Yannick Morel

Vincent Lebastard

Frédéric Boyer

Neural-based underwater surface localization through electrolocation

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I. INTRODUCTION

In underwater robotics, the electric sense ( [START_REF] Neveln | Biomimetic and bio-inspired robotics in electric fish research[END_REF]) is rapidly emerging as an effective complement to more classical navigation instruments, such as those based on acoustics, including SOund Navigation And Ranging systems (SONAR, [START_REF] Knight | Digital signal processing for SONAR[END_REF]) and Doppler Velocity Loggers (DVL, [START_REF] Larsen | High performance doppler-inertial navigation: Experimental results[END_REF]), or Inertial Navigation Systems (INS, [START_REF] Woolsey | Review of marine control systems: Guidance, navigation, and control of ships, rigs and underwater vehicles[END_REF]). This sensorial modality is based on manipulation of electric fields. Typically, the system applies a known electric field to its direct environment, then carefully monitors its variations. Presence of foreign objects in the system's vicinity will lead to perturbations of the emitted electric field. By detecting, measuring and interpreting these perturbations, it becomes possible to infer a range of information on the object(s) detected and more generally on the system's direct environment ( [START_REF] Neveln | Biomimetic and bio-inspired robotics in electric fish research[END_REF], [START_REF] Mciver | Towards a biorobotic electrosensory system[END_REF]).

The most common application of this sensing technology consists in underwater obstacle detection, localization and avoidance ( [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF], [START_REF] Dimble | Computationally efficient underwater navigational strategy in electrically heterogeneous environments using electrolocation[END_REF]), but a range of additional useful functionalities are achievable, including localization and movement coordination of teams of mobile robots ( [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF], [START_REF] Chevallereau | Electric sensor based control for underwater multi-agents navigation in formation[END_REF]), object shape recognition ( [START_REF] Boyer | Exploration of objects by an underwater robot with electric sense[END_REF]), and short range communication ( [START_REF] Wang | Underwater electric current communication of robotic fish: Design and experimental results[END_REF]). Such functionalities have been achieved using a Y. Morel is with the Commissariat à l'Energie Atomique et aux Energies Renouvelables (CEA Tech Pays de la Loire), Technocampus, 44340 Bouguenais, France, ymorel@gmail.com V. Lebastard and F. Boyer are with the IRCCyN Laboratory, Mines de Nantes, 44307 Nantes, France, {vincent.lebastard, frederic.boyer}@mines-nantes.fr

The work presented was performed as part of the strategic partnership between CEA Tech, Mines Nantes and ARMINES, funded by the région Pays de La Loire, in the frame of the Regional Technology Transfer Platform (PRTT) of CEA Tech. wide range of processing techniques. A number of results are based on techniques inspired from those in Electric Impedance Tomography (EIT, [START_REF] Snyder | Underwater object tracking using electrical impedance tomography[END_REF]), by which one attempts to exploit available electric measures to reconstruct the impedance map of the system's near environment. In general however, such techniques require expansive sets of measures and heavy computations ( [START_REF] Cheney | Electrical impedance tomography[END_REF]), and the estimation problem is oftentimes ill-conditioned ( [START_REF] Calderón | On an inverse boundary value problem[END_REF]). However, if instead of attempting to reconstruct a detailed impedance map, one assumes most of the environment is homogeneous and features only a limited number of foreign objects, it becomes possible to significantly alleviate the computational burden. In particular, one is then able to parameterize the scene using a relatively concise set of variables, for instance describing the objects' shape, electric properties and position in the scene. It has been shown that it is possible to reconstruct (or, more specifically: observe) these variables using electric measures together with a forward model, describing the relationship from scene geometry to achieved electric measures ( [START_REF] Lebastard | Localization of small objects with electric sense based on Kalman filter[END_REF], [START_REF] Lebastard | Environment reconstruction and navigation with electric sense based on a Kalman filter[END_REF]). This scene reconstruction or estimation problem can be addressed using different approaches, including well established techniques such as Kalman filtering ( [START_REF] Lebastard | Localization of small objects with electric sense based on Kalman filter[END_REF], [START_REF] Lebastard | Environment reconstruction and navigation with electric sense based on a Kalman filter[END_REF]). As an alternative to such methods, we have proposed in [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF] an approach based on the development of a neural networkbased forward model ( [START_REF] Haykin | Neural Networks: A Comprehensive Foundation[END_REF]). In particular, for a given type of scene (i.e. presence of a pair of cooperating mobile robots in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF], or for a spherical object in [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF]), we collect series of electric measures descriptive of this scene (for a range of acceptable relative robot positions or sphere locations, respectively). Such data sets can be generated using either analytical forward models (such as that in [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]), numerical models (such as that based on the Boundary Element Method in [START_REF] Porez | Multi-physics model of an electric fish-like robot: Numerical aspects and application to obstacle avoidance[END_REF], see [START_REF] Liu | CPG driven locomotion control of quadruped robot[END_REF] for a discussion of BEM), or an experimental setup (such as that described in [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]). Once such a model is constructed, it can be exploited to reconstruct the scene based on available electric measures. The inversion method proposed in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF] (also used in the following, in [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF], and comparable to the beamforming technique presented in [START_REF] Nguyen | Sparse beamforming for active underwater electrolocation[END_REF]) consists in presenting to the forward model a range of candidate scene structures, as described by the relevant variables (e.g. relative position of a robot, or location of a sphere). Then, comparing the neural model's output to the current available measure, it becomes possible to infer relative merit of different candidates. Specifically, owing to the forward map's continuity (as discussed in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF], and as illustrated by existing numerical and analytical models), and assuming that the forward neural model is faithful, outputs of the neural model for candidates close to the actual scene (as measured in terms of distance in the relevant variable space) will remain close to the available measure (distance in the measure space). Conversely, as candidate scenes become more distant from the actual one, the model outputs in general become more dissimilar to the measures. Accordingly, representing over the domain of definition of possible scenes (or of candidate scenes) the similarity or dissimilarity between corresponding model outputs and the available measure(s) allows to discriminate possible or likely candidate scenes from unlikely ones ( [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF], [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF]).

One should note, however, that the forward map that we attempt to invert using the aforementioned procedure is not necessarily invertible (in practice, even in ideal conditions, it is only at most locally invertible due to range limitations). Invertibility depends on a number of factors, including the number of available independent measures and complexity of the scene (i.e. number of independent variables necessary to describe it). In addition, impact of the presence of a foreign object on measures rapidly decreases with distance. More specifically, the corresponding perturbation on measures typically becomes indistinguishable from measurement noise at a distance roughly equivalent to the physical length of the dipole producing the electric field (specific detection range depends on the quality of electronics used for signal acquisition, dimensions and electric properties of the object to be detected, exposed surface and material of the electrodes, and power used to produce the electric field, as discussed in [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]). Hence, once an object is far enough from the sensor, it becomes impossible to infer its situation based on electric measures. Accordingly, attempting to directly construct or identify an inverse model (from measures to scene) can be expected to prove, at the very least, problematic, if not outright impossible. Instead, the proposed approach has several benefits. In particular, precisely because it is not attempting to reconstruct the inverse map, it circumvents issues of ill-definition. Typically (as discussed in [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF]), in an area of scene configuration space in which the mapping is invertible, a single scene configuration candidate will emerge as a best match (that is, when used as input to the neural model, yielding the neural output that provides the closest match to the measure available). Conversely, in a noninvertible area, such as when the sensor is to far away from the object to be detected and localized, entire swathes of the scene configuration space will emerge as equally likely. That situation makes it impossible to discriminate one single most likely location (as expected, since it is not possible to do so from available data). Yet, it remains possible to: a) come to a realization that there is ambiguity and that it is impossible to highlight a particular scene configuration candidate (in other words, we know that we do not know), b) still discriminate likely areas (instead of specific candidates) from unlikely ones. Typically, when located too far from the target object, it still remains possible to determine the area of space around the sensor within which the object is unlikely to be located, versus areas further away from the sensor where it may be.

In [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF], we applied the approach delineated above to the case of a scene featuring a single spherical object. In the following, we apply the same approach to a case in which the sensor is placed in the presence of a planar, vertical wall. For ease of exposition, the problem is treated in a horizontal plane of reference, in such a manner that, assuming the considered wall is infinite, two variables are sufficient to entirely describe the scene; a distance d ∈ R + , and a relative orientation ψ ∈ (-π, π] descriptive of the orientation of the sensor with respect to the wall. Definition of both variables is provided in Figure 1, which also provides a schematic view of the type of sensors considered hereafter: a slender, cylindrical electric probe composed of an insulating plastic shell (light grey in Figure 1) and of four pairs of steel electrodes (in a darker shade of grey, two hemispherical pairs at the probe's extremities, two additional cylindrical ones on the fore half). In the following, we show how, using the approach discussed in [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF], we are able to capture a neural model descriptive of the relationship between a scene such as that in Figure 1 and electric measures performed by a sensor (e.g. the probe represented in this same figure). We then employ the aforementioned inversion procedure to estimate the value of d and ψ corresponding to a particular electric measure performed. Finally, we show that the information obtained is sufficient to close the motion control loop and allow movements of the sensor in good intelligence with its environment, including regulation of the probe's position at given distances and relative orientations from the wall.

Contribution of the presented work stems from a number of factors. In particular, the results presented illustrate the generalizability of the proposed approach, which had previously successfully been applied to solve relative positioning and coordination problems (in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF]) and sphere localization problems ( [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF]), and is here used to address the problem of localizing a continuous planar wall. In addition, the presented work provides a novel algorithmic tool to solve the wall localization problem in electrolocation, which had previously been addressed using for instance Kalman filters ( [START_REF] Lebastard | Environment reconstruction and navigation with electric sense based on a Kalman filter[END_REF]), but never using the proposed neural approach. Finally, an interesting feature of the proposed approach, differentiating it from alternative techniques, lies in its capacity of handling scenes more complex than those used to construct the direct model it relies on. Specifically, the neural model we construct is based on the presence of a single planar wall. However, it can be used in cases in which several (possibly intersecting) walls are present, forming combinations of corners and corridors, and allow safe navigation of one such scene without requiring any adjustment. By comparison, alternate approaches such as that in [START_REF] Lebastard | Environment reconstruction and navigation with electric sense based on a Kalman filter[END_REF], while still able to handle corners and corridors, will typically require additional dedicated models.

In Section II, we provide a brief description of the experimental setup we used to collect data, present a simple analytical model used at latter stages to test the proposed approach, and discuss the neural model constructed based on data collected. Then, in Section III, we show in what manner we are able to exploit the neural model of Section II to infer position of a planar wall from performed electric measures. We also show how this same model can be used to navigate corners and corridors. Section IV concludes this paper.

II. EXPERIMENTAL SETUP AND COLLECTED MEASURES

The problem addressed in the following is that of estimating the relative position of a mobile underwater system with respect to an insulating wall (or more generally plane surface) using electrolocation. This electrolocation approach follows a simple principle; the system, using a set of electrode located on its surface, applies an electric field to its surroundings. These same electrodes are then used to measure relevant electric variables, such as for instance difference of electric potential between electrodes (as in [START_REF] Bai | Finding and identifying simple objects underwater with active electrosense[END_REF]), or electric currents flowing through the electrodes (see for instance [START_REF] Lebastard | Environment reconstruction and navigation with electric sense based on a Kalman filter[END_REF]). Based on such measures, one is able to infer a range of information descriptive of the system's direct environment. In particular, presence of foreign objects, whose electric conductivity is distinct from that of the surrounding fluid, will perturb the field emitted by the system. Detecting such perturbations in the measures performed can allows to estimate a range of information regarding the foreign objects in question, including relative position in space with respect to the system, size, shape, and conductivity ( [START_REF] Lebastard | Environment reconstruction and navigation with electric sense based on a Kalman filter[END_REF], [START_REF] Bai | Finding and identifying simple objects underwater with active electrosense[END_REF]).

A. Experimental Setup

Hereafter, we apply the neural-based approach introduced in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF] (where it is used to estimate relative position of underwater mobile systems) and later used in [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF] to locate spherical objects underwater. Building upon a series of measures characterizing electric current patterns in presence of an insulating wall, it becomes possible to construct a forward model, descriptive of the relationship from scene geometry to electric measures performed. As previously alluded to, in the case considered, which is that of a single mobile system in the presence of an infinite wall, and considering motion in the horizontal plane exclusively, the scene can be described using two variables; the distance d ∈ R + between wall and mobile system, and the orientation ψ ∈ (-π, π] of the mobile system with respect to the wall. The proposed approach will be tested using data collected on an experimental setup, described in detail in [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]. This setup is composed of 1m×1m×1m glass tank filled with fresh water. The setup uses electric probes (qualitatively representing a mobile vehicle equipped with the electric sense), which are composed of a cylindrical, non-conductive plastic body (of length 22cm, 2cm diameter), and feature four pairs of steel electrodes located along the length of the probe (three pairs on the fore half, one pair aft), with each pair of electrodes following a strict left-right symmetry (probe shown in Figure 1). In practice, a 22.5KHz sinusoidal signal is applied to the aft-most pair of electrodes. Then, the peak-to-peak value of the current flowing through the three fore-most pairs of electrodes (connected to a common ground) is measured. These probes are mounted on a three Degree of Freedom (DoF) actuation system, allowing movement in the horizontal plane (two translations, one rotation; see [START_REF] Lebastard | Localization of small objects with electric sense based on Kalman filter[END_REF], [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF] for additional details on the experimental setup).

Position of the probe relative to a particular wall is described by the aforementioned distance d and relative orientation ψ. Specifically, and as shown in Figure 1, we define d as the distance between the probe's center and the nearest point of the wall, and assign ψ = 0 to the relative orientation for which the probe's fore is pointing away from the wall. Below, we discuss the relationship between scene geometry, as described by d, ψ, and the current measures obtained by the probe. Then, data collected using the experimental setup described here is used to train a neural network providing an alternate model of the same relationship, and comparisons between experimental measures and models are performed.

B. Analytical Model

For ease of interpretation, currents measured by the probes described in Section II-A are summed and subtracted for each electrode pair. Qualitatively, looking at the sum of currents for each pair of electrodes on the probe (that is, for each pair, the sum of current flowing through the leftside and the right-side electrodes) provides insights into the position of a perturbing object in the longitudinal direction of the probe (that is, along its longitudinal axis of symmetry). Conversely, considering left-right differences (difference between the current flowing through the left-side electrode and that flowing through the right-side one) helps in determining whether objects are located port or starboard. In the following, we note the current sums i lo (t) ∈ R 4 , t 0, where lo stands for longitudinal, and i la (t) ∈ R 4 , the left-right differences, where la stands for lateral. These currents can be computed as follows (see details in [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]),

i lo (t) = C 0 K(t)C 0 u, t 0, (1) i la (t) = (1 + S ⊥ )L(t)C 0 u, ( 2
)
where the conductance matrix C 0 ∈ R 4×4 is a function of probe geometry and of the surrounding fluid's conductivity, the column vector u ∈ R 4 describes the electric potentials assigned to the four pairs of electrodes, S ⊥ ∈ R is a scaling factor, and K(t), L(t) ∈ R 4×4 are matrices computed as follows,

k ij = 1 4πγ∥r ij (t)∥ , i, j = 1, . . . , 4, t 0, (3)
l ij = - Āi 4π∥r ij (t)∥ 3 [ -sin(ψ(t)) cos(ψ(t)) ] r ij (t), (4)
where k ij (t), l ij (t) ∈ R represent the i th line-j th column entries of K(t) and L(t), respectively, γ ∈ R describes the medium's conductivity, Āi ∈ R is a scaling factor function of the surface of the i th pair of electrodes, r ij (t) ∈ R 2 is the vector describing the position of the i th electrode pair with respect to a mirror image of the j th pair (specifically, the j th pair of electrodes of a virtual probe, mirror image of the actual one across the wall's surface, see [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]), computed as follows,

r ij (t) = [ 2d(t) 0 ] + [ (x i + x j )cos(ψ(t)) (x i -x j )sin(ψ(t)) ] , t 0, (5)
where x k ∈ R, k = 1, . . . , 4, represents the position of electrode pair k along the longitudinal axis of symmetry of the probe. Using (1)-( 5), one is able to compute the value of lateral and longitudinal currents for given scene geometries, as described by values of d(t) and ψ(t). Finally, note that while the above model provides i lo (t) and i la (t), t 0, for all four pairs of electrodes (meaning i lo (t), i la (t) ∈ R 4 ), due to the fact that whatever current flowing into the front three pairs is flowing out of the aft-most pair of electrode (or emitter, [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]), the overall set of measures contains information redundancy. Accordingly, it is typical to only consider measures on the front three electrode pairs, as it allows to reduce dimension of the problem with no loss of information. Therefore, in the following, we will only consider īlo (t), īla (t) ∈ R 3 , which describe currents flowing through the first three pairs of electrodes only.

C. Neural Model

We follow the approach introduced in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF], and approximate the relationship from scene geometry, described by d, ψ, to collected measures of the current (row) vector i(d, ψ)

[ īT lo (d, ψ) īT la (d, ψ) ] ∈ D i ⊂ R 6
, for a given (fixed) fluid conductivity γ, using the following neural formalism, [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF] where î(d, ψ) ∈ D i provides an estimate of the current vector i(•, •) for a scene configuration given by d, ψ, the row vector φ(d, ψ) ∈ R p contains polynomial functions of d and ψ chosen a priori, and W ∈ R p×6 is a weight matrix, to be selected in such a manner that î(d, ψ) provides an accurate estimate of i(d, ψ) over the domain of definition of admissible scenes D s .

î(d, ψ) φ(d, ψ)W, (d, ψ) ∈ D d × D ψ D s ,
To train the above model, that is, to select an appropriate weight matrix W in (6), we use sets of data representative of the scene we intend to capture. This data can be generated using either analytical (or numerical) models, such as that in Section II-B, or experimental setups such as that discussed in Section II-A. In the following, we will work using the latter: experimental data captured on the electric bench setup of Section II-A. This data was collected by adjusting the position and orientation of an electric probe with respect to one of the tank's four walls. To reduce impact of the presence of the other three walls on measures performed, movements of the probe were constrained to a line perpendicular to the wall of interest, and crossing the center of the tank. Keeping the probe between the tank's center and the wall of interest, we are able to maintain the probe at a distance greater than half a meter from other walls, which is generally sufficient to ensure the impact of their presence on measures remains largely negligible. Measures i m (d, ψ) ∈ R 6 were collected for the front three pairs of electrodes (a measure i m (d, ψ) corresponding to the actual currents i(d, ψ) ∈ R 6 plus measurement noise). These measures were collected along straight trajectories starting at the tank's center and ending at the wall of interest. Movement speed of the probe and sampling frequency were adjusted to obtain samples about every 2mm along these trajectories. The process was repeated for relative orientations of the probe with respect to the wall ψ spanning (-π, π] with steps of π/36rad (5deg), for a total of 72 trajectories.

We used the collected data to determine an appropriate value for the weight matrix W . In particular, we built a current measure matrix

I m (d s , ψ s ) [ i T m (d 1 , ψ 1 ) . . . i T m (d q , ψ q ) ] T ∈ R q×6 ,
where i m (d j , ψ j ) is the current measure collected at distance d j from, and relative orientation ψ j with respect to the wall, j = 1, . . . , q, and d s , ψ s , are vectors of distances and relative orientations containing all values at which measures were collected. We then built the corresponding polynomial matrix

Φ(d s , ψ s ) [ φ T (d 1 , ψ 1 ) . . . φ T (d q , ψ q ) ] T ∈ R q×p , where φ T (d j , ψ j ) is a polynomial function of its inputs. Typically, we use φ(d j , ψ j ) = [ d 0 j ψ 0 j . . . d 0 j ψ j k2 d 1 j ψ 0 j . . . d j k1 ψ j k2 ] ∈ R p , where k 1 , k 2 ∈ N are φ's polynomial orders and, by construction of φ(•, •), p = (k 1 + 1) × (k 2 + 1)
. Then, we select the neural weight matrix W m ∈ R p×6 corresponding to (or descriptive of) the collected set of measures as follows (computed as the most-likely, a posteriori estimate, based on collected measures, [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF]),

W m = ( Φ T (d s , ψ s )Φ(d s , ψ s )+Σ m ) † Φ T (d s , ψ s )I m (d s , ψ s ), (7)
where the design constant Σ m ∈ R p×p allows to account for the presence of measurement noise in the data, and where explicit mention of W m 's dependency on (d s , ψ s ), was omitted for brevity. Then, the captured neural model We observe that, over the data set used for training, the mean error between the analytical model in Section II-B and experimental measures is 2.83e -2 mA (with a standard deviation of about 7.95e -2 mA). Scaled by each of the six considered currents maximum amplitude (specifically, for each current in īlo and īla we compute the amplitude of the experimental data as the difference between maximum and minimum current value measured, we then divide the mean error obtained for each of these six currents by this amplitude to better contextualize the error), we obtain an average error of 1.003% of each current's respective maximum amplitude (standard deviation of 2.6%). For the neural model, the mean error is of about 1.93e -3 mA (0.068% scaled), with a standard deviation of 4.08e -3 mA (0.14% scaled). A comparison between experimental data and output of the neural model is provided in Figure 2, where the copper surface corresponds to the collected set of data for the 3 rd lateral current (noted i la3 (d, ψ)). Output of the neural model for a range of (d, ψ) coordinates is marked with •'s in Figure 2. Consideration of Figure 2 gives rise to a number of points. First, the neural model appears to provide a reasonable approximation. However, such a judgement of value is fairly difficult to perform when considering Figure 2 alone. In particular, if currents' amplitude is of the order of 1mA near the wall, it very rapidly drops (decreases in 1/d 2 , as seen in ( 4)). To the extent that beyond d = 0.1m, it becomes difficult to discern anything as currents appear to be identically null. In reality, there remains notable and exploitable variations in currents well beyond d = 0.1m. To highlight that fact, we apply a logarithmic transformation to the data (as well as to the output of the neural network) as follows,

i lj (d, ψ) = sign(i j (d, ψ)) ( log(|i j (d, ψ)| + b j ) -log(b j ) ) max (d,ψ)∈Ds ( log(|i j (d, ψ)| + b j ) -log(b j ) ) , j = 1, . . . , 6, (8)
where i j (d, ψ) represents the j th entry of the current vector i(d, ψ), i lj (d, ψ) is the logarithmically transformed current corresponding to i j (d, ψ), and b j represents an offset used to avoid the logarithmic singularity at zero. Applying this transformation to the same data shown in Figure 2, it becomes easier to discern changes in currents at greater distance from the wall, as seen in Figure 3.

III. WALL LOCALIZATION

In the following, we employ the same inversion approach as that in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF], [START_REF] Morel | Neural-based underwater spherical object localization through electrolocation[END_REF], which is also comparable to the beamforming technique in [START_REF] Nguyen | Sparse beamforming for active underwater electrolocation[END_REF]. Specifically, we attempt to a) detect presence of a wall, and b) estimate its location and orientation relative to that of the probe. To achieve these objectives, we exploit electric measures performed by the probe together with the neural model discussed in Section II-C. In particular, when a measure of current i m (d, ψ) is made available, we use the developed neural forward model to compute possible current measures i c (d c , ψ c ), corresponding to a range of candidate relative distances d c and orientations ψ c . Then, comparing candidate measures i c (d c , ψ c ) to the actual measure i m (d, ψ), it becomes possible to discriminate areas of the scene space D s likelier to contain the actual wall position (d, ψ), from areas unlikely to contain it. Specifically, numerically computing the prediction error e p (i m , d c , ψ c ) ∥i m -i c (d c , ψ c )∥ over a given candidate range (d c , ψ c ) ∈ D c ⊂ D s , areas of D c featuring (relatively) high values of e p are less likely to contain the actual wall position, whereas areas featuring lower error are more likely to contain it.

A. Wall Localization over Open-loop Probe Trajectory

To test the proposed approach, we consider a case in which the probe is in the presence of a single wall. The probe is made to move along a quarter circle. Trajectory of the probe's center is represented with a thin blue line in Figure 4, with an initial position marked with an ×, and a final position with a ⋄. The shape of the probe itself is also shown in its final position. The wall is marked with a thick black line (along the x = 0 line). Orientation of the probe in its initial position is such that it is pointing towards the wall (ψ = πrad, in accordance with Figure 1), and is made to continuously decrease to reach ψ = π/2rad at the final position (probe parallel to the wall, as shown in Figure 4), in such a manner that the probe's longitudinal axis of symmetry remains tangent to the xy-trajectory of the probe's center. To estimate position of the wall, we used the forward neural model discussed in Section II.

Model inversion was performed using the iterative two-step procedure presented in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF]. In a first step, we compute predicted current measures on a grid of candidate positions

(d c , ψ c ) ∈ [1e -2 m, 0.4m] × [0, 2π)
Dc1, with steps of 5cm in the d direction and π/12rad in the ψ direction. A first estimate of the wall's relative position ( d1 , ψ1 ) is selected as the candidate position for which the current prediction error is minimal over the candidate grid; that is,

( d, ψ) = argmin (dc,ψc)∈Dc e p (i m , d c , ψ c ). (9)
Then, with use a finer candidate grid, centered about the previous estimate ( d1 , ψ1 ), with a grid width of 0.1m in d and π/12rad in ψ, and grid steps of 2.5e -3 m in d and π/180rad in ψ. This second grid's candidate with lowest prediction error is selected as the estimated relative position of the wall. We represented this estimated position as a thick blue line in Figure 4, with initial estimated wall position marked with an ×, and the final one with a •. The current measures used to estimate position of the wall as shown in Figure 4 were obtained using the reflection model in Section II-B (note the probe's final position reflection, represented to the left of the wall in Figure 4, with a • marking its center). Reconstructing the estimated wall (closest with respect to the probe) position (in thick blue in Figure 4), and comparing it to its actual (closest) position over the quarter circle trajectory considered, we obtain an average position error of about 7e -3 m with standard deviation 6.2e -3 m.

In other words, and as is apparent from Figure 4, the approach is able to reconstruct the wall's position, based on the neural model and provided current measures, with reasonable accuracy. The reconstruction error for the final probe position is shown in Figure 5, computed over a wide range of candidate distances and orientations. The resulting surface's lowest point designates the most likely candidate position. Trajectory of this estimate in the d-ψ plane is shown in blue; the actual relative position is in green. Fig. 5. Error surface for the final probe position, the surface's minimum represents the most likely candidate position and is selected as an estimate of this position. This estimate's trajectory is shown as a blue line, whereas the actual wall relative position is shown in a green line.

B. Closing the Motion Control Loop

Exploiting the information provided by the wall position estimate, it becomes possible to guide the probe's movements in such a manner that it avoids colliding with the wall, or even remains constrained at a consistent distance from this wall. More specifically, we assign the probe simple nonholonomic kinematics of the usual form,

ẋ(t) = v(t) cos(ψ a (t)), x(0) = x 0 , t 0, (10) ẏ(t) = v(t) sin(ψ a (t)), y(0) = y 0 , (11) ψa (t) = ω(t), ψ a (0) = ψ a0 , (12)
where x(t), y(t) ∈ R, represent the probe's center position, ψ a ∈ (-π, π] its orientation with respect to the xy-frame of reference (with ψ a = 0 when the probe points towards the x-axis' positive direction), v(t), ω(t) ∈ R, are the longitudinal displacement speed and angular velocity, which act here as control inputs to system (10)- [START_REF] Snyder | Underwater object tracking using electrical impedance tomography[END_REF]. These inputs are computed as the sum of two terms, v(t) = v c (t) + v j (t), ω(t) = ω c (t) + ω j (t), t 0, where v j (t), ω j (t) ∈ R, are obtained from a joystick's output, allowing the user to direct movements of the probe, whereas v a (t), ω a (t) ∈ R, are computed using wall position estimates as follows,

v c (t) = min(max(| d(t) -d d | -d t , 0), v s ), t 0, ( 13
)
ω c (t) = g ω ( sign(v j (t)δ ψ (t))ψ dm tanh ( d(t) -d d d d π ) +δ ψ (t) -sign(δ ψ (t))π/2 ) , (14)
where v s = 0.1m/s represents the maximum assignable velocity, d d = 0.1m is the distance from the wall at which we intend to regulate the probe, d t = 0.01m is a distance threshold, δ ψ (t) π -ψ(t) describes orientation of the probe with respect to the wall, ψ dm = π/4 represents a maximum desired relative heading, g ω = 0. v c (t) is by construction saturated at v s = 0.1m/s, whereas we saturate the value computed by ( 14) at π/5rad/s. In addition, value of v j (t) and ω j (t) is obtained by scaling the joystick's output signals on the front-back and left-right axes from 0 with the stick at the center, up to 0.05m/s and 0.35rad/s at maximum tilt on either axis, respectively. The closed-loop is simulated using MathWorks R ⃝ Simulink R ⃝ , together with a USB joystick. The neural model and inversion procedures are the same as in Section III-A, the electrical model used to produce electric measures is that in Section II-B. A wall is placed parallel to the y-axis at x = -0.5m, the probe is initially located at (x, y) = (-0.2m, -0.2m) with orientation ψ a = πrad. Control signals are shown in Figure 6; the corresponding probe trajectory is represented in blue in Figure 7. Initially, the user is providing a positive velocity command (in blue in Figure 6, up). The regulating command (13) (in red, up) is also pushing the probe forward, until it reaches the desired distance from the wall d d = 0.1m. Once the probe has reached its desired position with respect to the wall (at fixed relative distance and orientation), the command (13)-( 14) is able to maintain this desired position, using only estimates of distance and orientation provided by the electric measures and estimation algorithm discussed above. At around 8s, the user is attempting to wrestle orientation of the probe away from its regulated value. However, the regulating signal compensates and is able to maintain the orientation at a value near the desired one, as seen in the probe last position shown in Figure 7. Movements parallel to the wall (along the green line in Figure 7)

Finally, we reproduce the same simulation, using the same neural forward model, descriptive of the presence of a single planar wall, but now place the probe in presence of combinations of walls, forming either a corner or a corridor. We use the same control inputs ( 13)-( 14) to regulate distance from the wall and orientation of the probe. Results are shown in Figure 8 and Figure 9. In both cases, the estimation procedure is able to reasonably well estimate position of (the closest) wall. In the case of the corner, distance is underestimated in a given area due to the concurrent (and significant) influence of both walls on measures. Nevertheless, the method generalizes well in that it is capable of safely regimenting movements of the probe along the walls. Note in addition that it did not require new or additional modeling considerations to account for the presence of several walls.

IV. CONCLUSION

In the work presented, we have applied the electrolocation algorithm (first introduced in [START_REF] Morel | Estimation of relative position and coordination of mobile underwater robotic platforms through electric sensing[END_REF]) to estimate the relative distance and orientation of an underwater planar surface from an electric sensor. The approach is based on the development of a neural forward model, describing the map from scene geometry to electric measures. This neural network can be trained using either experimental data (as was the case here), or numerical data obtained from analytical of numerical models. To invert this forward model and estimate scenerelative information from measures, a series of candidate relative wall positions are fed to the neural network. Output for each candidate is then compared to the available electric measure. Based on similarity between model outputs and measures, it becomes possible to discriminate likely scene configurations from unlikely one and, in fine, reach an estimate. The approach was shown to work using a forward model obtained from experimental data. Numerical simulation results show that the information reconstructed is sufficient to enforce virtual movement constraints on a system and regulate its distance and orientation with respect to the detected wall (while leaving the degree of freedom of displacement along the wall free). Using a single forward model descriptive of the presence of a single wall, it is possible to successfully enforce such virtual constraints for combinations of walls, including corners and corridors. Further generalization of the proposed approach to localize non-planar walls (slanted, curved, or of arbitrary shape) is ongoing.

Fig. 1 .

 1 Fig. 1.Definition of scene configuration variables: distance d and orientation ψ; wall marked as a thick black line to the left.

  can compute predictions î(d, ψ) = φ(d, ψ)W m of current values i(d, ψ), for (d, ψ) ∈ [0, 0.5m] × (-π, π], domain over which experimental data was collected to train the model. To compare accuracy of the analytical model in Section II-B and of the above neural approach, we constructed one such neural model, using k 1 = k 2 = 12, and Σ m = 1e -6 I dp , where I dp ∈ R p×p is the p-dimensional identity matrix. The numerical value of parameters in the analytical model in Section II-B are u = [ 0 0 0 10V ] T , S ⊥ = 3.51, γ = 0.04S/m, Ā = [ π/2 2 2 π/2 ] T r 2 e m 2 , the probe radius is r e = 0.01m, positions of the electrode pairs' centers along the probe longitudinal axis of symmetry are x = [ 0.105 0.068 0.03 -0.1 ] T m (positive in the fore direction, negative aft), and the conductance matrix has the following value,

Fig. 2 .

 2 Fig. 2. Comparison between experimental data (copper surface), and output of a neural model captured using the represented data, shown with •'s.

Fig. 3 .

 3 Fig. 3. Comparison between experimental data (copper surface) and neural output (•'s) with logarithmic scaling.

Fig. 4 .

 4 Fig. 4. Electric probe moving in an open-loop trajectory; probe trajectory represented with a thin blue line, trajectory of the wall position estimate shown with a thick blue line, initial probe position marked with an ×, final position with a ⋄, final position of the probe reflection marked with a •.

Fig. 7 .Fig. 8 .

 78 Fig.7. Closed-loop probe trajectory, with virtual constraint at 0.1m from the wall.

Fig. 9 .

 9 Fig. 9. Closed-loop probe trajectory in a corner.

  2 is a control gain, and sign(•) is the usual signum function. Value of Fig. 6. Control signals, v j (t), ω j (t), t 0, shown in blue, vc(t), ωc(t), in red.

		0.1
	[m/s]	0.05
	vc(t), vj(t)	0
	-0.05
			0	2	4	6	8	10
						Time [s]
		1
	[rad/s]	0.5
	ωc(t), ωj(t)	-0.5 0
		-1	0	2	4	6	8	10
						Time [s]


Sylvain Lanneau

Vincent Lebastard

Frédéric Boyer

Object shape recognition using electric sense and ellipsoid's polarization tensor

This paper deals with the geometrical properties of an ellipsoidal object (aspect ratio, volume, orientation) estimation with an underwater sensor inspired by the uncommon sense of the electric fish. The proposed method first locates the object independently of its geometrical properties thanks to the MUSIC (MUltiple SIgnal Classification) algorithm and then, estimates the geometrical properties using an optimization method and the object's electrical response model. The simulation results show the relevance of the method.

I. INTRODUCTION

In the field of underwater robotics, a big challenge consists in giving autonomy to robots navigating in turbid water and confined or post catastrophic environment. In such conditions, usual solutions such as vision and sonar are not efficient enough for trajectory control or cartography. But an alternative is developing since the early 2000's, by taking inspiration of the electric sense, a way of environment perception used by some fish which live in muddy waters and hunt by night (Gnathonemus petersii in Africa or Eigenmannia virescens in South America). This original sense has been discovered by biologists in the 50's [START_REF] Lissmann | The mechanism of object location in Gymnarchus Niloticus and similar fish[END_REF]. The fish polarizes its body thanks to a dedicated organ and a surrounding electric field is thus created. The electrical field response of the objects in the neighborhood to that field is caught by the fish with a high density distribution of electroreceptors, spread out over its skin. The fish decodes this electric image [START_REF] Assad | Electric organ discharges and electric images during electrolocation[END_REF] to recover the environment features. By observation and comprehension of these biological mechanisms, researchers in bio-inspired robotics designed new kinds of sensors directly inspired by electric sense. Simply composed of two emitting and two receiving electrodes at the beginning [START_REF] Solberg | Active electrolocation for underwater target localization[END_REF], researchers made these systems evolve toward more complex sensors. Nowadays, the sensors are cylindrical multi-electrodes devices using either U-I technology (voltage polarization and current measurements) [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF], [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF], [START_REF] Snyder | Underwater object tracking using electrical impedance tomography[END_REF] or U-U technology (voltage polarization and voltage measurements) [START_REF] Bai | Sensing capacitance of underwater objects in bio-inspired electrosense[END_REF]. The U-I sensor, which is the purpose of this paper, is able to accurately estimate a sphere's location and size thanks to a Kalman filter algorithm [START_REF] Lebastard | Environment reconstruction and navigation with electric sense based on a kalman filter[END_REF]. With this technique, location and size are estimated at the same time. Indeed, we know the object's shape and position informations, both contained in the measured currents, to be intimately related to each other in a non-linear relationship. It is clearly shown by the Rasnow's potential perturbation model for a sphere of radius 1,2,3 Sylvain Lanneau, Vincent Lebastard and Frédéric Boyer are with IRCCyN, Ecole des Mines des Nantes, 4 Rue Alfred Kastler, 44300 Nantes, France {sylvain.lanneau,vincent.lebastard, frederic.boyer}@mines-nantes.fr a and of center O [START_REF] Rasnow | The effects of simple objects on the electric field of Apteronotus[END_REF]:

δφ(M ) = χ a r 3 E 0 .r (1)
where δφ(M ) is the potential perturbation evaluated at M point, χ is the contrast factor, representing the relative object conductivity with respect to that of the water, E 0 is an external and uniform electric field in the whole sphere, r = --→ OM and r = r . A recent work [START_REF] Bai | Finding and identifying simple objects underwater with active electrosense[END_REF] showed the feasibility, in the case of an ellipsoidal object, of location and geometrical informations extraction using the U-U technology SensorPod probe. The algorithm estimates the object's features in a sequential process, by an active alignment of the sensor with respect to the object's main axis and suitable models for parameters estimation. Ellipsoidal objects are in fact of particular interest. The ellipsoid is the next shape after the sphere (and including it) in term of complexity. Moreover, [START_REF] Khairuddin | Fitting ellipsoids to objects by the first order polarization tensor[END_REF], [START_REF] Ammari | Polarization and moment tensors, with applications to inverse problems and effective medium theory[END_REF] showed that the response of a compact object to a uniform electric field can be approximated, at first order, by that of an ellipsoid. We propose here a new method for estimating the geometrical parameters of a prolate ellipsoid (one long axis and two shorter axis), using our U-I technology probe. It consists in the estimation of three parameters. The two first ones are those which uniquely define the ellipsoid. They can either be the major semi-axis a and the minor semi-axis b, or the aspect ratio η = a/b and the volume V . The third parameter is the pose angle between the sensor main axis and the ellipsoid semi-major axis, denoted θ 0 (see figures 1b and 1c). The approach is based on methods recently introduced in the field of applied mathematics [START_REF] Ammari | Modeling active electrolocation in weakly electric fish[END_REF], in which electrolocation and shape recognition is numerically addressed with a dense array of electroreceptors. It consists in separating position and geometry informations from the sensor's input currents. This is performed in two steps : 1) locate the object independently of its intrinsic properties using the MUSIC (MUltiple SIgnal Classification) algorithm [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF], [START_REF] Devaney | Super-resolution Processing of Multi-static Data Using Time Reversal and MUSIC[END_REF] ; 2) knowing the position, estimate its geometrical properties with a least square optimization algorithm and a suitable equations system solving method. See a scheme of the process on figure 3. The results introduced in this paper, obtained in simulation, demonstrate that the method can be implemented with our sensor, even though it has much less electrodes compare to [START_REF] Ammari | Modeling active electrolocation in weakly electric fish[END_REF]. These results are particularly encouraging and constitute a first step before their experimental implementation. After having described in details the sensor and the test bed in section II, the currents models in section III and the assumptions about the sensor and the objects in section IV, results on localization are presented in section V and on shape recognition in section VI. Section VII deals with future works for improvements and experimental implementation of the approach.

O c i c j c E 1 E 2 E 3 E 4 O 0 u v θ 0 y c 0 x c 0 Ω V (

II. THE ELECTROLOCATION TEST BED

A. Sensor

The sensor is called slender probe due to its large aspect ratio and is composed of four macro-electrodes spaced by cylindric insulating parts, see figure 1a. Each macroelectrode can be set under an electric potential with respect to the others and this potential difference leads to an electric current flowing through all the conducting medium. In particular, the electric current flowing through the individual electrodes that compose each of the macro-electrodes can independently be measured with a remote Dspace R acquisition device (12 bits resolution, 5Hz sampling frequency).

B. Tank and Cartesian robot

The probe described above is immersed into a 1m sided cube tank. On the top of this tank, a Cartesian robot allows controlled translations along horizontal x and y axes and Fig. 2: The water tank topped with the Cartesian robot, permitting a planar motion of the sensor. A scene is composed of the sensor and objects lying in the same plane. rotation about the vertical z axis, see figure 2. The motion control is performed with a DSpace R system with custom program. The sensor being linked to this robot thanks to a rigid epoxy fiber tube, its motion in a plan can be controlled. For a complete description of the robot (electronics schematics, noise level, positioning precision, sensitivity to temperature) refer to [START_REF] Servagent | Electrolocation sensors in conducting water bio-inspired by electric fish[END_REF]. A scene can be composed by placing some objects of different shapes, sizes and materials at different locations in the tank.

III. CURRENTS MODELS

The model of currents describes the direct problem, which consists in knowing the electric currents flowing through the electrodes for a given scene, defined by the sensor's, the objects' and the medium's characteristics. In fact, we have at our disposal two models for the direct problem. They are both entirely described in [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF] and we briefly present them in this section. The first one, called BEM (standing for Boundaries Elements Method), is derived from the electrostatics theory and the currents are computed thanks to a discretization in finite elements of all the boundaries of the scene. It provides a numerical solution of the direct problem [START_REF] Porez | Multi-physics model of an electric fish-like robot: Numerical aspects and application to obstacle avoidance[END_REF], which will be used as the sensor input currents in this paper (input data of the process, as shown on figure 3). The second one, known as the analytical model, describes the measured currents with simple explicit mathematical expressions deduced from a reduction process described in [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]. The methods used for the localization and shape recognition are based upon this analytical model, see figure 3.

A. BEM Formulation of the direct problem

An electric potential difference is imposed between the electrodes of the sensor, so there exists an electric potential φ in the surroundings, satisfying Laplace equation : ∆φ = 0. All over the scene, Ohm's law is locally described by

Measured currents

From BEM computation Localization Based on analytical model

Estimated object's center

Shape recognition

Based on analytical model Estimated shape's parameters Fig. 3: Scheme of the process. The BEM computed currents are used as input. The methods for localization and shape recognition are based upon the analytical model. j = γ∇φ, j being the current density (A/m 2 ) and γ the medium conductivity (S/m). Boundary conditions on the sensor and the objects are known, depending on their material dielectric properties (conducting or insulating). The current flowing through the boundaries of each electrode is defined as the flux of the normal (with respect to the surface) current density. Thanks to the discretization of all the boundaries of the scene, the currents flowing through each electrode is numerically computed. BEM computed currents faithfully estimate the measured currents with the true sensor's geometry [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]. BEM computed currents will serve as input currents in this paper, in preparation to their substitution by future experiments.

B. Analytical model

Let us now add some assumptions: 1) the sensor has the slender shape and the electrodes are arranged as described in section II-A ; 2) each object is small enough to assume the electric field to be uniform over its whole volume. In practice, constraint 2) is justified when the lateral distance between sensor main axis and object's center is equal or greater than three sensor radius (y c 0 ≥ 3R on figure 1) and when the size of the object is at most of the order of the sensor radius (a ≤ R) [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]. Under these assumptions, the following analytical model for the currents is established. It describes a linear relationship between the vector U ∈ R 4 , being defined as the electric potentials we can set onto macro-electrodes, and two currents vectors denoted I ax ∈ R 4 and I lat ∈ R 4

I ax = C 0 + δC U (2)
I lat = 1 4π P ⊥ HP c G T C 0 U. (3)
Each component of the vector I ax (resp. I lat ) is the "axial current" (resp. "lateral current") of the corresponding macroelectrode [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF]. In particular, the model of the I ax vector (2) shows a decomposition into two independent contributions. First, I 0 = C 0 U is the axial currents vector measured in the case where there is no object in the scene. The 4 × 4 real matrix C 0 represents the conductivity of the surroundings which with no object only depends on the sensor's geometry and the medium's conductivity γ. It is obtained once for all either by numerical computation or by a calibration phase. Second, δI ax = δCU is the axial perturbation vector induced by the presence of the object and the 4 × 4 real matrix δC is the conductance variation matrix that encodes the surroundings' conductance variation due to the presence of the object. A model for δC is given in [START_REF] Boyer | Model for a sensor inspired by electric fish[END_REF] and here rewritten as

δC = 1 4πγ C 0 GP c G T C 0 . (4)
The 4 × 3 real matrix G is a non-linear function of the object's coordinates in the sensor frame, whereas the matrix P c depends on object's properties only (detailed in section IV). In the signal array processing literature [START_REF] Devaney | Super-resolution Processing of Multi-static Data Using Time Reversal and MUSIC[END_REF], δC is known as the multistatic response matrix. On the other hand, the analytical expression (3) of the I lat vector handles a 4 × 4 real matrix P ⊥ which depends on the sensor's only. It describes how the macro-electrodes are laterally polarized under the effect of the electric field reflected by the object. The 4 × 3 real matrix H, which, as the matrix G, is a non-linear function of the object's coordinates in the sensor frame. In practice, the components of I ax (resp I lat ) are obtained by computing the sum (resp. the difference) of the currents flowing across the two corresponding left and right electrodes. Thus, a numerical value of the matrix δC for a given scene can be obtained. Remarking that the measured vector δI ax for a polarization vector U = (1 0 0 0) T equals to the first column of δC, the four columns can be recovered by successively setting each macro-electrode to 1V and the three others to 0V .

IV. HYPOTHESES AND REQUIREMENTS

A. Additional assumptions

In addition to the basic assumption of the analytical model (previous section), let us add further restrictions on the sensor motion and on the objects' features : 1) the sensor always moves in straight line, along its main axis : rotation velocity Ω shown on figure 1c is set to zero ; 2) there is only one object at each time in the scene and it lies in the same plan as the sensor ; 3) the object's material properties (perfectly conducting or perfectly insulating) and the side of the sensor on which the object is, are determined by an algorithm that use the signs of the currents δI ax and I lat , described in [START_REF] Boyer | Underwater reflex navigation in confined environment based on electric sense[END_REF] ; 4) the object is assumed to be ellipsoidal (or cubic, in order to assess the robustness of the method for other shapes) ; 5) the ellipsoids used in this paper are supposed to be prolate, that is, their rotation axis corresponds to their semi-major axis ; 6) the aspect ratio is η = a/b, a being the semi-major axis, and b, the semi-minor axis, ranging from 1 (sphere) to 2.5 ; 7) all the objects' surfaces are smooth, with no cracks. The objects used in this paper are described in table I. The problem being symmetric with respect to the sensor's main axis, the objects are placed on the left side of the sensor only.

B. A model for the object's electrical response

According to [START_REF] Khairuddin | Fitting ellipsoids to objects by the first order polarization tensor[END_REF], [START_REF] Ammari | Polarization and moment tensors, with applications to inverse problems and effective medium theory[END_REF], the response of an ellipsoidal object to a uniform electric field is characterized by the first order polarization tensor which, in the object's principal basis, can be written as a 3 × 3 diagonal matrix, denoted P . An analytical expression of P is given in [START_REF] Ammari | Polarization and moment tensors, with applications to inverse problems and effective medium theory[END_REF]. For objects with large conductivity (conducting) or small conductivity The objects' representations and their corresponding features. (x 0 , y 0 ) are the coordinates of the object in the fixed frame. For the cubes, semi-axis a and b are defined as half of the edge length.

(insulating) compare to that of the water, P only depends on the object's volume and aspect ratio through the expression

P = V   f (η) 0 0 0 g(η) 0 0 0 g(η)   , (5)
f = 1 A , g = 1 B for a conducting object f = 1 A-1 , g = 1 B-1
for an insulating object with functions A and B being defined as

A, B : R → R A(η) = η -2 +∞ 1 1 t 2 (t 2 -1 + η -2 ) dt, B(η) = η -2 +∞ 1 1 (t 2 -1 + η -2 ) 2 dt.
Relatively to the sensor main axis, the ellipsoid is rotated of an angle θ 0 about the vertical axis. So, in the sensor frame, the matrix P becomes

P c = R θ0 P R T θ0 , (6)
p c 22 = p 11 sin 2 (θ 0 ) + p 22 cos 2 (θ 0 )., (8)
The component p c ij (reps. p ij ) denotes the i th row and j th column component of P c (resp. P ). With the symmetry properties of the problem, we have p c 21 = p c 12 , while p c 33 = p 33 and plays no role in the problem. The others components equal to zero. For a straight line motion of the sensor sweeping past the object 5 (table I), and with a constant polarization vector U = (0 0 0 1) T , the BEM Even though there are amplitudes differences, the analytical model faithfully describes the BEM computed currents. Now the context of the study being fixed, the description of the shape recognition process can start.

V. LOCALIZATION

As explained in introduction, the first step before shape recognition is the localization of the object independently of its intrinsic properties. This is in agreement with biologists' studies who showed the ability of the fish to localize objects independently of their shapes [START_REF] Der Emde | 3-dimensional scene perception during active electrolocation in a weakly electric pulse fish[END_REF]. Moreover, recent results on electric fish numerical modeling showed the feasibility of object's localization separately from of its intrinsic properties by using the MUSIC algorithm [START_REF] Ammari | Modeling active electrolocation in weakly electric fish[END_REF], [START_REF] Boulier | Modélisation de l'électro-localisation active chez les poissons faiblement électriques[END_REF]. This section details the implementation of the MUSIC algorithm to our sensor.

A. Model-based localization : the MUSIC algorithm

MUSIC is a model-based sensor array signal processing algorithm [START_REF] Krim | Two decades of array signal processing research: the parametric approach[END_REF]. It has been developed since the 80's to estimate the direction of arrivals of electromagnetic and sound waves onto a passive antenna network [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF]. In the 2000's, [START_REF] Devaney | Super-resolution Processing of Multi-static Data Using Time Reversal and MUSIC[END_REF] showed that the principle could be extended to an active antenna network in order to localize scatterers in an homogeneous medium. By sequentially emitting a polarization wave with each antenna of the network and by measuring the polarized scatterers' responses with the whole network , the multistatic response (MSR) matrix is computed. In our case, the MSR matrix is the conductance variation matrix δC (4). Then, a Singular Value Decomposition (SVD) Fig. 5: Geometric representation of the MUSIC principle in dimension 3 (inspired by fig. 1 in [START_REF] Schmidt | Multiple emitter location and signal parameter estimation[END_REF]). A basis of the noise subspace N is obtained with the SVD of the measured matrix δC. A set of steering vectors, corresponding to a set of candidate space coordinates are computed, defining trajectories represented in dotted lines. In virtue of the orthogonality between S and N , the steering vectors which projection onto N is 0 belong to S. The corresponding space coordinates are taken as an estimation of the object's center location. of this matrix is performed [START_REF] Horn | Matrix Analysis[END_REF]. The singular vectors associated to non-zero singular values form a basis of the signal subspace S whereas the singular vectors associated to zero eigen values form a basis of its orthogonal complement, the noise subspace N . Figure 5 represents these two subspaces in dimension 3 (the dimension is in fact 4 in our case because we have 4 components currents vectors). On the other hand, from the analytical model of the MSR matrix, one can exhibit particular vectors, called steering vectors in the literature [START_REF] Devaney | Super-resolution Processing of Multi-static Data Using Time Reversal and MUSIC[END_REF]. In fact, they correspond to the Green function vectors of the background medium. Steering vectors have two important properties: first, they are functions of the three space coordinates and they are independent of scatterers' intrinsic properties ; second, when evaluated in the scatterers' coordinates they are known to lie in the subspace S. So, in virtue of the orthogonality between the subspaces S and N , the projection onto N of the steering vectors evaluated in the object's coordinates tends toward 0. Figure 5 describes how MUSIC makes good use of this fact in practice. Moreover, an implementation of the MUSIC algorithm is precisely described in [START_REF] Song | Estimating source locations of unexploded ordnance using the multiple signal classification algorithm[END_REF] for the localization of buried unexploded ordnance. Even though the information carrier is a magnetic (not electric) field and the sensor array is planar (not linear), the analytical model for our respective MSR matrices are of identical structures. So, by using a similar process, we implemented the MUSIC algorithm with Fig. 6: MUSIC localization error map in percentage of sensor length (22cm) for a 1cm radius conducting sphere placed inside the green dashed line delimited area. Blue dotted line roughly defines an area in which the error is less than 5%.

our sensor model according to the main following steps:

• Measure the MSR matrix δC by switching the polarized macro-electrode one after the other, as described in section III-B. • Compute the SVD of the matrix δC.

• Select the singular vectors associated with the lowest singular values in order to form a basis of the noise subspace N . • A set of candidate coordinates are defined by discretization of an area around the sensor, in the plan in which the sensor and the object lie. • For each node of this grid, evaluate the steering vectors in the coordinates of that node and compute the inverse of their orthogonal projection onto N . • Once all nodes have been treated, select the node for which the inverse of the projection is the highest and take as the object location the coordinates of that node.

B. MUSIC algorithm localization performances

To evaluate the performance of the MUSIC algorithm for localization, an error map was computed for a 0.01m radius conducting test sphere (object 6 in table I), see figure 6. This map is obtained in the following manner. First, the sensor is placed so that the sensor's frame and the fixed frame coincide. An area of size 0.4m × 0.07m is defined on the left side of the sensor (see green dashed rectangle on figure 6). Then, this area is discretized by a 0.01m square grid and the sphere is placed at each node of this grid. For each sphere position on the grid, the MUSIC algorithm is applied with a 1mm square discretized search area. Finally, for each sphere position a relative location error , in percentage of the sensor's over all length is computed according to

= 100. (x c 0 -x c 0 ) 2 + (ŷ c 0 -y c 0 ) 2 /0.22, (10)
(x c 0 , y c 0 ) and (x c 0 , ŷc 0 ) being the true and the estimated coordinates of the center of the object in the sensor's frame, respectively. One can observe that the best results are obtained in a longitudinal area ranging from -0.02m to -0.08m, in which the error is less than 2%. More generally, the area 2.3% error 4.1% error 2.7% error 1.4% error 3.2% error Fig. 7: Localization error in percentage of sensor diameter for several cubes and ellipsoids using MUSIC algorithm. Solid lines : true objects' locations, dotted lines : estimated objects' locations. Blue dotted line is the same as that of figure 6. Note that even though they are represented on the same figure, the objects were localized one after the other, not at the same time.

delimited by the blue dashed line defines a zone in front of the sensor where the localization error is less than 5%. Recall that a 5% error corresponds to an absolute error of 1.1cm. If the sphere is outside of this area, the error can reach 10% or more (white zones). Then we have tested the algorithm with the other objects described in table I. The results on figure 7 and in table II show that the location errors are of identical magnitude compare to the case of the sphere. This confirms the ability of the MUSIC algorithm to estimate the position of the center of an object independently of its shape with our sensor design. In other terms, the location and geometrical informations have been separated, as expected.

VI. SHAPE RECOGNITION

Once the object is located with the method described above, the purpose of this section is the estimation of the objects properties. The strategy for shape recognition will consists of two steps : first, we estimate the components of the matrix P c with measured currents ; second, we estimate the parameters θ 0 , η and V using the model of the components of P c , described in [START_REF] Bai | Sensing capacitance of underwater objects in bio-inspired electrosense[END_REF] to [START_REF] Rasnow | The effects of simple objects on the electric field of Apteronotus[END_REF].

A. Matrix P c estimation

As mentioned in [START_REF] Devaney | Time-reversal-based imaging and inverse scattering of multiply scattering point targets[END_REF], [START_REF] Ammari | Modeling active electrolocation in weakly electric fish[END_REF], our typical structure for the analytical model of δI ax and I lat (section III-B) allows rewriting the currents vectors as a system of equations linear with respect to the components of P c . In the literature [START_REF] Ljung | ): theory for the user[END_REF], this model structure is known as linear regression, and a solution for the components of P c can be estimated with an optimization algorithm such as the well known least squares. This method requires a collection of measurement points. For the sake of simplicity, the straight line trajectory have been chosen, because it doesn't require any other motion but constant forward speed. The trajectory is 0.4m long sweeping past the object and at a constant lateral distance. The polarization vector is constant along this trajectory: U = (0 0 0 1) T and the currents vector I lat is measured every centimeter. Let us denote P c the estimate of P c obtained this way. A comparison between BEM currents (solid lines) and modeled currents using the estimated matrix P c (dashdot lines) is shown in figure 8. The identification process provides a P c matrix that makes the model to fit the input data (BEM) very well. In next section, we will see how the object's properties estimation can be performed, given an estimated matrix P c .

B. Object's parameters estimation

Having an estimation P c , the parameters θ 0 , η and V can be computed by solving the system of three nonlinear equations in three unknowns [START_REF] Bai | Sensing capacitance of underwater objects in bio-inspired electrosense[END_REF] to [START_REF] Rasnow | The effects of simple objects on the electric field of Apteronotus[END_REF]. For this, any numerical system as the function fsolve of the Matlab R software is appropriate. However, our intention being to implement this algorithm on autonomous system in the future, we propose another method requiring no particular computing software. The idea is to extract and estimate, by using the analytical model of P c , each unknown one after the other. First, according to equations ( 7) to [START_REF] Rasnow | The effects of simple objects on the electric field of Apteronotus[END_REF], and supposing the component p c 12 = 0 in a first step, the ratio = h(θ 0 ), h being defined as

h : - π 2 ; 0 ∪ 0; π 2 → R , θ 0 → 1 -tan 2 (θ 0 ) tan(θ 0 ) .
Function h having two solutions over -π 2 ; 0 ∪ 0; π 2 (see figure 9), the sign of the component p c 12 is used to resolve this indetermination. In the particular case where the estimated component pc 12 is null, the matrix P c is diagonal and corresponds to either a sphere or an ellipsoid with an angle of 0 or π/2. As a result, the simple following algorithm ensures the discrimination between the three cases:

Next, from the tensor model ( 5), the ratio p11 p22 is a function of the aspect ratio η only, and is independent of the volume V . Function f /g having only one solution over the range [1; 2.5] (see figure 9), the aspect ratio can be unambiguously estimated from this function (remind that the conductivity is already known, as mentioned in section IV, so there is no doubt about which curve has to be used). Finally, volume V can be estimated using p11 or p22

V = p11 f (η) or V = p22 g(η) . (12)
The shape recognition results are presented on figure 10 in the case where the localization error have not been introduced in the process. The cubes are approximated by ellipsoids with aspect ratios close to 1, ie. spheres with a radius roughly equal to the half-diagonal of the cube face (ie. a radius of √ 2 ≈ 1.41 cm). This is in accordance with [START_REF] Khairuddin | Fitting ellipsoids to objects by the first order polarization tensor[END_REF] in which a 2 cm sided cube is theoretically estimated by a 1.32 cm radius sphere. In spite of an over estimation of the sizes (except for the sphere), the ellipsoidal objects are faithfully estimated. This proves, at least in simulation, that the geometrical parameters estimation algorithm works if the location is well known. The over estimated volumes for the ellipsoids and cubes can be attributed to the purely dipolar object's response hypothesis of the analytical model. The Fig. 11: Comparison between real objects (solid lines) and estimated objects (dashed lines and dash-dot lines) with MUSIC algorithm localization error taken into account in the shape recognition process. Dashed lines represent same ellipses as the dotted ones but placed at the true object's location for shape recognition error visualization only. Colors correspond to those of the table I. All dimensions in meter. BEM formulation does not make such assumption, resulting into BEM currents that contain additive and more complex components. This is depicted by the amplitude difference between BEM and analytical currents on figure 4. Thus, because the shape recognition process estimates the tensor components that makes the model to best fit the measured data (BEM), it necessarily over estimates them. It directly impacts the volume V that is factored out, as shown in (5). To complete this discussion, the shape recognition results including the localization errors of the MUSIC algorithm are shown on figure 11 and in table II. The ellipsoidal objects (numbered 3 to 6 in table I) are faithfully estimated. This shows the ability of the algorithm to estimate the geometrical properties of such objects. On the other hand, the ellipsoids that estimate the cubes are more significantly effected by the localization error. II: Absolute values of all estimated parameters. For parameters, η, V and θ0 , values correspond to estimation with localization errors in the shape recognition process. For the sphere, the true location is (x c 0 , y c 0 ) = (4cm, 7cm).

VII. CONCLUSION

We have applied to our U-I technology sensor a method proposed in the field of applied mathematics for estimating the location and the geometrical properties of an ellipsoidal object with the electric sense [START_REF] Ammari | Modeling active electrolocation in weakly electric fish[END_REF]. The simulation results show the applicability of this method in the context of robotics with a very few measurements. Indeed, the principle of localization and geometrical informations separation works. It follows that the ellipsoidal objects' locations and intrinsic properties are well estimated. To a lesser extent, more complex shapes objects can also be estimated by an ellipsoid. Moreover, compare to the algorithm proposed in [START_REF] Bai | Finding and identifying simple objects underwater with active electrosense[END_REF], the method describe in this paper doesn't require any iterative motion loops. Also, it is based on analytical models, instead of numerically identified abacus. Now, these encouraging results have to be confirmed in experimental conditions. In addition, many other alternatives exist for the application of the method and have to be studied. For example, we could use the model of I lat instead of that of δI ax for the localization with the MUSIC algorithm. Other trajectories that the straight line for the identification phase could also be studied. The estimated positions and geometrical properties of an object with this method could serve as an initial values for a filter in the context of object tracking.

  Top view of the object

  Fig. 1: (a) The sensor with the polarization and measurement schematic and dimensions in meters. R denotes the sensor radius. (b) The ellipsoid and its related local frame (O o , u, v). O o coincides with the geometrical center of the object. Vectors u and v are aligned with the semi-major axis (a), and the semi-minor axis (b), respectively. (c) The sensor, the object and their respective frames. The coordinates of the geometrical center of the object are denoted (x c 0 , y c 0 ) in the sensor frame. Its orientation relative to the sensor axis is θ 0 . Vector V is the longitudinal velocity and vector Ω is the rotation velocity about the vertical axis. The macroelectrodes are numbered from E 1 to E 4

  the subscript T denoting the transposition and R θ0 being the rotation matrix of angle θ 0 about the vertical axis. The matrix product (6) leads to the following expression for the components of P c p c 11 = p 11 cos 2 (θ 0 ) + p 22 sin 2 (θ 0 ), (7) p c 12 = (p 11p 22 ) cos(θ 0 ) sin(θ 0 ),

7 7 4 Fig. 4 :

 7744 Fig.4: Currents I lat (top) and δI ax (bottom) for a straight line motion of the sensor, with constant polarization vector U = (0 0 0 1) T , going past the object numbered 5 in table I. Solid lines : BEM currents. Dashed lines : currents calculated with the analytical model.

7 7 4 Fig. 8 :

 7748 Fig.8: Currents I lat (top) and δI ax (bottom) for a straight line motion of the sensor, with constant polarization vector U = (0 0 0 1) T , going past the ellipsoid represented with black line in table I . Solid lines : BEM currents. Dash-dot lines : currents calculated with the analytical model in which P c = P c , P c being estimated with the least squares method.

Fig. 9 :

 9 Fig. 9: Functions f , g and f /g for a conducting object (top left) and an insulating object (top right) over the interval [1; 2.5]. Function h over the interval [-π 2 ; π 2 ] (bottom).

TABLE I :

 I

  Fig.10: Comparison between real objects (solid lines) and estimated objects (dashed lines) with no localization in the shape recognition process. Colors correspond to those of the table I. All dimensions in meter.

	0.08			0.1			0.09
							0.08
	0.06			0.08			0.07
							0.06
	-0.02 0.04	0	0.02	0.06	0.1	0.12	0.05	0.06	0.08
							0.09
	0.1			0.06			0.08
	0.09						0.07
	0.08			0.04			0.06
	-0.06 0.07	-0.04	-0.02		0.12	0.14	0.02 0.05	0.04	0.06
	0.08			0.1			0.09
							0.08
	0.06			0.08			0.07
							0.06
	-0.02 0.04	0	0.02	0.06	0.1	0.12	0.05	0.06	0.08
							0.09
	0.1			0.06			0.08
	0.09						0.07
	0.08			0.04			0.06
	-0.06 0.07	-0.04	-0.02		0.12	0.14	0.02 0.05	0.04	0.06

TABLE

	0.1
	0.1
	0.15 .1 .2 0.05 -0.05 -0.1 -0.15 0 y (m) -0.2 -0.15 -0.05 0 x (m) -0.1 xc 0 (cm) ŷc 0 (cm) η y (m) -0.1 x (m) 0 0.05 -0.05 x (m) 0.05 0.1 0 -0.2 -0.15 0 0.05 0.1 0.05 y (m) 0 11.1 5.5 7.1 1.4 1.2 -0.05 0 8.6 6.9 V (cm 3 ) -0.2 -0.15 0.05 0.1 0.15 -0.1 7 6.4 1.4 0 1.1 -0.1 0.05 0.2 0.15 0.1 -0.05 -4 8.7 2.4 1.2 -0.05 x (m) 0.2 0.15 0 x (m) 0.05 0.2 13 4.1 4.3 6.5 2.9 1.1 0.1 0.15 0.49 3 0 0.05 0.2 x (m) θ0 (deg) 81.9 5.8 -12.3 -74 29.2 -	0.1	0.1	0.15 0.15	0.2	0.2

ACKNOWLEDGMENT

This work was supported by the European Union, by funding the Project: EU H2020 FET-Proactive project "sub-CULTron", no. 640967. The authors would like to thank Gilles Chabert and Saïd Moussaoui for their precious help on MUSIC algorithm implementation.