Sonia Akkoul
email: sonia.akkoul@univ-orleans.fr

Adel Hafiane

Olivier Rozenbaum

Eric Lespessailles

Rachid Jennane

3D Reconstruction of the Proximal Femur Shape from Few Pairs of X-Ray Radiographs

Keywords: 3D shape reconstruction, stereo-radiography, proximal femur

This paper presents a 3D reconstruction method of the proximal femur shape based on contour identification from pairs of 2D X-ray radiographs. The aim is to reconstruct the 3D surface of the proximal femur from a limited number of projections. Our approach is based on the reconstruction of several 3D contours, which are meshed to obtain a 3D shape. Our proposed algorithm is based on different processing steps to obtain the 3D personalized model. Three approaches are proposed. The first technique consists of contour extraction, matching the points of these contours and calculation of a 3D contour using an original algorithm. The second and third techniques use the results of the previous method as well as new points chosen by an operator from a reference model of femur to improve accuracy either globally or only in sensitive areas. The proposed method was evaluated on 10 cadaveric proximal femurs using the 3D CT-Scan models. Obtained results show good performance and promising perspectives for 3D shape reconstruction of the femur from only a few pairs of radiographs.

INTRODUCTION

Personalized three dimensional (3D) patient bone models provide an interesting analysis tool for the diagnosis and the follow-up of many bone diseases and play an important role in preoperative surgery planning and improved guidance during surgical interventions such as total hip replacement, modeling and simulation.

Accurate 3D anatomical models can be achieved using conventional 3D imaging modalities such as Computed Tomography (CT-Scan) and Magnetic Resonance Imaging (MRI).

However, the use of these modalities is restricted to few specific procedures, due to constraints of radiation risk, cost and availability. In addition, the majority of these modalities require the patient to be lying down, thus involving a lengthy acquisition protocol with a positioning, which does not allow clinical evaluation of physiological function.

Thus, the diagnosis and planning of many interventions still rely on two-dimensional radiographic images, where the surgeon has to mentally visualize the 3D anatomy of interest.

As an alternative to current 2D radiographs, direct 3D imaging needs to be developed in order to assist clinicians in their medical tasks [START_REF] Gamage | Diagnostic radiograph based 3D bone reconstruction framework : Application to the femur[END_REF]. However, reconstructing a 3D bone shape from a small number of 2D X-ray images is a challenging problem, especially when reconstructing a patient bone specific surface model.

Studies on the reconstruction of bone models based on information collected through 2D

imaging modalities can be divided into two groups: methods that use only 2D images for 3D model reconstruction [START_REF] Caponetti | Computer-aided simulation for bone surgery[END_REF] and methods based on prior knowledge of the 3D anatomical structure in addition to 2D images [START_REF] Gamage | Diagnostic radiograph based 3D bone reconstruction framework : Application to the femur[END_REF]. Most of these studies require prior knowledge of the 3D anatomy model to compensate for the lack of information obtainable from the 2D images [START_REF] Gamage | Diagnostic radiograph based 3D bone reconstruction framework : Application to the femur[END_REF]. This information can be provided by incorporating a generic geometrical shape of the bone structure [START_REF] Laporte | A biplanar reconstruction method based on 2D and 3D contours: application to the distal femur[END_REF], [START_REF] Bras | 3D reconstruction of the proximal femur with low-dose digital stereoradiography[END_REF], [START_REF] Kim | Dense femur reconstruction from two x-ray images using generic 3D model with twist correction[END_REF], by Statistical Shape Models (SSM) [START_REF] Baka | 2D-3D shape reconstruction of the distal femur from stereo X-ray imaging using statistical shape models[END_REF], [START_REF] Whitmarsh | Reconstructing the 3D Shape and Bone Mineral Density Distribution of the Proximal Femur From Dual-Energy X-Ray Absorptiometry[END_REF], [START_REF] Zheng | A 2D/3D correspondence building method for reconstruction of a patient-specific 3D bone surface model using point distribution models and calibrated X-ray images[END_REF], [START_REF] Karade | 3D femur model reconstruction from biplane X-ray images: a novel method based on Laplacian surface deformation[END_REF], by Parametric Models [START_REF] Baudoin | Parametric subject-specific model for in vivo 3D reconstruction using bi-planar X rays: application to the upper femoral extremity[END_REF], [START_REF] Chaibi | Fast 3D reconstruction of the lower limb using a parametric model and statistical inferences and clinical measurements calculation from biplanar X-rays[END_REF], [START_REF] Humbert | 3D reconstruction of the spine biplanr X-rays using parametric models based on transversal and longitudinal inferences[END_REF], or by using independent shape and appearance models that are learned from a set of training data to encode the information of the bone [START_REF] Yu | Personalized x-ray reconstruction of the proximal femur via a nonrigid 2D-3D registration[END_REF]. To reduce radiation exposure, some authors have proposed 3D reconstruction methods from bi-planar Xray images based on stereovision techniques [START_REF] Bras | 3D reconstruction of the proximal femur with low-dose digital stereoradiography[END_REF], [START_REF] Boussaid | 3D model based femur reconstruction of the proximal femur from low-dose biplanar X-ray images[END_REF], [START_REF] Zheng | 3D reconstruction of a patient-specific surface model of the proximal femur from calibrated x-ray radiographs: a validation study[END_REF]. Our work is based on the same principle, i.e. to propose an alternative method using pairs of X-ray images taken in an upright position.

Our proposed approach for the 3D proximal femur surface reconstruction is based on a pseudo stereo matching technique. An original mathematical model and a limited number of 2D X-ray images taken from different orientations are employed for this purpose. The proposed scheme comprises two main stages. In the first one, two successive projections are used to compute the coordinates of a 3D contour. After extracting the contour of the proximal femur from the 2D X-ray images, the matching process is carried out. To match the points of two contours, three distances were compared: the City-block, the Chess board and the Euclidian 2D spatial distances. The estimated point pairs are then used to compute a set of 3D points. The surface model is reconstructed from this cloud of points by a meshing technique based on Poisson's equation which enables a closed 3D surface to be obtained. In the second stage, a 3D reference surface of the proximal femur is used to extract points and inject them locally into the reconstructed surface to compensate for the lack of information and improve the accuracy of the reconstructed 3D surface. To assess the accuracy of the proposed shape reconstruction framework, our results are compared to real 3D CT-scan data. Ten cadaveric proximal femurs were used as the anatomy of interest throughout this study.

The paper is organized as follows: Section 2 explains the method and the proposed projection model. Section 3 presents the data, reports the experimental results and presents the global and local correction shape methods. The last section concludes this paper.

PROPOSED METHOD

This section aims to describe the proposed method for the 3D reconstruction of the femur form CT scanner images. The projection model and the different processing steps are presented.

The 3D model

The protocol of acquiring the radiographic images is shown in Figure 1. The proximal femur is placed between the X-ray source and the rotating sensor. An X-ray of the specimen is acquired at each angle α. Let the pairs of points (P1, O1) and (P2, O2) represent the projection of a point P of the object and the projection of the center of rotation O on the sensor before and after rotation, respectively. From the coordinates of the points P1 and P2 and using our proposed mathematical model (detailed in Section 2.2), the 3D coordinates of the point P can be determined.

Since our aim is to use a minimum number of radiographic images, it was necessary to assess the impact of the choice of the angle between two radiographs as well as the influence of the angle between two pairs of radiographs on the quality and the accuracy of the 3D reconstructed shape. To do so, the angle  between two images of the same pair was fixed at 4 degrees and the angle  between the pairs of radiographs was set at different values [START_REF] Humbert | 3D reconstruction of the spine biplanr X-rays using parametric models based on transversal and longitudinal inferences[END_REF][START_REF] Frenkel | Curve matching using the fast marching method. Energy Minimization Methods in Computer Vision and Pattern Recognition[END_REF][START_REF] Kruskal | On the shortest spanning subtree of a graph and the travelling salesman problem[END_REF], 45 and 60 degrees). For each angle , the error between the 3D reconstructed shape and the ground truth was estimated and the optimal orientation  o giving the minimum error was retained. Keeping the angle  o , the next step consisted in varying the angle  from 4 to 32 degrees by power of 2, in order to find the best combination ( o ,  o ) that reconstructs the desired 3D shape close to the ground truth (GT) [START_REF] Akkoul | 3D Femur Reconstruction Using X-ray Stereo Pairs[END_REF]. Figure 2 illustrates different pairs of Xray projections separated by the angles  and .

Computing 3D points

This section describes how the 3D coordinates are computed from the 2D points selected [START_REF] Akkoul | 3D Reconstruction Method of the Proximal Femur and Shape Correction[END_REF].

Using the diagram in Figure 1, the bone is placed between the source and the sensor, the Ycoordinate (height) is fixed, and the source and the sensor are rotated around the object along the Y axis.

Let R(X, Z) be the reference system and P 1 (with (P 1 x, P 1 y) as coordinates), the projection onto the sensor of the point P. Knowing the distance between the X-ray source S 1 and the sensor Sr 1 , and the distance between the bone and the source and using the Thales theorem it is possible to calculate the Y coordinate of the point P (Figure 3).

To compute the Y coordinate one can use:

(1)
Once the Y coordinate has been calculated, the resolution of the system of two equations obtained using the straight lines (S 1 P 1 ) and (S 2 P 2 ) (Figure 1) is used to calculate the coordinates X and Z.

The first equation represents the straight line D 1 (Figure 1) connecting the X-ray source S 1 to the point P 1 which represents the projection of the point P on the sensor before rotation of the source-sensor system.

(2)

where OO 1 is the distance between the bone and the sensor, OS 1 is the distance between the bone and the X-ray source, and P 1 x is the X coordinate of the projected point P on the sensor.

The second equation shows the same straight line D 1 linking the X-ray source and the sensor through the point P after rotation of the source-sensor system by an angle α.

As the object and source-sensor distances are fixed, the source-object and object-sensor distances are invariant. Thus, we obtain:

OO 1 = OO 2 and OS 1 = OS 2 (3)
The equation of the second straight line D 2 (Figure 1) can then be written as follows:

(4)

with,



Substituting equation [START_REF] Humbert | 3D reconstruction of the spine biplanr X-rays using parametric models based on transversal and longitudinal inferences[END_REF] in equation ( 9), Z can be written as:

with, where S 1 is the X-ray Source, Sr 1 is the Sensor, P is a point of space belonging to the object, P 1 (P 1 x, P 1 y) is the projection of the point P on the sensor, OS 1 is the source-object distance and O 1 S 1 is the sensor-source distance.

The solution (X, Z) of the system of equations ( 2) and ( 4) is the coordinates of the point P.

Thus, for each pair of projections separated by an angle α, it is possible from these equations to estimate the coordinates of a 3D contour. Using several pairs of images, a 3D points cloud is obtained.

OS1 S1 Sr1 P P1 (P1x, P1y) O1S1 P1y Y O O1

3D reconstruction process

This section presents the processing flowchart (Figure 4) and the different steps of the proposed method for the 3D shape reconstruction of the proximal femur. After data acquisition, we extract the contours and match them by pairs. Then, using our proposed algorithm the 3D points are calculated, registered and meshed to generate the 3D shape form [START_REF] Akkoul | 3D Reconstruction Method of the Proximal Femur and Shape Correction[END_REF]. A description of each step is described in the following sections:

Block based contour extraction

For edge detection, we used active contour models (ACM), because unlike conventional methods, active contours provide closed and smooth contours, which are suitable for our application. In addition, the contours are more regular and less sensitive to noise, because they are subjected to certain rigidity. To this end, we used the Chan and Vese [START_REF] Chan | Active contours without edges[END_REF] model. For the initialization step, different configurations were tested (square and circle), and the circle was retained, because it provides better convergence and better fitting of the contours. The initialization was done from the center of the image. The required parameters by the method (regularization and fitting parameters) were fixed empirically.

However, extracting the contours can be complex. In the case of radiographs taken at 90 degrees, for example, there is occlusion of different regions of the proximal femur. The great trochanter is superimposed to the intertrochanteric region. In this radiograph, the intertrochanteric area is the Region Of Interest (ROI) and when the edge detection method is applied on the entire image, it is the outer contour that is detected rather then the contour of the ROI (Figure 5.b).

To improve the contour detection, we divided the image into three blocks to isolate the great trochanter into the middle block and process each block independently. For our images, using ACM enabled obtaining precise contours. Thus, no post processing was needed to assemble the otained contours. This way, it is possible to better define the area of interest and reduce occlusion effects.

Matching between 2D contours

Once the 2D contours are extracted, matching procedure is applied to determine the 3D coordinates. The objective is to establish a correspondence between two input images from the same pair separated by an angle α. For a given point on the first X-ray image, the goal is to identify a corresponding point on the second X-ray image [START_REF] Akkoul | A 2D Matching Method for Reconstruction of 3D Proximal Femur using X-Ray Images[END_REF]. Several techniques exist for matching contours [START_REF] Cui | Curve matching for open 2D curves[END_REF], [START_REF] Frenkel | Curve matching using the fast marching method. Energy Minimization Methods in Computer Vision and Pattern Recognition[END_REF], [START_REF] Park | Contour matching: a curvature-based approach[END_REF]. For the present study, the matching between the points of Since we are using binary images, matching using contour 1 (C1) towards contour 2 (C2) is not symmetric and the points obtained may be different if C2 is used towards C1. Therefore, the 3D reconstruction shape may be different. To discuss this issue and study its influence on the 3D reconstructed shape, the following configurations were tested [START_REF] Akkoul | 3D Reconstruction Method of the Proximal Femur and Shape Correction[END_REF]:

 Configuration 1: C1 towards C2.

 Configuration 2: C2 towards C1.

 Configuration 3: the intersection between the results of configurations 1 and 2. Here the goal is to keep the pairs of pixels that are common to both configurations.

 Configuration 4: the union of the results of configurations 1 and 2. Here the goal is to keep the maximum number of pixel pairs.

The obtained results are presented in the experimental section (Section 3).

3D rigid registration

Before meshing the obtained 3D point cloud, a rigid registration with a 3D point cloud extracted from the GT is needed. Using Meshlab, the 3D points cloud from the GT was extracted as a meshed surface. Then, all the vertexes were retained as 3D points cloud.

Finally, by registering the 3D point cloud in the same environment as the GT, the error between the two surfaces can be estimated. For the registration, we used the Iterative Closest Point (ICP) algorithm [START_REF] Besl | A method for registration of 3-D shapes[END_REF], [START_REF] Almhdie | 3D Registration Using a New Implementation of the ICP Algorithm Based on A Comprehensive Lookup Matrix : Application to Medical Imaging[END_REF], [START_REF] Andette | An Algorithmic Overview of Surface Registration Techniques for Medical Imaging[END_REF], [START_REF] Tam | Registration of 3D Point Clouds and Meshes: A Survey From Rigid to Non-Rigid[END_REF], which is employed to minimize the difference between two clouds of points. The first step of the ICP algorithm is based on the search for pairs of nearest points between the two sets (reconstructed and ground truth). The second step includes the estimation of the optimal rigid transformation that aligns the two data sets. In the last step, the rigid transformation is applied to the points of the reconstructed data. The procedure iteratively revises the transformations (translation, rotation) needed to minimize the distance between the points until convergence is achieved.

The registration step is followed by the meshing which enables reconstructing the 3D surface of the proximal femur.

Meshing

Next comes the meshing and the reconstruction of the computed 3D shape. This task was achieved using Meshlab [25] which is an advanced 3D software system for the processing and editing of unstructured 3D triangular meshes.

Our aim is to regularize surfaces and reduce noise and other artificats, thus our choice focused on the Poisson surface reconstruction algorithm proposed in Meshlab. The method requires the 3D points cloud as well as their corresponding normals.

The computation of the normals requires 3 steps:

 "Locating tangent planes": makes it possible to find the tangent planes for the approximation of the local surface. For this, the center of gravity of each vertex of the triangles constituting the mesh is computed as the average of the set of the K-closest neighbors.

 "Building Riemannian graph": enables to construct the graph of Riemann, which helps creating the structure, where for each vertex, the borders of its closest K-neighbors are included in the construction.

 "Minimum spanning tree": applies the Kruskal's algorithm [START_REF] Kruskal | On the shortest spanning subtree of a graph and the travelling salesman problem[END_REF] on the Riemann graph to weight the normals.

To sum up, the algorithm computes the normal of a 3D point as a weighted product of its closest K-neighbors.

Experiments and results

Using a few X-ray image projections, our task is to extract 3D information and reconstruct the 3D shape of the femur. For this purpose, ten cadaveric proximal femurs (F1, F2, …, F10)

were completely scanned using a CT scanner. This imaging modality provides both the projections and the resulting 3D volume and avoids calibration of the input images. Two different devices were used for acquisition, with no impact on our proposed method for the reconstruction of the 3D proximal femur shape. X-Ray tomographic images of three proximal femurs (F1, F2 and F3) were obtained using a high quality Computed Tomography (CT) scanner (eXplore CT120). The specimens were imaged with an isotropic pixel size of 50 µm.

Each sample was placed between the X-ray source and the rotating sensor (Figure 1). The distance between the X-ray source and the sensor was 45.265 cm and each proximal femur was placed 39.776 cm from the source. The seven other proximal femurs (F4-F10) were scanned using the Nanotom (Phoenix) scanner with a resolution of 50 μm. The distance between the X-ray source and the sensor was 50 cm and each proximal femur was placed 25 cm from the source. Thus, two different devices were used for acquisition, with no impact on our proposed method for the reconstruction of the 3D proximal femur shape.

The 3D volumes were reconstructed using an implementation based on Feldkamp's cone beam reconstruction algorithm [START_REF] Feldkamp | Practical cone-beam algorithm[END_REF]. Examples of the ex vivo proximal femur radiographs are shown in Figure 7 for different orientations (0°, 90°, 25° and 315°). The 3D models of the reconstructed proximal femurs are used as ground truths to compare the performances of our proposed algorithm.

0° 90° 225° 315°

Figure 7: X-ray images of the ex vivo proximal femur at different orientations.

Influence of the contour matching distance

First, to validate our acquisition protocol and our proposed approach, one extra cadaveric sample (different from F1,... , F10) was used to fix the optimal distance and angles. After performing various tests, the results showed that the optimal angles for the 3D reconstruction of the proximal femur shape are ( o ,  o ) = (4°, 45°) which led to use 16 projections.

For the matching of the contours, our processing chain was used to reconstruct the first 3D proximal femur with the three distances and the results were compared. The distance which gave the minimum error after the 3D reconstruction was selected. The comparison of the three distances showed that the Euclidean distance gives the best results (Table 1). It was therefore retained for the experiments in the present study. Table 2 compares the results using the four configurations for the asymmetric distance presented in section 2.3.2. As can be seen, the contours matching of configuration 1 provides the minimum errors, whatever the used distance. This configuration is therefore better suited for our application.

The estimated point pairs are then used to compute a set of 3D points. Once our approach have been validated, it was applied on the ten proximal femurs to test its robustness. The next section presents the obtained results.

Estimation of the error of reconstruction

For each sample, the optimal parameters for contour extraction, angles, etc. were settled.

Figure 8 shows visual examples of the 3D surfaces obtained for different proximal femurs and the estimated errors compared to the GT. These errors were evaluated using the Metro tool on simplified surfaces [START_REF] Cignoni | Metro: measuring error on simplified surfaces[END_REF]. The error is estimated in term of mean symmetric distance using the Euclidean distance to evaluate the difference between two triangular meshes. For this purpose, the "Point-to-surface distance" approach available in Metro was used.

The red and blue colors in the 3D reconstructed shapes represent the high and the low error values, respectively. For each sample, the low error value (blue color) is near the zero value, while the high error value is expressed by red color. Table 3, summarizes the reconstruction error of the ten proximal femurs. It noticed that, the errors obtained are mainly due to an occlusion effect which leads to poor edge detection and poor matching at the cavity between the femoral neck and the greater trochanter which is the most difficult area to rebuild.

Front view

Back view obtained for the 3D reconstructed proximal femurs. As can be seen, results demonstrate that whatever the processed sample (acquired using the different devices, eXplore CT or Nanotom), our approach gives stable and encouraging results with equivalent errors, demonstrating its robustness for 3D reconstruction. The next section presents our proposed method to improve the obtained results.

Global and local shape correction

As shown in Table 3 and Figure 8 our approach enabled 3D reconstruction of the proximal femur surface with a satisfactory accuracy. However, a precise analysis of the results shows that there is some residual error in the upper part of the femoral neck on the side of the greater trochanter. The following sections present two new approaches to improve the reconstructed 3D shape and overcome the lack of information in this area.

Global correction

To improve the 3D reconstructed shape, we propose to densify the reconstructed point cloud.

The first attempt called global approach [START_REF] Akkoul | 3D Reconstruction Method of the Proximal Femur and Shape Correction[END_REF] consists in extracting a percentage of points randomly from a reference volume and injecting them into the shape to be reconstructed (Figure 9). The extraction of the points was achieved using Meshlab tool as a meshed surface.

Then, all the vertexes were retained as 3D points cloud. The 3D Euclidean distance was used as a criterion between the points of the reference volume and those of the rebuilt cloud. To select only relevant points, i.e. points different from existing ones (already computed points), a threshold value on the number of points to retain was fixed. The threshold was chosen according to the percentage of points to be inserted in the 3D surface to be reconstructed. The percentage was calculated using the number of points of the reconstructed shape. For the test, several percentages of points were considered (5 %, 10 %, 15% and 20%). To validate our approach, we used two proximal femurs, the volume CT-Scan of the proximal femur F1 as the reference volume and the proximal femur F2 as the volume to be reconstructed and vice versa (i.e. the proximal femur F2 as the reference volume and the femur F1 as the volume to be reconstructed). Before the extraction points, a registration was realized between the two femurs (3D reference surface and the reconstructed one). This way, we ensure that the two femurs are comparable in size and are set in the same environment.

The results are shown in Table 4. These results show that the estimated errors decrease with some percentage of injected points from the reference volume. This finding is consistent with our expectations. The maximum error also decreases substantially and the shape of the great trochanter is better reconstructed with less error. However, beyond 15% of injected points, the average and the RMS errors increase slightly. This is mainly due to the global shape of the reconstructed femur which conforms to the shape of the reference volume. This demonstrates the importance of combining the two approaches: stereo reconstruction and shape constraint by completing the reconstructed volume from a reference model [START_REF] Akkoul | 3D Reconstruction Method of the Proximal Femur and Shape Correction[END_REF].

Local correction

As the global approach constrains the reconstructed volume to conform to the shape of the reference volume, here we propose a new approach that consists in inserting points from a 3D

reference shape only at specific locations that need to be improved. The targeted areas are those that reduce the precision of the reconstruction and increase the errors. These areas generate largest errors, because of occlusion problem. A semi-automatic procedure with human-machine interaction was adopted to reduce errors, thereby removing aberrant 3D points and injecting in the same zones points from the 3D reference shape. To do so, first a 3D/3D registration is conducted to register the two 3D shapes in the same environment. Then, a 3D points cloud is extracted from the reference shape. The points to extract are selected massively by the operator from the targeted area. All the points are kept and then fused to the 3D shape to be reconstructed. Figure 10 shows the points extracted from the reference shape and inserted into the reconstructed shape. To validate our approach, we used the CT-Scan volume of the proximal femur F1 as the reference volume to reconstruct the proximal femur F2 and vice versa. The results obtained for the two proximal femurs F1 and F2 are shown in Table 5. These results show that targeting the areas to be improved reduces the errors and increases the precision of the 3D shape reconstruction.

Figure 1 :

 1 Figure 1: Data acquisition process.

Figure 2 :

 2 Figure 2: Illustration of the angle  between two pairs of radiographs as well as the angle  between two radiographs of the same pair. For this example, α = 4 and β = 100 degrees.

Figure 3 :

 3 Figure 3: Calculation of the Y coordinate using the Thales theorem.

Figure 4 :

 4 Figure 4: The proposed processing flowchart.

Figure 5 :

 5 Figure 5: Improved contour detection. a) X-ray image taken at 90°. b) Obtained contour using the entire image. c) Obtained contour after subdivision of the image.

  two contours was performed by computing three different distances: the City-block, the Chess board and the Euclidian 2D spatial distance. Comparison of these three distances is presented in section 3.1.

Figure 6 :

 6 Figure 6: Extracted contours. (a) Contour1 (red), extracted contour from the first image, (b) Contour2 (green), extracted contour from the second image with a gap of  = 4 degrees.

Figure 8 :

 8 Figure 8: Reconstructed 3D surfaces for different proximal femurs and estimated errors using Metro with Euclidean distance. The low error value (equal to zero) correspond to the blue color, while the maximale error is represented in red.

Figure 9 :

 9 Figure 9: Ilustration of the global approach: extraction of points from the GT shape to be injected inside the reconstructed shape.

Figure 10 :

 10 Figure 10: Illustration of the local shape correction method.

Table 1 :

 1 Estimated maximum, average and RMS errors for the three distances.

		City-block	Chess-board	Euclidean distance
		(mm)	(mm)	(mm)
	Maximum Error	17.568	14.073	15.434
	Mean Error	2.615	2.592	2.528
	RMSE	3.758	3.556	3.506

Table 2 :

 2 Maximum error, Mean error and RMS obtained using the three distances and the four configurations.

		City Block (mm)	Chess Board (mm)	Euclidean Distance (mm)
		Max Err Mean Er	RMS	Max Err	Mean Er	RMS	Max Err	Mean Er	RMS
	Conf. 1	17.568	2.615	3.758	14.073	2.592	3.556	15.434	2.528	3.506
	Conf. 2	15.173	2.768	3.949	14.706	2.585	3.598	14.434	2.689	3.785
	Conf. 3	13.898	2.856	3.933	13.980	3.154	4.170	15.454	3.053	4.151
	Conf. 4	16.180	3.078	4.299	16.526	3.308	4.578	15.657	2.916	4.087

Table 3

 3 presents the maximum error, average error and RMSE (Root Mean Square Error)

Table 3 :

 3 Estimated maximum, average and RMS errors for the ten proximal femurs.

	Proximal femur	Maximum error (mm)	Mean error (mm)	RMS (mm)
	F1	9.421	0.889	1.369
	F2	9.237	1.259	1.753
	F3	10.498	1.291	1.883
	F4	10.490	0.940	1.441
	F5	9.378	0.913	1.384
	F6	9.657	1.344	1.784
	F7	7.709	1.379	1.869
	F8	7.740	1.327	1.859
	F9	7.197	0.860	1.280
	F10	8.001	0.926	1.283

Table 4 :

 4 Estimated errors for the proximal femurs F1 and F2 using the global approach.

	Percentage	Maximum (mm)	Mean (mm)	RMS (mm)
	of injected
	points	F1	F2	F1	F2	F1	F2
	0%	9.421	9.237	0.889	1.259	1.369 1.753
	5%	9.241	8.791	0.906	1.296	1.359 1.728
	10%	8.775	8.630	0.891	1.253	1.301 1.657
	15%	8.654	7.491	0.933	1.207	1.328 1.592
	20%	8.660	7.615	0.969	1.223	1.358 1.605

Table 5 :

 5 Estimated errors for the reconstructed proximal femurs F1 and F2 using the local shape correction method.

		Maximum error (mm) Mean error (mm)	RMS (mm)
		F1	F2	F1	F2	F1	F2
	No correction	9.421	9.237	0.889	1.259	1.369	1.753
	Local shape correction method	3.908	5.624	0.792	1.102	1.042	1.433

Acknowledgment

We gratefully acknowledge the Region Centre, France, for its support through the French APR Project (FRACTOS). The authors would also like to acknowledge Stéphanie Lerondel and Julien Sobilo from TAAM-CNRS UPS44 for the acquisition of the CT images.

CONCLUSION

In this paper, we have presented a new method for the 3D reconstruction of the femur shape from a relatively small number of X-ray projections. Our proposed method combines an Xray stereo model and contour points matching. Cadaveric proximal femurs and their CT scans were used to assess the feasibility of our study.

This work shows that it is possible to reconstruct the 3D surface of the proximal femur from a relatively small number of X-ray projections with an average error of 0.89 mm and an RMS of 1.37 mm. The proposed reconstruction scheme combines an X-ray stereo model and shape constraint. The accuracy of the reconstructed shape was improved by using several 3D points extracted from a reference model obtained from CT-Scan images. The proposed technique to densify the 3D point cloud improves the results, since the average error and the RMS are reduced to 0.79 mm and 1.04 mm, respectively. Our proposed approach for 3D surface reconstruction of the proximal femur seems to be an interesting and promising technique.

Further experiments using 2D classical X-ray images should be conducted to evaluate the effective potential of our method. For real clinical applications, a challenging point would be the contour extraction due to soft tissue and occlusion effects. As intra-operative assistance plays an important role in planning and supporting various computer assisted surgical procedures, we believe that our proposed approach is promising for clinical applications.

Competing interests:

None declared

Ethical approval:

Not required

Highlights

 3D reconstruction method of the proximal femur shape using few 2D


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Vincent Lebastard

Exploration of objects by an underwater robot with electric sense

In this article, we propose a solution to the underwater exploration of objects using a new sensor inspired from the electric fish. The solution is free of any model and is just based on the combination of elementary behaviors, each of these behaviors being achieved through direct feedback of the electric measurements. The solution is robust, cheap and easy to implement. After, stating and interpreting it, the article ends with a few experimental results consisting in exploring small and large unknown objects.

Introduction

In spite of its high potential interest for applications as the exploration of deep seas or the rescue missions in catastrophic conditions, underwater navigation in confined unstructured environments wetted by turbid waters is till today a challenge for robotics. Obviously due to the fluid opacity, vision cannot be used while the jamming caused by the multiple interfering reflections as well as the diffraction by small floating particles considerably increase the problems posed to echolocation by sonar. Pursuing a bio-inspired approach in robotics, we can question nature to learn which solutions could be implemented to solve this difficult problem. In fact, evolution has discovered an original sense well adapted to this situation: the electric sense. Developed by several hundreds of fish who have evolved in parallel on both African and South-American continents, electric sense have been discovered by Lissman in 1958 [START_REF] Lissmann | The mechanism of object location in gymnarchus niloticus and similar fish[END_REF]. In the African fish Gnathonemus Petersii for instance (pictured in figure 1-(a)), the fish first polarizes its body with respect to an electric organ discharge (EOD) located at the basis of its tail. This polarization which is applied during a short time-pulse, generates around the fish a dipolar shaped electric field which is distorted by the objects present in its surroundings (see figure 1-(b)). Then, thanks to many electro-receptors distributed along its body, the fish "measures" the distortion of the electric fields and infers an image of its surroundings. Thus, electric sense has a quite narrow but relevant niche since none other sense as vision or sonar can work in these conditions. Thus, understanding and implementing this bio-inspired sense on our technologies would offer the opportunity to enhance the navigation abilities of our today under-water robots. Based on these potential interests, Mc. Iver and co authors have recently exploited a sensor based on the measurement of the electric potential through electrodes in order to address the problem of electrolocation of small objects through off-line particle filtering [START_REF] Solberg | Active electrolocation for underwater target localization[END_REF]. In this case, the sensor reduces its body to a so a small surface (two points electrodes between which the difference of potentials is measured) that it does not perturb the electric field produced by another pair of punctual (emitting) electrodes between each the voltage is imposed. In Angels1 [3], another technological solution is proposed to the electric sensing. In this case, the sensor is embedded on a realistic 3D body on which each electrode can be polarized with respect to the others through a given vector of voltage U. The electric field distortions are then measured through the vector I of the currents flowing across the electrodes and the measurement mode is then qualified of U -I. The first letter standing for the emission (here, a vector of voltage U), the second, for the reception (here a vector of currents I), and distinguished from the U -U mode of [START_REF] Solberg | Robotic electrolocation: Active underwater target localization[END_REF]. In the article here presented we address the problem of the underwater exploration of objects electrically un-transparent with respect to the ambient water. Based on the morphology of the sensor (its symmetries in particular), the solution does not require any model of the environment. At the end, it consists of sensor based feedback loops, or in the language of neurobiology of sensory-motor loops. Each of these loops ensure to the sensor a simple behavior whose the combination allows it to find and explore the objects. By exploration, we here means an orbiting motion around the object. Finally, the article is structured as follows. In section 2 we introduce our experimental set up including a family of sensor said slender and the electro-location test-bed on which the algorithms are developed. In section 3, the electric model of the slender sensors is given, while a control strategy suited to object exploration is dealt with in section 4. Finally, experiments based on this control law are detailed in section 5. The article ends with a conclusion in section 6.

The electrolocation test-bed

Sensor

Based on the U -I measurement principle previously explained, we built a first generation of sensors named "slender probes" due to their high aspect ratio (length/thickness) morphology. On these probes, the macro-electrodes Eα are rings or hemispheres which are azimuthally divided into an even number of identical measurement electrodes. As an illustrative example, Fig. 2 shows one of this probes where each of the macro-electrodes (except E0 located in the tail) is divided into a pair of two identical left-right measurement electrodes (it is consequently named the 7-electrode probe and such that E1 = e1 ∪e2, E2 = e3 ∪e4, E3 = e5 ∪e6. In all the following, E3 will be named the head electrode, E2, the neck electrode while E0 is the tail electrode.

Tank and cartesian robot

In order to test our electrolocation sensors and algorithms in controlled and repeatable conditions, an automated test bench consisting in a tank of one cubic meter side with insulating walls and a three-axis cartesian robot has been built (see Fig. 3). The robot fixed on top of the aquarium allows probes positioning in translation along X and Y with a precision of 1/10mm and the orientation in the (X, Y ) plane is adjusted in 0.023 • abs using an absolute yaw-rotation stage. All probes tested are positioned in the aquarium at adjustable height using a rigid glass epoxy fibre tube (⊘14mm). This vertical insulating tube allows the passage of electrical cables dedicated to the signals coming from the electrodes to readout electronics (analogue chain + ADC board) without compromising the measurements. The maximum speed available is 300mm/s (≃ 1km/h) for both translations and 80 • /s (13.5tr/min) for rotation.

Objects and scenes

To investigate navigation algorithms in quite complex scenes using electric sense, a set of test objects has been fabricated with conductive and insulating materials. By insulating (or conductive), we mean an ideal material with a conductivity γ such that γ/γ0 = 0 (or γ/γ0 = ∞), γ0 being the conductivity of the ambient fluid (for instance, an ordinary tap water). In practice, plastics (metals) are a good approximation of insulating (conductive) materials. To these simple shaped bodies, we can also add some removable insulating walls. Finally, in the following any scene will be constituted of a combination of these objects and removable walls arbitrarily configured between the four fixed (insulating) walls of the tank.

Model

We here restrict our investigations to the case of resistive phenomena. In these conditions, the electric state of a scene can be entirely parameterized by an electric potential φ satisfying Laplace equations with boundary conditions imposed on the sensor and wall boundaries as well as crossing boundaries through the objects. In these conditions, reconsidering the principle of the sensor, we first polarize the Eα (α = 0) with respect to the ground electrode E0. This is done by imposing the vector of voltages U = (U0, U1...Um) T = (0, U, U...U ) T with U0 = 0 fixing the ground and U defining the controlled voltage. Then, we measure the vector I of the currents I k crossing the e k for k = 0, 1, 2...n. According to the laws of electrostatics [START_REF] Jakson | Classical Electrodynamics 3rd edition[END_REF], they detail as:

I k = γ0 e k ∇φ.n ds (1)
where we used the Ohm law relating the vector field of currents in the medium j with the electric field -∇φ and where by convention n being the inward normal to the sensor, one current is considered positive when it flows out from the sensor. To this first decomposition of the total currents vector we can add another one which is based on the morphology of the sensor. Indeed, thanks to the symmetries of the sensor, the vector of total currents can be decomposed as a sum of two components named "lateral currents" and "axial currents" as follows:

I = Iax ⊕ I lat . (2)
In ( 2), Iax is axi-symmetric whereas the component I lat is axi-skewsymmetric. In other words, for any of the ei belonging to a same Eα, the components Iax,i are identical whereas for a couple of opposed (ei, ei+1) on a same ring Eα, we have I lat,i = -I lat,i+1 . From these considerations of symmetry, it will be easy to extract I lat and Iax from I using the relations:

Iax = D+P+I, (3)
Where P+ projects the currents crossing the e k on those crossing the Eα, by simply adding all the I k of a same ring Eα, that being done ring by ring. Regarding D+, it is the (n + 1) × (m + 1) matrix defined by D +(iα) = Ai/Aα if ei ⊂ Eα and D +(iα) = 0 otherwise, with Ai and Aα the area of ei and Eα respectively. This matrix allows one to equidistribute the Iα onto the e k with e k ∈ Eα.

In the same manner, we can write:

I lat = D-P-I. (4)
where, the projection matrix P-is such that once applied to I = (I0, I1, I2, ...In) T , it gives the reduced vector : (0, I2 -I1, I4 -I3, ...In -In-1) T , in the case (always verified) where the numbering of the electrodes ei (i > 1), is such that the (ei, ei+1) are opposed pairs in the same Eα. In addition, D-allows one to recover the vector of currents I lat by distributing the |Ii+1 -Ii| onto the (n + 1) ei slots.

From an expansion in perturbations with respect to the aspect ratio (radius of the sensor over its length) we obtained in [START_REF] Boyer | Model for a sensor bio-inspired by electric fish[END_REF], the model of the lateral currents as following:

I lat ≃ P ⊥ ∇ ⊥ Φ1, (5)
with:

∇ ⊥ Φ1 = e 0 ∇φ1.n ⊥ ds, ..., en ∇φ1.n ⊥ ds T ( 6
)
the vector of lateral excitation fluxes in which φ1 is the potential reflected by the object once polarized by the sensor, n ⊥ is the lateral component of the normal to the sensor, and P ⊥ is a tensor named tensor of lateral polarizability of the sensor which is diagonal positive at the leading order. As regards the axial currents, we have shown in [START_REF] Boyer | Model for a sensor bio-inspired by electric fish[END_REF], that we also have at the leading order:

Iax ≃ I (0) -D+C (0) Φ1. (7)
with: Φ1 = (φ1(xc0), ...φ1(xcm)) T , the vector of iso-potentials applied by the reflected field by the object on the sensor's electrodes; I (0) the vector of basal currents, i.e. with no object in the scene; C (0) the conductivity matrix of the sensor with no object in the scene (i.e. such that I (0) = D+C (0) U). Finally, I (0) , P ⊥ , and C (0) are data intrinsically related to the sensor and the ambient medium with no object. As such, they can be computed once before all through a numerical code as the Boundary-Elements-Method or directly through a preliminary off-line calibration of the sensor. On the other hand, ∇ ⊥ Φ1 and Φ1 are vectors depending on the geometry of the object and its situation (position-orientation) with respect to the sensor frame.

Control law

In this section, we present a control strategy allowing the sensor to explore the objects in its surroundings. This strategy is stated in this section, while we postpone to the next section its physical interpretation.

General architecture

The general architecture of the underwater robot is decomposed into three elementary blocks as depicted on figure 4. The first block is directly related to the electric measurements of the electric sensor. The second is related to the navigation, and is consequently named "navigator". The last one controls the locomotion of the underwater vehicle on which the sensor is embarked. As a result, it is named "the locomotor". In the following we will concentrate our attention onto the second block which allows the robot to compute the desired (axial) linear and angular (lateral) velocities V and Ω ⊥ as a function of the electric measurements I. The navigator, can be decomposed into the following three sub-blocks: -1 • ) A sub-block "axialisation" which extracts the axial component of the currents Iax from the vector of total currents I.

-2 • ) A sub-block "lateralisation" which extracts the lateral component of currents I lat from the vector of total currents I.

-3 • ) A sub-block "memorization" which memorizes certain values taken by the measurements along the motion of the sensor.

Case of the 7-electrode sensor

Before going on with the implementation of this general structure, let us illustrate these three operations with the 7-electrode sensor of figure 2.

In this case, the axialisation operator consists of:

Iax = D+P+I = (I5 + I6, I5 + I6, I4 + I3, I4 + I3, I2 + I1, I2 + I1, I0) T (8)
Alternatively, we also introduce the vector of reduced axial currents:

Iax = (I5 + I6, I4 + I3, I2 + I1, I0) T (9)
where the over-barre indicates that the concerned currents are those flowing across the m rings Eα. The lateralisation operator allows to obtain the vector of lateral currents :

I lat = D-P-I = (I6 -I5, I5 -I6, I4 -I3, I3 -I4, I2 -I1, I1 -I2, 0) T (10)
As a first application of the "memorization block" the basal component of the currents I (0) is memorized in a preliminary calibration phase. By virtue of the symmetry properties of the sensor, this component is purely axial. Thanks to this preliminary calibration, we can compute at any time of the experiment the perturbative axial currents δIax:

δIax = Iax -I (0) , (11)
which only result from the presence of an object in the scene. We introduce δIax the reduced vector of perturbative currents flowing across the rings. In the same way, the vector of lateral perturbative currents is directly given by δI lat = I lat since, due to the lateral symmetry of the sensor, the vector of basal currents has no lateral component, i.e.: D-P-I (0) = 0. With these definitions in hand, we are now going to introduce a simple control law ensuring the exploration as previously defined.

Exploration control strategy

Using the three elementary operations previously introduced, one can derive a control strategy consisting in exploring the objects present in the scene. By "exploring" we here mean "to seek the objects" and "to follow their boundaries without touching them". Such a law is the natural resultant of the combination of three basic behaviors:

-1 • ) Seek any object electrically contrasted with respect to the ambient medium.

-2 • ) Flee from the electric influence of the object.

-3 • ) Follow the boundaries of the object. Now, let us reveal how these three behaviors can be easily obtained through simple sensory-motor loops. The two first ones are in fact produced by applying a feedback control law of the head lateral currents of the general form:

V = cte > 0, Ω ⊥ = K δI lat,1 , (12)
with K a gain ensuring the sensor to be attracted by any un-transparent object when K = k/δIax,1 and to be repulsed by any object when K = k/ | δIax,1 | with k > 0 in both cases. As regards the third behavior, it is simply obtained by applying the control law:

V = cte > 0, Ω ⊥ = k ′ (δIax,1 -δI (mem) ax,1 ), ( 13
)
where the axial velocity V is ruled by the same constant value as in (12), while k ′ > 0 is a constant gain. In (13), δI

(mem) ax,1
represents a value of the axial head current stored by the memorizing block of figure 4 at a time of the motion where it has a remarkable evolution (typically when it attains an extremum). Thus, beyond this time, δI

(mem) ax,1
represents a desired value that the sensor will track along its motion.

Application to the 7-electrode sensor

We are now going to see how we can combine the three behaviors of section 4.3 in order to achieve the exploration task. Indeed, it suffices to sequentially order the three behaviors along with the three following phases symbolized through the graph of transitions of figure 5-(a) whose the application is schematized on figure 5-(b): -First phase (from A to B in figure 5): It consists in seeking an electrically non-transparent object by applying the attractive behavior (12) with K = k/δIax,1 and k > 0.

-Second phase (from B to B ′ in figure 5): This second phase corresponds to the initialization of the orbiting motion of the sensor around the object. It is obtained by applying the repulsive behavior, i.e. (12

) with K = k/ | δIax,1 | and k > 0.
-Third phase (from B ′ to C in figure 5): This is the orbiting phase obtained by applying the law (13), in which δI

(mem) ax,1
is the value of δIax,1 measured at the last time of the previous phase. As indicated on the sequential graph of figure 5-(a), the commutation between these phases is ruled by the following events which only depend time variation of the measurements (and not of their magnitude). Indeed, the commutation from phase 1 to phase 2 is activated by the change of sign of δIax,2, i.e. when the current flowing across the second ring (neck ring) change of sign. The second transition (from phase 2 to phase 3) is activated when | δIax,1 | attains its minimum value.

Interpretation

Let us now interpret the previous law. First, since due to equations 5 and 6, the lateral current δI lat,1 = I lat,1 is proportional to the lateral flux of the electric field reflected by the object, any law of the general form (12) ensures the sensor to align its head on the electric field lines. This first condition is ensured while the sensor moves with the constant axial velocity. As a result, since all the electric lines (integral of the reflected electric field) emanate from the objects close to the sensor, the sensor is attracted or repulsed by them depending of the sign of the gain K in (12). Finally, the presence of |δIax,1| in (12) allows to normalize the control as the sensor approaches a targeted object. When it is applied in the first phase, the sensor is attracted by the objects. When getting closer and closer to an object, the sensor faces it and more and more electric field lines are captured by the object if it is conductive. Thus, the electric field lines are funneled on the front (head) electrode. On the contrary, if the object is insulating the electric field lines are pushed backward along the sensor axis. As a result, when the sensor approaches an object, the current lines go from the head to the neck electrode if the object is and insulator and from the neck to the head if it is a conductor (see figure 6). Thus, in both cases it happens a time when δIax,2 changes of sign. This is at this time that the control law commutes from the attractive to the repulsive behavior. When this occurs, the sign of K in (12) is changed and the sensor starts to avoid the object, this is done till the | δIax,1 | attains its minimum which means that the tail (E0) and the head electrodes are equidistant from the object. Indeed, when 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 | δIax,1 | is larger (see figure 6). Finally, as this condition is satisfied, δIax,1 is memorized as a reference that the sensor try to maintain. That corresponds to follow an iso current around the sensor, this last condition ensuring the following of the object boundaries.

Experimental results

In this section, we report some of the experimental results illustrating the previous exploration control strategy. The electric feedback loops (12) and (13) are tuned once for all with k = 50V and k ′ = 2, 5V , and then applied to all the tests. The experimental conditions are those described in section 2. For each experiment the sensor first starts and seek the object, second, it flees from the electric influence of the object and third, follows the boundaries of the object. Many trials have been successfully achieved with this single simple law. We here report three of them, the first one illustrating the exploration of a small object while the second and the third are applied to the exploration of large objects. In the first case, the explored object is a conducting ellipsoid. Figure 7 (Top) displays the trajectory of the sensor when the sensor orbits around the ellipsoid. Note that while the sensor trajectory is near to be elliptic, this elliptic orbit displays an angular shift with respect to the elliptic boundary of the object. This shift is probably due to the initial conditions of the orbit while the aspect ratio of the elliptic trajectory (great over small axis) should be an image of that of the object. Finally, the case of the exploration of large objects is illustrated on the two last examples figure 8, the first object is an insulating wall placed in the middle of the tank and the second is simply the bound walls of the tank. In the last case, while the sensor is orbiting around the tank, a piece of insulating wall is placed on one of the tank corners (bottom-left). When moving past this corner, the sensor does follow the new piece of wall.

Conclusion

In this article we have addressed the problem of object exploration using a sensor bio-inspired of electric fish. The solution is free from any model of the sensor electric interactions with the surrounding. It allows seeking any object electrically contrasted with respect to the ambient water and then to turn around. This behavior will be exploited in future to infer a geometric model of the object. Today, it has shown its robustness in many situations with many different objects. In spite of these first success, many other questions today remain. In particular, since the use of the model is replaced by the morphology of the sensor, the following question arises. In which proportion the sensor morphology determines its behavior with respect to the explored objects. For instance, the distance between the head and neck rings determines the distance at which the sensor detects the presence of an object.

Fig. 1 .

 1 Fig. 1. (Left) The electric fish Gnathonemus Petersii. (Right) The electric field is distorted by the presence of an object (for instance, an insulating cylinder pushes the field lines away whereas a conductive cube funnels them).

Fig. 2 .

 2 Fig. 2. Picture (left) and schematic view (right) of a 7-electrode sensor organized in 4 polarizable rings, 3 of them being divided in two half rings allowing two lateral (left and right) current measurements.

Fig. 3 .

 3 Fig. 3. Electrolocation test bench.

Fig. 4 .

 4 Fig. 4. (Left) The general architecture of the underwater robot. (Right) The navigator.

Fig. 5 .

 5 Fig. 5. (a) Sequential graph of exploration task. (b) typical scenario of the object exploration.

Fig. 6 .

 6 Fig. 6. (Top) The electric lines are pushed forward and backward depending if the object facing the sensor is conducting or insulating. (Bottom) When the head and tail electrodes are equidistant of the object, the head currents attain a minimum.

Figure 8 (

 8 Bottom) shows the time evolution of the measured currents along the sensor trajectory in the case where it follows the tank walls. The two time transitions of the sequential graph (5-a) are pointed by B and B ′ as in figure (5-b).

Fig. 7 .

 7 Fig. 7. Exploration of a conducting ellipsoid.

Fig. 8 .

 8 Fig. 8. (Top) Exploration of large objects. (Bottom) Time evolution of the currents δIax,1 (blue) and δIax,2 (red) along the sensor path when the sensor follows the tank walls.

The ANGELS project is funded by the European Commission, Information Society and Media, Future and Emerging Technologies (FET) contract number:


Vincent Lebastard

Frédéric Boyer

Christine Chevallereau

Noël Servagent

Underwater electro-navigation in the dark

This article proposes a solution to the problem of the navigation of underwater robots in confined unstructured environments wetted by turbid waters. The solution is based on a new sensor bio-inspired from electric fish. Exploiting the morphology of the sensor as well as taking inspiration from passive electro-location in real fish, the solution turns out to be a sensory-motor loop encoding a simple behavior relevant to exploration missions. This behavior consists in seeking conductive objects while avoiding insulating ones. The solution is illustrated on experiments. It is robust and works even in very unstructured scenes. It does not require any model and is quite cheap to implement.

I. INTRODUCTION

In spite of its high potential interest for applications as the exploration of deep seas or the rescue missions in catastrophic conditions, underwater navigation in confined unstructured environments wetted by turbid waters is till today a challenge for robotics. Obviously, due to the fluid opacity, vision cannot be used in these conditions while the jamming caused by the multiple interfering reflections as well as the diffraction by small floating particles considerably increase the problems posed to echolocation by sonar. Moreover, electric sense can be exploited in order to perceive some differences between objects depending on their electric properties. In particular, it would be relevant for underwater exploration to design an autonomous underwater vehicle capable of seeking conductive objects while avoiding insulating ones. This difference in electric nature could for instance instantiate the difference between dead (insulating) and alive (conductive) objects, or metallic (conductive) and other minerals (insulating)... Pursuing a bio-inspired approach in robotics, in the EU funded project Angels [START_REF]The angels' project[END_REF], we began to study and reproduce a sense used for navigation and communication by several families of fish whose activity is principally nocturnal and whose natural habitat is the particle-charged waters of equatorial forests: the electric sense [START_REF] Lissmann | The mechanism of object location in gymnarchus niloticus and similar fish[END_REF]. In mormyrid fish, for instance, active "electrolocation" has a range of 1 body length and is based on the emission of an electric field by polarization of the body relative to an emitter-organ in the tail [START_REF] Der Emde | Electric fish measure distance in the dark[END_REF]. The high internal conductivity of the fish's body relative to the surrounding environment focuses the emitted field lines, obliging current to flow through the specialized electroreceptive skin. Discrete receptor organs distributed over the epidermis capture an instantaneous 3-dimensional "electric image" of the near environment, comparing the trans-epidermal currents that are expected in the absence of any obstacles with the sensory image that is actually measured. This mode of sensory perception, which is by nature "active", has up to now been very little studied in robotics, despite its natural adaptation for the detection of obstacles in conditions of poor visibility. Based on these potential interests, Mc. Iver and co authors have recently exploited a sensor capable of measuring the difference of potentials between two punctual electrodes, in order to address the problem of electrolocation of small objects through off-line particle filtering [START_REF] Solberg | Active electrolocation for underwater target localization[END_REF]. In this case, the sensor's body reduces to a so small surface (two points electrodes between which the difference of potentials is measured) that it does not perturb the electric field produced by another pair of punctual (emitting) electrodes between which, the voltage is imposed. In Angels [START_REF]The angels' project[END_REF], another technological solution is proposed to the electric sensing. In this case, the sensor is embedded on a realistic 3D body on which each electrode can be polarized with respect to the others through a given voltage U . Thus, the sensor sets up an alternating dipole electric field around its body from an emitting electrode on it. This field induces a current in the measurements electrodes. And when, one introduces an object into the media around the sensor, it perturbs the field such that there is a measurable current change in the receiving electrodes. In short, the electric field distortions are then measured through the vector I of the currents flowing across the electrodes and the measurement mode is then qualified of U -I. In this terminology, the first letter stands for the emission (here, a voltage U ), the second, for the reception (here a vector of currents I). Thus, in the following we will use the U -I sensor recently proposed in [START_REF] Servagent | Bioinspired sensor for electrolocation and navigation in conducting media[END_REF] that we distinguish from the U -U sensor of [START_REF] Solberg | Active electrolocation for underwater target localization[END_REF]. With this sensor, we will address the problem of the underwater electro-navigation in confined environments, a problem till today unsolved by robotics. For that purpose, we propose a solution based on our sensor's morphology and a sensory-motor loop whose parametrization allows one to achieve conductive objects seeking and insulating obstacles avoidance in a very robust manner without any model as those required by the classical planned approaches [START_REF] Baffet | Biomimetic localization using the electrolocation sense of the electric fish[END_REF], [START_REF] Lebastard | Underwater robot navigation around a sphere using electrolocation sense and kalman filter[END_REF]. The article is structured as follows. First we will present the sensor technology (section 2) as well as the electric test-bed (section 3) on which the final experiments will be carried out. Then, we propose in section 4 a solution to our starting navigation problem. The solution is then implemented on the experimental test-bed in section 5. Lastly, the article ends with some concluding remarks and perspectives in section 6.

II. THE SENSOR

For more details about the sensor, we refer the reader to [START_REF] Servagent | Bioinspired sensor for electrolocation and navigation in conducting media[END_REF] and here just remind some basic elements about its physical principle as well as its technology.

A. The sensor principle

From an external point of view, the sensor is constituted of a body with insulating wetted boundaries on which are fixed a set of m + 1 macro-electrodes noted E 0 , E 1 ...E m (Fig. 1). In general each macro-electrode E α is sub-divided into an array of smaller measurement electrodes denoted e 0 , e 1 , ...e N except e 0 which always verifies e 0 = E 0 . Thanks to a voltage generator, the potential on E α=1,2...m is imposed to be the common ground with respect to which E 0 is put under the controlled voltage U . Once the sensor immersed in a fluid, this active device produces electric field lines, or equivalently, from Ohm law, current lines which flow through the fluid surrounding the sensor from E 0 to the other macro-electrodes E α=1,...m (cf. Fig. 1). These current lines are closed through an electric circuit embarked inside the sensor. This circuit measures the vector of the currents I = (I 1 , I 2 , ...I N ) flowing across the e i=1,2...N . When an object is placed around the sensor, it generates perturbations of the sensor electric current lines portrait that are measured by this current measurement circuit. The entire device can be embarked on a mobile body as typically an underwater motion controlled vehicle. From the internal point of view, the principle of the U -I sensor is schematized on Fig. 1, where the squares represent the measurement electrodes. The controlled voltage U is generated by a wave generator G (amplitude 5V, frequency 22.5 KHz). The currents flowing across the e i are measured with ammeter-like circuits A. Finally, the sensor has a range of about its body length when working in active mode [START_REF] Servagent | Bioinspired sensor for electrolocation and navigation in conducting media[END_REF]. This range is that observed on fish of the mormyride family.

B. Sensor morphology

Based on this principle, we built a first generation of sensors named "slender probes" due to their high aspect ratio (length/thickness) morphology. On these probes, the macroelectrodes E α are rings or hemispheres which are azimuthally divided into an even number of identical measurement electrodes. As an illustrative example, Fig. 2 shows one of this probes where each of the macro-electrodes (except E 0 located in the tail) is divided into a pair of two identical left-right measurement electrodes (it is consequently named the 7electrode probe and such that E 1 = e 1 ∪e 2 , E 2 = e 3 ∪e 4 , E 3 = e 5 ∪ e 6 , with E 3 the head electrode).

III. THE ELECTROLOCATION TEST-BED

A. Tank and cartesian robot

In order to test our electrolocation sensors and algorithms in controlled and repeatable conditions, an automated test bench consisting in a tank of one cubic meter side with insulating walls and a three-axis cartesian robot has been built (see Fig. 3). The robot fixed on top of the aquarium allows probes positioning in translation along X and Y with a precision of 1/10mm. The orientation in the (X, Y ) plane is adjusted in 0.023 • abs using an absolute yaw-rotation stage. All probes tested are positioned in the aquarium at adjustable height using a rigid glass epoxy fibre tube ( 14mm). This vertical insulating tube allows the passage of electrical cables dedicated to the signals coming from the electrodes to readout electronics (analogue chain + ADC board) without compromising the measurements. Its verticality is ensured by a micrometric adjustment base. Our test also allows dynamic trajectories management on a significant range of speed relatively to the aquarium size. The maximum speed available is 300mm/s ( 1km/h) for both translations and 80 • /s (13.5tr/min) for rotation. The three axes robot is motion-controlled using simulink software with the dSpace system. The electrolocation signals coming from the electrodes of the probe are, first processed by analogue chain (amplification and filtering) then converted by a 16 bits ADC, DS2004 card, with a resolution of 0.3mV/bit (range from ±10V). This card allows us to convert the 16 channels at the same time with a maximum speed of 1.25 MHz.

B. Objects and scenes

To investigate navigation algorithms in quite complex scenes using electric sense, a set of test objects has been fabricated with conductive and insulating materials. By insulating (or conductive), we mean an ideal material with a conductivity γ such that γ/γ 0 = 0 (or γ/γ 0 = ∞), γ 0 Fig. 3. Electrolocation test bench.. being the conductivity of the ambient fluid (for instance, an ordinary tap water). In practise, plastics (metals) are a good approximation of insulating (conductive) materials. To these simple shaped bodies, we can also add some removable insulating walls. Finally, in the following any scene will be constituted of a combination of these objects and removable walls arbitrarily configured between the four fixed (insulating) walls of the tank.

IV. FROM CLASSICAL TO EMBODIED APPROACH

Starting from classical approaches where the navigation is planned in a geometric map of the scene around the sensor [START_REF] Baffet | Biomimetic localization using the electrolocation sense of the electric fish[END_REF], [START_REF] Lebastard | Underwater robot navigation around a sphere using electrolocation sense and kalman filter[END_REF], we have progressively shifted to a more embodied approach where navigation is achieved without any conceptual representation of the world, just through behaviors encoded into simple sensory-motor loops. Such an embodied solution has been achieved in two steps. The first step is based on the exploitation of the sensor morphology, the second, on the observation of living fishes.

A. Exploitation of the sensor morphology

Even if the embodied navigation does not use any model [START_REF] Porez | Multy-physics model of an electric fish-like robot: Numerical aspects anbd application to obstacle avoidance[END_REF], the modeling of the electric sensor is crucial since it gives the insights helping to guide the seeking of the solution. In particular, in [START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF] an original modeling approach of the electro-location is proposed. It is based on the method of reflections as it is today developed in the community of diphasic fluid dynamics at low Reynolds numbers [START_REF] Happel | Low Reynolds number hydrodynamics[END_REF]. In this approach the electric interactions between the sensor and any object, which are parameterized by the electric potential φ, appear as a super-imposition of successive reflections:

φ = φ 0 + φ 1 + φ 2 + ..., (1)
each φ i representing a signal traveling from the sensor to the object or conversely (see Fig. 4). As a consequence, each reflection produces a contribution to the total vector of currents, which can be detailed as I = I (0) + I (1) + I (2) ..., with

I (i) = (I (i) 1 , I (i) 2 , ...I (i)
N ) T and:

I (i) k = γ 0 ek ∇(φ i ).n ds, ( 2
)
where n is the inward normal to the sensor boundaries and a point denotes a scalar product. Now, due to the rapid attenuation of the reflections with the distance, it is shown in [START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF] that the vector of total currents can be evaluated with a sufficient precision under the approximation of the second reflection, i.e. with the truncated series expansion:

I I (0) + I (1) + I (2) , (3)
where I (0) represents the measured currents with no object in the scene (said vector of "basal currents"), I (1) is the vector of the currents reflected by the object (with no sensor), i.e. it is a purely geometric component representing the net flux of the field reflected by the objet across the geometric surfaces of the electrodes e k . At last, I (2) represents the electric response that the sensor generates in order to recover its electric equilibrium under the excitation of the first reflection φ 1 . In [START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF], we have exploited the symmetry properties of the sensor to write a second decomposition of the total currents as the exclusive summation of a lateral and an axial component:

I I ax ⊕ I lat , (4)
where the axial component (or vector of axial currents) I ax is axi-symmetric while the lateral component (or vector of the lateral currents) is axi-skew-symmetric. In the first case, the currents measured by any two measurement electrodes of same geometry on a same ring (macro-electrode) are equal. In the second case, the currents flowing across two geometrically identic electrodes azimuthally opposed on a same ring (macro-electrode) are of same strength but of opposite sign (see Fig. 2). Thus, it is easy to extract I ax and I lat from the measured currents I, since adding the currents on a same macro-electrode removes the lateral component while subtracting the currents of two azimuthally opposed measurement electrodes removes the axial component. For instance, for the 7 electrode of Fig. 2 subtracting the left currents with the right ones give their lateral components while adding them give their axial one. This is essentially due to the fact that the axial and lateral currents are defined as the flux (2) of a radial 1 and a uniform electric field respectively. Finally, it is shown in [START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF] that due to the slender morphology of the sensor, we have at the leading order I ax = I coefficient of proportionality strictly positive. But then, from (2) (with "∼" meaning "proportional to"), we have: with n = n ⊥ + n the decomposition of the normal into an axial and a lateral component. Now, since the third member of ( 5) is nothing but the lateral flux of the electric field reflected by the object, the exploitation of lateral currents for electrolocation is particularly simple. Indeed, the lateral currents obey to a luminous analogy, the surface of the electrodes laterally facing a conducting object is lit by currents flowing in, while the electrodes on the opposite side of the sensor are in the shadow of the outward currents.

I lat ∼ I (1) lat ∼ (5)
Conversely, with an insulating object the nearer side of the sensor is shadowed while the opposite side is in the light.

As regards the axial perturbative currents, as shown in [START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF] they are due to the variations of the total resistance of the scene. For instance, if the resistance of the scene globally increases2 , then the axial currents decrease while if the scene increases its conductivity, the axial currents increase. These properties are summarized on the table I, which indicates how the perturbative currents δI ax = I ax -

I (0)
ax and δI lat = I lat -I (0) lat behave in presence of a single object (see [START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF] for more details). From this table, it is worth noting that δI ax and δIax > 0 for a conducting object δIax < 0 for an insulating object for a conducting object on the left of the sensor δI lat > 0 or for an insulating object on the right of the sensor for a conducting object on the right of the sensor δI lat < 0 or for an insulating object on the left of the sensor δI lat have a complementary role since the first can be used to determine if the object is conductive or insulating while, once we know the electric nature of the object, the second can allow one to determine if it is on the left or on the right hand side of the sensor. Finally, the axial and lateral components of currents can be easily (in real-time) extracted from the total currents, since by symmetry, subtracting the currents measured by two azimuthally opposed electrodes only retains the lateral currents, while summing all the currents measured by the electrodes of a same ring, kills the lateral currents and only retains the axial ones.

B. Coming back to nature

The second step of the embodied solution is based on the observation of nature. In fact, it is known from biologists after Hopkins [START_REF] Shieh | Short-range orientation in electric fish -an experimental study of passive electrolocation[END_REF] that when an electric fish is placed alone in an empty tank preliminarily equipped with an active dipole, while the dipole is switched off, the fish peacefully navigates without any apparent objective. On the contrary, as the active dipole is fired so generating an exogenous electric field in the tank, the fish, drastically changes its behavior, now seeking the dipole actively. Furthermore, in order to seek the dipole the fish does not go in straight line but follows the electric lines of the active dipole (see Fig. 5). Although well studied by biologists in the case of the passive sense, this strategy of navigation has only been conjectured in active electro-location [START_REF] Caputi | The electric image in weakly electric fish. ii physical images of resistive objects in gnathonemus petersii[END_REF] where the fields are primarily produced by the fish itself. In this case, the active dipole field of biological experiments should be replaced by the secondary dipoles re-emitted by the objects once polarized by the sensor. Finally, this bio-inspired strategy plays a key role in the embodied solution to the navigation problem.

C. Sensory-motor loops and behaviors

In order to extend the previous bio-inspired strategy to the active case, we first of all have to remove from the total currents, the basal component I (0) and only consider the perturbative component δI = I -I (0) . As a result, this operation allows the sensor to measure the presence of the objects in its surroundings as if they were active unsteady dipoles 3 . Then, by applying a control law of the general form (3D) :

V = cte and:

Ω ⊥ = K δI lat , (6)
with K a matrix of control gains, one can easily force the sensor to move forward along its axis at the constant scalar velocity V (> 0), while aligning its axis on the secondary electric lines of φ 1 through the lateral rotation of its body Ω ⊥ . This self alignment is ensured by ( 5) as K = 0 in (6). Now, with the morphology of the 7-electrode sensor of Fig. 2 and all the objects assumed to be in the horizontal equatorial plane of the sensor, we can instantiate the general law ( 6) with the simpler one:

V = cte and:

Ω ⊥ = K(I 5 -I 6 ), (7)
where K, and Ω ⊥ are now two scalars, and where the feedback lateral currents reduces to the left minus right difference of the head currents. Now, from the table I, and with our convention on the sign of the currents and angular velocities, taking K > 0 in [START_REF] Lissmann | The mechanism of object location in gymnarchus niloticus and similar fish[END_REF] ensures that when a conductive object is on the right (respectively on the left), the sensor turns to the right (respectively to the left). While, when a conductive object is in front of the sensor, the sensor goes forward without changing its orientation 4 . Thus, this law ensures the sensor to be attracted by any conductive object. Now, and still from table 1, if the object is insulating and on the right (respectively the left), this control law makes it react as if there was a symmetric conductive object on the left (respectively the right). Thus, the feedback law [START_REF] Lissmann | The mechanism of object location in gymnarchus niloticus and similar fish[END_REF] with K > 0, ensures the sensor to be attracted by the conductive virtual object symmetric of the real insulating one with respect to the sensor axis, i.e. the sensor is repulsed by the insulating object. Finally, this first law achieves the attraction by conductive objects and the repulsion by insulating ones as it is observed in real fish when they are seeking conductive objects as small leaving preys [START_REF] Der Emde | Finding food : senses involved in foraging for insect larvae in the electric fish gnathonemus petersii[END_REF]. In the next section, this law is implemented on experiments.

V. EXPERIMENTAL RESULTS

In the next subsections, we report some of the experimental results obtained by applying the sensory-motor loop ensuring the behavior: "attracted by conductive objects and repulsed by insulating ones". The experimental conditions are those described in section 3. The conductivity of water is about 0.041 S/m, i.e. enough high with respect to plastic and low with respect to metal, to allow considering the objects and walls as ideal insulators or conductors. We progress step by step, starting from the case of the tank with one object only, to the case of complex scenes where several objects are immersed in the tank.

A. Tank with one conductive object

In this first test, we introduce in the scene a conductive sphere as indicated on Fig. 6(a). From the sensor initial position A to the first one in doted line (point B), the robot escapes from the insulating corner towards the sphere. The conductive sphere being initially away from the sensor, the detection is first tiny (see Fig. 6(b) where the currents are very weak in A). Then, from A to B, the sensor slightly turns left since it is attracted by the conductive sphere. In B, the perturbation induced by the tank's insulating wall becomes predominant compared to that of the conductive object. Consequently, the sensor is repulsed by the wall. In the C sensor's configuration, the sensor avoids a first tank's wall. Being in a corner, the sensor has to avoid a second wall. The configuration denoted by C represents, for this path, the sensor position in which the effect of the corner (i.e. the two walls) is the weakest. From C to D, the sensor goes away from the corner and goes towards the most conductive region of the scene: the boundaries of the sphere. When being in the E configuration, the sensor is one length away from the object (from one center to the other), and it detects the conductive region (δI ax > 0). From E to the final position, the sensor comes closer to the conductive object (δI ax increases) up to the contact.

B. Large insulating object immersed in the tank

In order to illustrate the effect of large (insulating) objects, we put in the tank a removable wall of 0.38 m length, in plexiglass material. Its position in the tank is represented in Fig. 7(a). From the initial position (A) to its B position, the sensor seeks the most conductive region within its range. After this point, the robot initiates a cyclic trajectory (Fig. 7(a)) along which the current remains maximum (i.e. the resistance reach a local minimum). In the points C and D, the current (Fig. 7(b)) is the weakest of the closed sensor path. Indeed, in the two positions, the sensor goes through in the regions the most confined by the insulating walls. Let us remark that in C, the sensor is closer to the removable wall that to the tank's walls. This is due to the fact that in this point, the removable wall being sensed through its edge, it is less influential than in A, where the two walls have comparable effects on the measures. Indeed, in this case the sensor is near to be equidistant from the removable and fixed wall.

C. Several (insulating and conducting) objects immersed in the tank

In its initial position (A), the sensor is immersed in a resistive very confined scene (insulating sphere placed in a corner of the tank) (Fig. 8(a)). At point B, the sensor comes closer to the insulating sphere and the current strongly decreases (Fig. 8(b)). From B to C, the sensor get out of the confined resistive region. From C to D, the sensor seeks the most conductive region. From D to F , the sensor locks itself in a corner. The point E is the position in which the effect of the corner on the sensor is minimum. At point F , the lateral current increases (Fig. 8(b)) since the sensor senses the insulating sphere on its left hand side. From the point E to the point F , the sensor seeks the most conductive region of the scene. After the point F , the sensor senses the conductive object (δI ax > 0) that it finally touches.

VI. CONCLUSION AND PERSPECTIVES

In this article we opened new perspectives to solve the problem of underwater navigation in confined environments wetted by turbid waters, where vision and sonar cannot work. The solution is based on a sensor recently proposed in [START_REF] Servagent | Bioinspired sensor for electrolocation and navigation in conducting media[END_REF]. This sensor is inspired by a natural mode of perception named electric sense. In fish, electric sense is based on the measurement of the distortions by the surrounding of a selfgenerated electric field. These distortions are measured by a distribution of transcutaneous electro-receptors which can be considered as a retina or a skin depending if one insists on the visual, or the haptic modality, of the electric sense. Starting from model-based approaches where the navigation requires the construction of a map of the environment, the Angels project then operated a paradigmatic reorientation based on the concept of embodiment and on the observation of living fishes. From the point of view of embodied intelligence, the navigation problem is investigated as a low level reflex ability of the robot, while pursuing a bio-inspired approach, we implemented a navigation strategy of electric fish which follow the electric lines of active dipoles to seek their preys. In our case, these active fixed dipoles have been successfully replaced by the secondary unsteady dipoles induced by the active polarization of the surrounding by the moving sensor. Thanks to recent progresses in electric modeling and exploiting the morphology of the sensor (in particular its symmetries), it became intuitive to implement a simple sensory-motor loop on currents. This sensory-motor loop allows the robot to seek conductive objects while avoiding the insulating ones. This embodied solution requires no model and exploits the haptic modality of electric sense. From this point of view, electric sense can be considered as touch but with no contact. Indeed, what is interacting here with the surroundings is not the material body (as in haptic) but the electric field generated by the sensor (or the fish) which can be, as such, interpreted as an immaterial extension of the probe's body around it. Today, among the numerous perspectives of these first results, we can mention: the implementation of this behavior on a real autonomous underwater robot, its extension to other behaviors relevant for underwater exploration as "avoiding all objects" or orbiting around them, or again to the multi-agent navigation... In order to address more complex mission, we also study the combination of this embodied navigation with the modelbased approaches.

VII. ACKNOWLEDGMENT

The ANGELS project is funded by the European Commission, Information Society and Media, Future and Emerging Technologies (FET) contract number: 231845

Fig. 1 .

 1 Fig. 1. General principle of the U -I sensor from the external (left) and internal (right) point of view: the macro-electrodes are put under voltage with respect to the emitter. Each receiving macro-electrode is divided into a set of independent measurement electrodes which measure the currents.

Fig. 2 .

 2 Fig. 2. Picture (left) and schematic view (right) of a 7-electrode sensor organized in 4 polarizable rings, 3 of them being divided in two half rings allowing two lateral (left and right) current measurements.

Fig. 4 .

 4 Fig.4. The principle of the method of the successive reflections. First step (Step 0): we ignore the object (here a small sphere) and we solve the Laplace equation for φ0. Second step (Step 1): this time we ignore the sensor and we solve the Laplace equation for φ0 + φ1 where φ1 is the perturbation of φ0 induced by the object. Third step (Step 2): we solve the Laplace equation for φ0 + φ1 + φ2 ignoring this time the object, φ2 being the perturbation of φ0 +φ1. Each novel potential field can be represented as a new contribution (to φ) reflected by the object or the sensor. The method can be pursued but as the interactions fall with the inverse of a positive power of the distance, there is no interest of doing it.

  e. invariant by any rotation around the sensor axis.

e0 ∇φ 1

 1 .n ⊥ ds, e1 ∇φ 1 .n ⊥ ds, ..., eN ∇φ 1 .n ⊥ ds T ,

Fig. 5 .

 5 Fig. 5. From Hopkins [10] (left) Electric lines emanating and returning to an active dipole plunged in a tank with circular insulating boundaries. (right) Several fish trajectories recorded by a camera. At each of the trials, the fish seeks the active dipole by following its electric lines.

Fig. 6 .

 6 Fig. 6. Experimental results: Behavior "Attracted by the conductors and repulsed by the insulators" for a tank with a conductive sphere (left) Sensor path for initial position A (right) Time evolution of lateral (top) and axial (bottom) currents.

Fig. 7 .

 7 Fig. 7. Experimental results: Control law "Attracted by the conductors and repulsed by the insulators" for a tank with an insulating wall in its middle (left) Sensor path for initial position A (right) Time evolution of lateral (top) and axial (bottom) currents.

Fig. 8 .

 8 Fig. 8. Experimental results: Control law "Attracted by the conductors and repulsed by the insulators" for a tank with several objects: 3 insulating spheres and a conductive one (left) Sensor path for initial position A (right) Time evolution of lateral (top) and axial (bottom) currents.

TABLE I PROPERTIES

 I OF THE PERTURBATIVE CURRENTS δIax AND δI lat .

We should add here and after in the same context: "within the range of the sensor".

All the objects being at rest, the unsteady character of a dipole is due to the time variations (in a fixed point of space) of the exciting basal field moving with the sensor's body.

The sensor going forward in a steady scene, only these three cases (object facing the head on left, right or front) are significant since as shown in[START_REF] Boyer | Model for a sensor bio-inspired from electric fish[END_REF], the lateral currents are negligible in any other case.