero initial tilt angle. (b) As in (a) but with an initial tilt angle of 30 • .

Figure 12 .

 12 Figure 12. Aperture angle of the reactive attractive law as a function of the object aspect ratio.

Figure 13 .

 13 Figure 13. (a) Stopping distance ρ for a sensor of various values of ∆l as a function of α. Dashed lines show the limits of the sensor range and the aperture cone of the reactive attractive law (24), for an ellipsoid of maximal aspect ratio of 2. (b) Same, but magnified view of area of interest.

Figure 14 .

 14 Figure 14. Evolution of I ax3 as a function of the tilt angle of the probe µ when the probe initially points toward the object center. (a) Condition of the experimental test. (b) Evolution of I ax3 for an ellipsoid (a = 2, b = 1) tilted at different angles (0 • , 30 • , 60 • and 90 • ).

  (a) Paths of the sensor for different values of reactivity ξ ′ = k ′ /V starting from a probe pose perpendicular to the line between the probe and object centers. (b) Paths of the probe revolving around a sphere of high reactivity, starting from different tilted pose of the probe.

Figure 16 .Figure 17 .

 1617 Figure 16. Sensor controlled by the reactive law (32) in the presence of a conductive ellipsoid. (a) The law is activated from pose B to C. View from the top of the tank. (b) Magnified view of the path around the object.

Figure 18 .

 18 Figure 18. Inspection of the boundaries of an insulating sphere, ellipse and cube. (bottom -left)) Time evolution of the currents δI ax3 (blue) and δI ax2 (red) during inspection of the cube; (right) detailed view of (c) highlighting a phase transition.

Figure 19 .

 19 Figure 19. Inspection of large objects: (top) An insulating wall is placed in the tank center (left); removable corner (right). (bottom) Time evolution of the currents δI ax3 (blue) and δI ax2 (red) along the sensor path when the sensor follows the walls of the tank: (left) complete time evolution; (right) detailed view of (a) highlighting a phase transition.

Acknowledgment

This work was supported by the European Union, by funding the Project: EU H2020 FET-Proactive project 'subCULTron ', no. 640967


MD Quentin Scanvion

MD-PhD Guillaume Lefèvre

MD-PhD; Eric Hachulla

MD-PhD Pierre-Yves Hatron

MD-PhD. Marc Lambert
email: marc.lambert@chru-lille.fr

Subcutaneous immunoglobulin therapy prevents systemic capillary leak syndrome attack

To the Editor :

Systemic capillary leak syndrome is a life-threatening condition, characterized by capillary hyperpermeability [START_REF] Clarkson | Cyclical edema and shock due to increased capillary permeability[END_REF] with hypotension, edema, hemoconcentration and hypoalbuminemia [START_REF] Gousseff | The Systemic Capillary Leak Syndrome: A Case Series of 28 Patients from a European Registry[END_REF] . Prognosis remains poor as mortality reaches 25-34% after 10 years [START_REF] Gousseff | Syndrome de fuite capillaire idiopathique[END_REF][START_REF] Druey | Narrative Review: The Systemic Capillary Leak Syndrome[END_REF] . Intravenous immunoglobulins (Ig) reverse systemic capillary leak syndrome flare [START_REF] Lambert | High-dose intravenous immunoglobulins dramatically reverse systemic capillary leak syndrome[END_REF] and prevent recurrence [START_REF] Gousseff | The Systemic Capillary Leak Syndrome: A Case Series of 28 Patients from a European Registry[END_REF] .

Case report

We report the case of a 36-year-old woman successfully treated with subcutaneous_Ig infusions.

Systemic capillary leak syndrome was diagnosed on 8 recurrent attacks, including 5 requiring intensive care, with edema of all limbs (+6kg), hypotension (systolic blood pressure 60mmHg), hemoconcentration (hemoglobin 19g/dL) and hypoalbuminemia (30g/L).

Prophylactic treatment comprising theophylline 150mgx2/day and terbutaline sulfate 5mgx2/day decreased the severity, but not the frequency, of the flares. From 2010 on, 4 edematous attacks occurred, including one that required intravenous_Ig. In 2014, two months after a breast pexy, the patient developed edema with hypotension (systolic blood pressure 90mmHg), treated with intravenous_Ig (2g/kg). After two further recurrences, prophylaxis was replaced by intravenous_Ig infusion (1g/kg per month, i.e. 60g). As the following year remained recurrence-free and our patient could handle injections, we switched to a home-based therapy management with subcutaneous_Ig (0.25 g/kg/week over 2 days). Our patient hasn't relapsed and her quality of life has enriched.

Review

Intravenous_Ig is the recommended therapy to avert systemic capillary leak syndrome crises [2, 6] . Our case emphasizes the benefits of subcutaneous administration.

Efficacy and safety of Ig infusions subcutaneously and intravenously have been assessed [START_REF] Rezaei | Indications and safety of intravenous and subcutaneous immunoglobulin therapy[END_REF][START_REF] Jolles | New Frontiers in Subcutaneous Immunoglobulin Treatment[END_REF] .

Based on equivalent cumulative monthly-dose, several European studies report similar serum IgG levels after intravenous or subcutaneous infusion, although therapeutic plasma levels are earlier achieved intravenously. However, in primary immunodeficiency disorders, the transition from intravenous_Ig to subcutaneous_Ig maintains serum IgG within therapeutic range [START_REF] Skoda-Smith | Subcutaneous immunoglobulin replacement therapy in the treatment of patients with primary immunodeficiency disease[END_REF] . Furthermore subcutaneous_Ig administration yields more stable serum IgG concentration [START_REF] Skoda-Smith | Subcutaneous immunoglobulin replacement therapy in the treatment of patients with primary immunodeficiency disease[END_REF] . Although swelling, erythema, tenderness or induration commonly occur at injection-site, these local reactions last less than 24 hours [START_REF] Jolles | New Frontiers in Subcutaneous Immunoglobulin Treatment[END_REF] . Subcutaneous treatment has several positive implications: (i) it is home-based, possibly self-administered, allows flexibility in daily routine; (ii) it reduces social expenses linked to hospitalization rate for infusion, infection, fluid overload or hyperosmolality possibly triggering acute renal failure or thrombosis; (iii) sick leave is not required [START_REF] Jolles | New Frontiers in Subcutaneous Immunoglobulin Treatment[END_REF] .

Conclusions

To prevent recurrences in patients responding to intravenous_Ig treatment for acute systemic capillary leak flares, the subcutaneous infusion seems a valuable alternative that improves patient's quality of life and lowers socio-economic charges.

Acknowledgements

We thank Dr Laura Ravasi (MD-PhD) who provided medical writing services on behalf of the University of Lille, Nord de France.

Funding source -None

Conflicts of interest

Dr Guillaume Lefèvre (MD-PhD) reports royalty payments (consultancy) from LFB and grants from LFB, Octapharma and CSL Behring (<$20,000/year). Dr Eric HACHULLA (MD-PhD) reports receiving lecture fees from CSL Berhing, LFB and grants from CSL Berhing, LFB and BAXALTA.

Key words

Systemic capillary leak syndrome, Clarkson's disease, subcutaneous immunoglobulin therapy, prevention, quality of life.


Antoine Masse
email: antoine.masse@ign.frsidonie.christophe@ign.fr

Sidonie Christophe

Geovisualisation of animated tides in coastal area with an OpenSource OpenGL platform

Keywords: Geovisualisation, animation, coastal area, tides, Opengl

Introduction

Geovisualisation of coastal area and one of its major spatiotemporal phenomenontideis very challenging because:

(1) coastal area observation is difficult, underwater and upon water observation techniques are different and produce heterogeneous data, including airborne and bathymetric LiDAR surveys (Light Detection and Ranging); (2) coastal area elements are constantly in motion, in time and space, e.g. tides, waves, sands, implying uncertainty and imprecision in data; and (3) animation of spatiotemporal dynamics, particularly tide cycles, is tricky and brings difficulties to geovisualisation design adapted to uses and users of the coastal area.

In this paper, we propose and discuss about animated geovisualisation of the coastal area and its improvement by representation of water level dynamics in a realistic way. These improvements are based on, first, the use of close-toreality data such as Digital Elevation Model obtained with LiDAR data to depict more accurately coastal elements and thus enhance the perceived realism, and, second, on the manipulation of the rendering engine (style) with abstraction (maps) and realism (photorealism images) in an existing OpenSource and OpenGL platform, in 2D 1 and 3D 2 .

State-of-the-Art

Animated geovisualisation is related to the visualisation of continuous or pseudo-continuous geographical spatio temporal elements [START_REF] Andrienko | Geovisualization of dynamics, movement and change: key issues and developing approaches in visualization research[END_REF][START_REF] Bertin | Sémiologie graphique : Les diagrammes-Les réseaux-Les cartes[END_REF]. As an example of animated geovisualisation of the coastal elements, we can mention rising water levels simulation [START_REF] Douglas | New Mapping Tool and Techniques for Visualizing Sea Level Rise and Coastal Flooding Impacts[END_REF], and sandbank movement monitoring [START_REF] Gierlinger | Visualization of Marine Sand Dune Displacements Utilizing Modern Gpu Techniques[END_REF]. Animated visualisations are increasingly used due to new rendering capabilities (fluid visualisation and data quality). Leskens et al. [START_REF] Leskens | An Interactive Simulation and Visualization Tool for Flood Analysis Usable for Practitioners[END_REF] demonstrate animation importance, which enhances the good perception of the result of a 1 http://ignf.github.io/geoxygene/ last access 2016/05/09 2 http://www.itowns-project.org/ last access 2016/05/09 scenario. This demonstration is based on a user study on geovisualisation of a flood (crisis event). Resch et al. [11] conclude on the importance of the dynamic representation, including its temporal dimension from the evaluation of their 4D (3D+t) geovisualisation system of marine data. However, Rensink et al. [10] point out the limitations of animations in terms of information transmission efficiency, and especially on the perception of changes. In conclusion, animations have to be used sparingly to keep focus on the main purpose of the visualisation. None of these existing geovisualisation platforms brings together stylisation tools, animation functionalities, 2D/3D +t rendering engine. Moreover, we claim for OpenSource solutions.

Method

We introduce a methodology to improve the realism of animated tide geovisualisation in the coastal area, characterized by a seaside and a landside, separated by the sea/land interface. The major spatial and temporal phenomenon of this area is the tidal phenomenon, requiring information at any time t of the water depth (observation or prediction) and the position of the sea/land interface.

We have the following objectives: (1) to visualize and simulate the tidal phenomenon with water depth perception and (2) to obtain a coherent rendering over land and sea (same level of stylisation on both sides), in order to facilitate the understanding of the visualisation. We choose to explore rendering possibilities of water depth with inspiration of common styles corresponding to the levels of abstraction provided by nautical and topographic maps, or, the level of photorealism provided by ortho-imagery.

We have used and improved the OpenSource GIS platform and its OpenGL rendering engine developed within the project ANR Mapstyle 3 [START_REF] Christophe | Map Style Formalization: rendering techniques extension for map design[END_REF]. For water depth rendering, we have implemented and improved in the rendering engine the ColorMap parameter of RasterSymbolizer from the Symbology Encoding standard from the Open Geospatial The general methodology of our implementation is presented in Figure 1 and is based on well-known architecture [START_REF] Brasebin | A knowledge base to classify and mix 3D rendering styles[END_REF] with data to visualise (dark blue), stylisation (yellow), rendering (red) and user (light blue). We focus our work on the sequence of these different parts and especially on data selection, data processing and stylisation with inspiration sources (map abstraction or photo-realism) [START_REF] Masse | Homogeneous geovisualization of coastal areas from heterogeneous spatio-temporal data[END_REF].

Results and Discussion

We present three types of animated geovisualisation of water depth based on tide information: (1) a 2D cartographic map (abstraction), (2) a 2D ortho-photorealistic image (photorealism) and (3) a 3D immersion (3Drealism). Figure 2 shows these three results at different tidal stages (low, medium and high level). Demonstration videos are also available 4 . We can observe in (1) that we have a perception of water animation with displacement of water surfaces in a 2D cartographic map. In (2), we have more realism with better perception of submarine topography due to natural colour interpolation. Finally, in (3), we can note that the third dimension gives more interaction possibilities (displacement in 3D) to users and increases the comprehension of the tide phenomenon in producing more realistic sea animations.

During the workshop, it will be interesting to discuss about both methodological aspects, architecture and results, and especially the problem of temporal representation of the tide phenomenon: some thoughts will be provided about required tools to develop to give better interaction between users and

  3 http://mapstyle.ign.fr/ last access 2016/05/09 Geovisualisation of animated tides in coastal area with an OpenSource OpenGL platform Consortium [9] for assigning colours, colour palettes and colour gradients to values or interval of values, e.g. water depth values. The colour interpolation option of the ColorMap parameter has also been implemented and provides a continuous rendering across multiple depth values, contrary to layer tints, which are commonly used in existing visualizations but not realistic.

Figure 1 :

 1 Figure 1: General methodology of our geovisualisation system

Figure 2 :

 2 Figure 2: Geovisualisation examples at different tide stage, from left to right, at (a) 0m (low tide), (b) 4m and (c) 8m (high tide) and from up to down, (1) a 2D cartographic map, (2) a 2D ortho-photorealist image and (3) a 3D immersion.

https://www.youtube.com/playlist?list=PLqYkYbT96Aq1FjC4LKzB SyIij1lD5D3ix last access 2016/05/09 renderer to manipulate time (speed, etc.) and stylisation (colours, highlights, stream, etc.).

Acknowledgement

This work is supported by the French National Research Agency, MapStyle project [ANR-12-CORD-0025]. We would like to thank IGN and SHOM for providing data.