Julien Dupeyroux

Julien Diperi

Marc Boyron

Stéphane Viollet

Julien Serres

A novel insect-inspired optical compass sensor for a hexapod walking robot

A novel insect-inspired optical compass sensor for a hexapod walking robot* Julien Dupeyroux 1 , Julien Diperi 1 , Marc Boyron 1 , Stéphane Viollet 1 and Julien Serres 1

Abstract-In an outdoor autonomous navigational context, classic compass sensors such as magnetometers have to deal with unpredictable magnetic disturbances. In this paper, we propose to get inspiration from the insect navigational abilities to design a celestial compass based on linear polarization of ultraviolet (UV) skylight. To compute the solar meridian relative orientation, our 3D-printed celestial compass uses only two pixels created by two UV-light photo-sensors topped with linear polarizers arranged orthogonally to each other, in the same manner that was observed in insects' Dorsal Rim Area ommatidia. The compass was then embedded on our hexapod walking robot called Hexabot. We first tested the UV-polarized light compass to compensate for yaw random disturbances. We then used the compass to maintain Hexabot's heading direction constant in a straight-forward task, knowing the robot has important yaw drifts. Experiments under various meteorological conditions provided steady state heading direction errors from 0.3 • under clear sky conditions to 1.9 • under overcast sky, which suggests interesting precision and reliability to make this optical compass suitable for robotics.

I. INTRODUCTION

Navigation systems such as Inertial Measurement Units (IMUs) embedded on robots, smart-phones and so on, generally include magnetometers providing an absolute heading direction relative to the Earth's magnetic field. However local magnetic fields produced by ferrous materials can cause incorrect magnetometer measurements. The classical approaches to magnetic disturbances compensation rely on Kalman filtering and require sensor fusion from gyroscopes and accelerometers [START_REF] Bergamini | Estimating orientation using magnetic and inertial sensors and different sensor fusion approaches: accuracy assessment in manual and locomotion tasks[END_REF]. It is well known that gyroscopes and accelerometers are also prone to drifts. Therefore, the solutions currently proposed remain sensitive to multiple sources of disturbances. In complex outdoor environments where magnetic interferences are increasingly present and often unpredictable, it would be interesting to benefit from a new way to get a reliable measurement of heading direction.

Biomimetic approach has led to interesting models for optimization, control and automation in robotics. We propose here to get inspiration from the insects' skylight polarization compass [START_REF] Wehner | Visual navigation in insects: coupling of egocentric and geocentric information[END_REF] to provide a reliable measurement of the heading direction.

The polarization pattern of skylight is caused by the scattering phenomenon within the Earth's atmosphere. Indeed, solar radiations remain unpolarized until their entry into the atmosphere. Scattering interactions with atmospheric julien.serres@univ-amu.fr constituents induce the partially linear polarization of the skylight [START_REF] Coulson | Polarization and Intensity of Light in the Atmosphere[END_REF]. As shown in figure 1, the direction of the linear polarization of skylight at the zenith point is always perpendicular to the solar meridian. During daytime, the sun path makes the polarization pattern rotate around the zenith at an average speed of 15 • per hour. The sun path is not constant and depends not only on the position of the Earth about the sun, but also on the location of the observer. Studies showed the existence of ommatidia sensitive to the polarization of skylight in the insect's compound eye [START_REF] Labhart | Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye[END_REF]. Electro-physiological recordings from the cricket showed that these particular ommatidia are restricted to the upper side of the compound eye : the Dorsal Rim Area (DRA) [START_REF] Labhart | Polarization-opponent interneurons in the insect visual system[END_REF]. Many other insects exhibit the same polarization sensitivity, such as the desert locust, the monarch butterfly, the fruit fly [START_REF] Heinze | Polarized-light processing in insect brains: recent insights from the desert locust, the monarch butterfly, the cricket, and the fruit fly[END_REF], and honeybees [START_REF] Srinivasan | Honeybees as a model for the study of visually guided flight, navigation, and biologically inspired robotics[END_REF]. For all species whose DRA was identified, each ommatidium is sensitive to a unique direction of polarization and its orthogonal direction, though the exact anatomical structure differs between species. The spectral sensitivity of the DRA is generally in the UV light, except for a few species whose maximal sensitivity is in the blue range or in the green range. Many hypotheses have been proposed to explain why ommatidia are sensitive to UV light instead of other spectral ranges. The most reasonable assumption is that the skylight polarization remains strong in the UV under canopies and clouds [START_REF] Brines | Skylight polarization patterns and animal orientation[END_REF], [START_REF] Barta | Why is it advantageous for animals to detect celestial polarization in the ultraviolet? Skylight polarization under clouds and canopies is strongest in the UV[END_REF].

From a neural standpoint, for instance in the locust brain, the information perceived through the DRA is first integrated by polarization neurons, called POL-neurons, in the optic lobe, which show a high synaptic activity for three distinct angles (10 • , 60 • and 130 • ). In the central complex, POLneurons show a rather uniform synaptic activity for all polarization directions from 0 • to 360 • [START_REF] Labhart | Neural mechanisms in insect navigation: polarization compass and odometer[END_REF], [START_REF] Heinze | Linking the input to the output: new sets of neurons complement the polarization vision network in the locust central complex[END_REF]. A winner takes all mechanism can be used to understand how the heading direction is retrieved in the central complex of an insect [START_REF] Srinivasan | Honeybees as a model for the study of visually guided flight, navigation, and biologically inspired robotics[END_REF], [START_REF] Sakura | Polarized skylight navigation in insects: model and electrophysiology of e-vector coding by neurons in the central complex[END_REF]. DRA-based neural models mostly provide an estimated heading direction by computing the logarithmic difference between response of the ommatidium to a single polarization orientation, and the response of the same ommatidium to the corresponding orthogonal polarization orientation [START_REF] Labhart | Polarization-opponent interneurons in the insect visual system[END_REF].

End-use analyses of the celestial compass found in desert ants or in honeybees show that insects tend to refer to a unique, global polarization direction angle [START_REF] Wehner | Desert ant navigation: how miniature brains solve complex tasks[END_REF] in order to get their bearings. For instance, during a foraging trip in an unknown place, desert ants Cataglyphis integrate their heading direction through their celestial compass. Although their foraging trip consists of a slightly random exploration trajectory, their homing trajectory tends to be direct and straight to the nest [START_REF] Wehner | Desert ant navigation: how miniature brains solve complex tasks[END_REF].

In the late 1990s, Lambrinos et al. created the first wheeled robot, named Sahabot 1, integrating a celestial compass [START_REF] Lambrinos | An autonomous agent navigating with a polarized light compass[END_REF] which spectral sensitivity ranges from 400nm to 520nm (like in the DRA of the cricket). The project sought to test three different models to get the heading direction: (i) the scanning model uses only one polarization sensor and makes the robot rotate to find the highest sensor response and thus get the heading direction. However, the rotating phase can induce 2D displacements and therefore increase the final position error in a navigational context, (ii) the extended scanning model uses the same procedure as the scanning model but with three polarization sensors set at different orientations (0 • , 60 • and 120 • ). The heading direction is then computed simply by subtracting the sensor signals. This method provides more reliable results since peaks detected at the corresponding linear polarization angle are sharper than in the scanning model, (iii) the simultaneous model, uses three polarization sensors without rotating the robot. Logarithmic differences are computed between each sensor so that the heading direction can be correctly estimated. Tests were performed in the early morning and the average angular error was of 0.66 • using the simultaneous model, and 1.73 • using the simple scanning model. The simultaneous model was then applied to Sahabot 2 in order to implement ant inspired path integration models [START_REF] Lambrinos | A mobile robot employing insect strategies for navigation[END_REF].

Chu et al. developed a celestial compass based on the one integrated in the Sahabot projects, using the simultaneous model [START_REF] Chu | Construction and performance test of a novel polarization sensor for navigation[END_REF], [START_REF] Chu | Application of a novel polarization sensor to mobile robot navigation[END_REF]. The compass was embedded onto a wheeled robot using a fuzzy logic controller to follow a preprogrammed trajectory. Tests were performed at the end of the day to prevent any photo-sensors saturation. A miniaturized version of the celestial compass has also been proposed [START_REF] Chu | Integrated polarization dependent photodetector and its application for polarization navigation[END_REF] but no implementation onto a mobile robot has been recorded.

Another implementation of the celestial compass has been embedded on a small Unmanned Aerial Vehicle (UAV) [START_REF] Chahl | Biomimetic attitude and orientation sensors[END_REF].

Three polarization sensors, including effective directions and their corresponding orthogonal, were integrated in an ocelli based autopilot designed to control the UAV roll and pitch over ten seconds of flight.

It is still unclear how insects distinguish solar and antisolar angles. Sahabot simply integrated the position of the sun to avoid any ambiguity [START_REF] Lambrinos | An autonomous agent navigating with a polarized light compass[END_REF], [START_REF] Lambrinos | A mobile robot employing insect strategies for navigation[END_REF], but some suggest that insects use a circadian clock to both dispel the heading direction ambiguity and compensate the sun path [START_REF] Wehner | Insect navigation: use of maps or Ariadne's thread?[END_REF].

In this study, we propose to merge both scanning and simultaneous models proposed by Lambrinos et al. [START_REF] Lambrinos | An autonomous agent navigating with a polarized light compass[END_REF] into a UV-polarized light scanning model providing highly accurate measurement of the heading direction of our walking robot under various meteorological conditions and a low UV-index 1 . Section II presents the UV-polarized light compass. Section III describes the hexapod walking robot. Section IV examines two practical experiments of heading direction recovery using our celestial compass in real outdoor conditions and under various weather conditions.

II. THE UV-POLARIZED LIGHT COMPASS

A. The 3D-printed UV-polarized light sensor

The celestial compass uses two UV-light sensors (SG01D-18, SgLux) mounted below rotating UV linear sheet polarizers (HNP'B replacement) held by 70-teeth gears (fig. 2

.b).

Both gears are driven by a third one composed of 10 teeth and actuated by a stepper motor (AM0820-A-0,225-7, Faulhaber). Due to its symmetric properties, the two UV sheet polarizers holder gears turn in the same direction. The entire prototype was printed using PLA filament (polyactic acid). The angular resolution of the compass can be modified by changing the micro-step settings of the stepper motor. The angular resolution was arbitrarily set to 1.29 • for all the experiments introduced in this paper.

We call POL-sensor any polarization sensor system composed of a UV sheet polarizer and a UV-light photo-receptor. The left POL-sensor is called UV 0 and the right one is called UV 1 . Let x be the rotation angle of the UV sheet polarizer holder gears, and ψ the solar meridian direction angle. According to the polarization pattern in the skydome, UV 0 and UV 1 are π-periodic sinusoidal functions of x, as decribed below:

UV 0 (x) = A 0 + B 0 • cos(2(x + ψ)) UV 1 (x) = A 1 + B 1 • cos(2(x + ψ + π 2 )) (1)
where x ∈ [0; 2π] and ψ ∈ [0; π], A 0 and A 1 are offsets determined by the average UV-light radiance and inner bias of each photo-sensor, B 0 and B 1 are constants determined by the degree of polarization and inner gain of each photosensor. In case of bad weather conditions, B 0 and B 1 values are significantly reduced due to the weakening of the degree of polarization, implying heavy noise disturbances in POLsensors measurements (see fig. 3).

UV i = DFT(UV i ) 3: UV i [2 : length( UV i )/2] = 0 4:
UV n i = abs(rDFT( UV i ))

5:

UV nc i = UV n i -min(UV n i ) + ε 6: UV nc i = UV nc i /max(UV nc i )
where DFT and rDFT are respectively the direct and reverse Discrete Fourier Transform functions, and ε = 0.0001 is arbitrarily set to prevent from logarithm calculation failure.

The POL-unit response p(x) is defined as

p(x) = log 10 UV nc 1 (x) UV nc 0 (x) (2)
The solar meridian direction ψ is then calculated by locating the two local minimum values of the p function, the first one being in [0; π] and the second one in [π; 2π]:

ψ = 1 2 arg min x∈[0;π] p(x) + arg min x∈[π;2π] p(x) -π (3)
The knowledge of ψ is restricted to [0; π] due to the symmetry of the polarization pattern around the zenith point.

Classical methods to eliminate the ambiguity between ψ Solar and ψ Anti-Solar use the ambient radiance distribution [START_REF] Lambrinos | An autonomous agent navigating with a polarized light compass[END_REF], [START_REF] Lambrinos | A mobile robot employing insect strategies for navigation[END_REF]. As none of the tasks asked from the robot imply a turn back movement, there were no reason for ψ to change for ψ + π. Therefore, we always assumed that ψ ∈ [0; π]. Using the average value of the two minima of function p provides more accuracy in determining the solar meridian direction angle.

B. Sensor characteristics

The UV-light sensor is SG01D-18 (SgLux). Its active area is equal to 0.5mm 2 and its spectral sensitivity is between 200nm and 375nm with a maximum spectral response at 280nm. Due to the positioning of each component of the UV-polarized light compass, the full angular field of view of a POL-sensor is equal to 100 • . The refresh rate of each POL-sensor is 33.3Hz.

The UV linear sheet polarizer has a local maximum single (respectively parallel) UV-light transmission of 52% (respectively 27%) for wavelengths from 270nm to 400nm, with a peak transmission located at λ ≈ 330nm.

C. POL-unit signal processing

Typical signals acquired with our celestial compass are shown in figure 3. These results were obtained on Feb. 9, 2017 at 11:36 am. The UV-index was equal to 1 according to the French meteorological services and the whole sky was covered with clouds hidding the sun. Through a 42-second acquisition time, the signal magnitude does not remain constant as a result of little changes in weather. The degree of polarization varies in function of the height of clouds crossed by the light, or partial clearing of the sky. Under clear sky conditions, the peak-to-peak magnitude of the UV signals Figure 4 shows a comparison between the celestial compass and the magnetometer (LSM303D) in an outdoor area, next to our laboratory building where ferrous materials inside the walls and below the asphalt caused magnetic interferences. The robot was placed on the ground and successively turned by 10 • until it came back to its initial position. At each step, the new heading direction is computed by the celestial compass and the magnetometer. The data were acquired under clear sky conditions, with a UV-index of 6 in April 2017. The magnetometer was calibrated so that the inner electronic parts of the robot do not disturb its measurements. The Mean Squared Error (MSE) between ground-truth and sensor measurements have been computed; the MSE of the celestial compass was of 0.25 • while the MSE of the magnetometer reached 104.14 • (fig. 4). These results reveal the ability of the celestial compass to provide a precise and reliable orientation when magnetometers fail.

III. THE HEXAPOD WALKING ROBOT

A. The robot platform

We decided to employ Hexabot (fig. 5), a fully 3Dprinted, open source, six-legged walking robot based on Metabot, a four-legged walking robot [START_REF] Passault | Metabot: a low-cost legged robotics platform for education[END_REF]. Hexabot has three DYNAMIXEL XL-320 actuators per leg providing the ability to reach high walking speed (approximately 35cm/s in optimal conditions) and execute complex motions when crossing over an uneven terrain. Besides, six-legged robots show more stable walking movements than the four-legged ones since they can operate static gait (i.e. three to five legs remain in contact with the ground at any time). For instance, the tripod gait is a symmetric walking gait (three legs moving per walking step), and the wave gait consists in moving only one leg per walking step.

Previous ground-truth measurements were made in the Flying Arena of the Mediterranean (6m x 8m x 6m-height), equipped with 17 motion-capture cameras and showed: (i) small roll and pitch average disturbances of about respectively 9.0 • and 9.9 • when tested at maximal speed, (ii) Hexabot is prone to heavy instantaneous disturbances in yaw orientation (up to 28.4 • ) [START_REF] Dupeyroux | Hexabot: a small 3D-printed six-legged walking robot designed for desert ant-like navigation tasks[END_REF]. During a walking task, such yaw angle errors result in an important drift from the initial trajectory.

The overall weight of Hexabot, including batteries, of 925g, and has a maximum length of 360mm and a maximum height of 145mm (fig. 5). Its battery endurance, depending on the capacity, can last up to one hour.

B. Robot electronic architecture

Hexabot is controlled by the OpenCM9.04C microcontroller board (based on 32-bit ARM Cortex-M3) connected to a Raspberry Pi 2B board including a 32-bit quadcore ARM Cortex-A7 processor. The Raspberry Pi board executes sensor data acquisition and processing to compute high level orders which are to be sent to the robot controller. The UV-polarized light compass is embedded on the dorsal part of the robot (fig. 2.a). Communication with the Raspberry Pi board was implemented by means of I2C protocol (fig. 6). The refresh rate of each POL-sensor is 33.3 Hz and a full acquisition time takes 42 seconds for an angular resolution set at 1.29 • .

IV. EXPERIMENTS Hexabot was set to tripod gait for all navigational tasks since it provides an optimal compromise between high walking speed and moderate attitude disturbances. However, tripod gait tend to cause an important walking direction drift over time. We propose here to use our UV-polarized light compass to contain the drift occurring after each stride. All experiments were done between 02/02/2017 and 02/20/2017 in outdoor conditions, at any time of the day, and were located in an open-air car park in the Luminy campus (43 • 14 01.6 N ; 5 • 26 39.2 E) of Aix-Marseille University, Marseille, France.

A. Heading direction recovery under various meteorological conditions

The first objective was to measure and analyze simple heading reorientation tasks under different meteorological conditions at different time of the day. The task consisted in getting the initial heading direction of the robot, then rotating it by a random yaw angle, then getting the new heading direction and compare it to the first one in order to reorientate. Once the yaw command is executed, the new heading direction was measured and compared to the initial one. All UV-polarized light measurements are compared to ground truth magnetometer measurements. To prevent any yaw angle ambiguity since the sensor response is a π-periodic sinusoidal function, yaw disturbances were set between -70 • and +70 • . Such a restriction makes sense as yaw disturbances over a straight-forward walking task are systematically small and but oriented in the both direction due to interactions with the ground.

Figure 7 shows heading errors under three different weather conditions. Under clear sky conditions, the median value is of 0.4 • (n = 15), which is twice better than the results obtained in Sahabot 1 [START_REF] Lambrinos | An autonomous agent navigating with a polarized light compass[END_REF] whereas the weather conditions are highly different. The peak error measured is of -8.4 • corresponding to a yaw disturbance of -60 • . Under partially cloudy conditions, the median value is of -2.9 • (n = 11). The peak error measured is of -13.8 • (-65 • yaw disturbance). Finally, under cloud-covered sky conditions, the median value is of -1.9 • (n = 7) and the peak value is of -13.9 • . To highlight these results, it should be considered that Hexabot shows an average yaw turning angle inaccuracy per stride of 8.2 • with a statistical spread of 7.5 • due to interactions between legs and the ground. In view of this, the presented results exhibit great performance, especially under clear sky conditions. The decrease noticed under cloudy sky and overcast sky conditions stems from the low degree of polarization of the skylight [START_REF] Brines | Skylight polarization patterns and animal orientation[END_REF] which implies the overpowering of the Rayleigh scattering, thus disturbing the polarization pattern of the skylight. Considering the results obtained with Sahabot robot in [START_REF] Lambrinos | An autonomous agent navigating with a polarized light compass[END_REF], our UV-polarized light compass provide similar and even slightly better results under clear sky, and promising results under bad meteorological conditions such as clouds in the sky and a much lower level of ultraviolet radiance due to the period and the location of experiments.

B. Heading-lock over a straight-forward walking task

Depending on various parameters such as the type of ground, the walking speed and the power supply, Hexabot shows important drifts in yaw orientation. Drift measurements were made for five straight-forward walking tasks on a flat but rough terrain. Tests were conducted over six seconds at maximum walking speed. Results show an average heading direction disturbance of 28 • (magnetometer measurements in an outdoor terrain deprived of any magnetic field interference). In those conditions, Hexabot drifts from initial walking axis by an average length of one meter. Our UV-polarized light compass is used to limit the heading direction drift while applying yaw correction after each walking step. First, the initial direction is measured, then Hexabot executes a series of strides (a walking step) during two seconds and measures its new yaw orientation, the value of which is then compared to the initial one to compute the yaw angle correction to be applied. Hexabot executes the corresponding turning movement before moving to the next series of strides.

Due to supply limits and to prevent from the drift effect induced by the sun movements, data were acquired over two distinct days (02/18/2017 and 02/20/2017). As a result, experiments were all performed at the same time (2:00 pm) under perfectly clear sky conditions (UV index of 2). Initial heading direction was set to 220 • . Results for all experiments are shown in figure 8. An example of a walking step is shown in supplementary video. The average heading direction angle error calculated is of -0.3 • which is consistent with performances exhibited previously under clear sky conditions. The peak error measured is of 7.7 • , occurring during the ninth walking step. Since there were no clouds in the sky, the polarization pattern remained rather constant all over the experiments. As a consequence, the heading direction error is mainly caused by interactions between legs and the ground.

V. CONCLUSION

In this paper, a novel insect-inspired celestial compass was presented and embedded on a walking hexapod robot to provide a new way to get the heading direction in an outdoor, open-air context, under various meteorological conditions, a low UV-index range (from 1 to 2), and at any time of the day, possibly when magnetometers fail to provide any reliable orientation measurement.

The experiments performed showed highly precise and reliable results under clear-sky conditions, with an average steady state error as small as 0.4 • . Results under cloudysky conditions exhibits good performances as well, from 0.8 • under variable weather, to 1.9 • under overcast sky, but slightly less reliable due to the high variability of meteorological conditions. Future work will focus on the impact of the turning uncertainty of the robot on the heading direction. Residual heading-lock errors can be reduced by changing to a closedloop system and tweaking the turning parameters of the robot. The reliability of the computation of the heading direction may also be improved by optimizing the acquisition time of the celestial compass. Finally, we will perform further studies to show the suitability of this new optical compass sensor for autonomous robotic tasks such as homing, including to find a method to fix the angular ambiguity.

Fig. 1 .

 1 Fig.1. Three dimensional representation of the polarization of skylight seen from an observer (O). The red curve crosses zenith point (Z) relative to observer, and sun point (S). The solar meridian is the part of red curve that starts from the zenith and crosses the sun. The anti-solar meridian is the symmetric part of the curve. Orange lines' orientation and thickness show the direction and the degree of polarization respectively. Adapted from[START_REF] Wehner | Himmelsnavigation bei Insekten, Neurophysiologie und Verhalten[END_REF].

Fig. 2 .Algorithm 1

 21 Fig. 2. A. Top view of the Hexabot robot equipped with the UV-polarized light compass. (a) UV-polarized light compass; (b) MinImu-9 v.3 gyro, accelerometer & compass (Pololu) used for ground truth measurement of absolute heading direction; (c) Raspberry Pi 2B board. B. An exploded view of the UV-polarized light compass. (a) 3D-printed fixation (PLA, polyactic acid) for the UV sheet polarizer; (b) UV linear sheet polarizer (HNP'B replacement, UV grade 275 -750nm); (c) 3D-printed gears (PLA); (d) stepper motor AM0820-A-0,225-7 (Faulhaber); (e) ball bearing; (f) 3D-printed support (PLA) for gears, ball bearings and stepper motor; (g) UV-light sensor SG01D-18 (SgLux); (h) 3D-printed support (PLA) for UV-light sensor.

Fig. 3 .

 3 Fig. 3. Example of signals obtained during one acquisition from the UV-polarized light compass. (a) and (b) show normalized raw (UV n 0 and UV n 1 ) and filtered (UV nc 0 and UV nc 1 ) outputs for UV 0 and UV 1 photo-sensors. (c) shows POL-unit response for both normalized raw data and normalized filtered data. These data were acquired on 02/09/17 11:36 AM (UV-index 1, cloudy sky).

Fig. 4 .

 4 Fig. 4. Comparison between the celestial compass and the magnetometer in an outdoor context where magnetic field interferences happen. Weather conditions : clear sky, UV-index equal to 6, April 2017.

Fig. 5 .

 5 Fig. 5. Hexabot robot equiped with a pair of UV-polarized light sensors forming a celestial compass.

Fig. 6 .

 6 Fig.6.Robot electronic architecture. The dashed line marks out the robot controller and actuators. The magnetometer and UV-polarized light sensors (in yellow) are connected to the Raspberry Pi 2B board using I2C communication protocol.

Fig. 7 .

 7 Fig. 7. Heading direction angle errors in degrees in function of weather conditions. From left to right, the median heading direction angle error measured is equal to 0.4 • (n = 15), -2.9 • (n = 11), and -1.9 • (n = 7), n being the number of experiments. UV-index from 1 to 2 (Source: French Meteorological services). 42-second acquisition time per measurement.

Fig. 8 .

 8 Fig. 8. Evolution of the measured heading direction angle ψ before (in red) and after (in blue) angular correction using only the UV-polarized light compass during a straight-forward walking task. Walking steps measurements from 1 to 6 were acquired on 02/18/2017 while the next six measurements were acquired on 02/20/2017.

Experiments conducted with Sahabot were done in desert conditions whith high UV range (UV-index of 11 in Maharès, Tunisia, in August 1996). Source: http://www.temis.nl/uvradiation/UVindex.html

ACKNOWLEDGMENT

The authors would like to thank Grégoire Passault and Olivier Ly for their investment to the maintenance of the hexapod robot, and Alexandra Colombani for reviewing the English manuscript.


Keywords: Biomimetism, Biomimicry, Bionics, Biorobotics, Legged-robotics, Hexapod, Roll stability, Attitude stability

Over the last five decades, legged robots, and especially six-legged walking robots, have aroused great interest among the robotic community. Legged robots provide a higher level of mobility through their kinematic structure over wheeled robots, because legged robots can walk over uneven terrains without non-holonomic constraint. Walking robots' locomotion mode is now a well-known problem that permits to address navigation tasks issues over uneven terrains. We present here a six-legged walking robot, called Hexabot, which is a 3D-printed, low-cost, small and light structure developed at LaBRI as an open source project. We explain our choice of Hexabot over other interesting robotic platforms such as PhantomX in our navigation setup, and provide ground-truth measurements of both Hexabot and PhantomX dynamics stability when walking on smooth, flat terrain. Thanks to its geometrical structure and dynamic behaviour, Hexabot showed the lowest orientation disruptions and a remarkably stable walk. Precisely, Hexabot's orientation and walking values are similar to those of desert ants, considering the scale factor. Finally, we describe visual cues needed in order to complete desert ant-like navigation tasks.

Hexabot: a small 3D-printed six-legged walking robot designed for desert ant-like navigation tasks

Julien Dupeyroux * Grégoire Passault * * Franck Ruffier * Stéphane Viollet * Julien Serres *

INTRODUCTION

Great advances have been made in hexapod walking robotics over the past decades, since first six-legged walking robots were built, performing animal-like but straightline walking. From the end of the 60s to the begining the 90s, many hexapod walking machines were created. For instance, [START_REF] Gurfinkel | Walking robot with supervisory control[END_REF] from the Russian Academy of Sciences of Moscow developed Masha, [START_REF] Bartholet | The first functionoid developed by odetics inc[END_REF] from Odetics ITS Ing. developed Odex 1, and [START_REF] Koyachi | Hexapod with integrated limb mechanism of leg and arm[END_REF] from the Mechanical Engineering Laboratory, AIST, MITI (Japan), developed the two hexapods Melmantis I and II. Most of them share the same caracteristics: large dimensions (generaly more than one meter), important weight (up to more than 150 kg) and low mobility and speed.

The next generation of walking hexapod robots includes major efforts of miniaturization. For instance, from the mid 80s, NASA's Graduate Student Researcher Program Fellowship Jet Propulsion Lab provided huge support to develop Genghis [START_REF] Brooks | A robot that walks; emergent behaviors from a carefully evolved network[END_REF]) first and then Hannibal [START_REF] Ferrell | Robust and adaptive locomotion of an autonomous hexapod[END_REF]), two hexapod walking robots of small size (35 cm long) and weight (1kg). From the mid 90s, the FZI (Research Center for Information Technology, Germany) started the LAURON project (Legged AUtonomous RObot Neural Controlled), which led to several walking robots: LAURON I, II, III, IVc and V (see [START_REF] Roennau | Design and kinematics of a biologically-inspired leg for a six-legged walking machine[END_REF] and [START_REF] Roennau | Lauron v: a versatile six-legged walking robot with advanced maneuverability[END_REF]). For instance, first version of LAURON weighs only 12 kg and is 50 cm long by 60 cm wide.

Although the first walking machines were built to provide solutions to issues met by traditional wheeled systems, hexapod robots tend to be designed to address the thematics of locomotion among insects: optimization and adaptability of gait generation, anatomic organization of the insects' legs, complex motion skills in uneven environments. Apart from the LAURON project, which got inspired by the Indian stick insect Carausius Morosus, we can mention RHex, a 6-DoF (degrees of freedom) walking robot designed at Boston Dynamics by [START_REF] Saranli | Rhex: A simple and highly mobile hexapod robot[END_REF], whose legs are curved and mounted on rotational articulations, taking inspiration from cockroach to consider the leg morphology and walking gait. Another example of robots' design based on cockroach is WHEGS, developed by [START_REF] Schroer | Comparing cockroach and whegs robot body motions[END_REF] from Case Western Reserve University, which uses tri-spoke appendages that enable the robot to climb over small obstacles. Finally, [START_REF] Schneider | Hector, a bio-inspired and compliant hexapod robot[END_REF] developed Hector (HExapod Cognitive auTonomously Operating Robot, from the University of Bielefield) which embodies one of the latest example of bio-inspiration among walking robots, including impressive biomimicry such as reflex behaviour to prevent any fault, and a decision-making system to provide insectlike walking behaviour, and intended for leg control, gait generation and body motion tasks.

Nowadays, in hexapod walking robots, gait generation and leg control during walking and running tasks are wellknown problems, backed by five decades of research in this field of study. Consequently, we decided to develop a small walking, six-legged robot that will address challenging navigation tasks and benefit from the knowledge on the subject of gait control.

Homing tasks in desert ants have received great interest from entomologists. We now have plenty of information about what type of visual cues ants get and use while returning from feed source to their nest. To complete navigation tasks, desert ants rely on visual cues like optical flow, low resolution panoramic vision, and UVlight polarisation (see [START_REF] Collett | Spatial memory in insect navigation[END_REF] and [START_REF] Cheng | Path integration, views, search, and matched filters: The contributions of rüdiger wehner to the study of orientation and navigation[END_REF] for review). [START_REF] Raderschall | Head roll stabilisation in the nocturnal bull ant myrmecia pyriformis: implications for visual navigation[END_REF] studied head roll stability affected by locomotion and how oscillations spoil navigation performances. In their study, they found that bull ants experience average head roll of 10 • peak-to-peak. Similarly, [START_REF] Ardin | How variation in head pitch could affect image matching algorithms for ant navigation[END_REF] showed that Catagliphys velox 's body pitch causes head pitch oscillations (up to 60 • peak-to-peak when carrying food) that disrupts navigation performances. However, it seems that no mechanical compensation of pitch exists, and only a few compensation are made concerning head roll. One hypothesis [START_REF] Raderschall | Head roll stabilisation in the nocturnal bull ant myrmecia pyriformis: implications for visual navigation[END_REF] proposed is that ants acquire visual informations when body oscillations reach their lowest level. Roll and pitch stabilities of the visual information appear to be one of the milestone in hexapod robot to navigate over an uneven terrain.

WHY HEXABOT ?

Perhaps the very first question should be: why a six-legged walking robot instead of a four-legged one? Hexapod walking robots have many advantages over quadruped ones: static gait (three to five legs are in contact with the ground at any moment), increased walking pace and more stability. Besides, hexapod robots can still walk with interesting performances if a damage occurs to one leg, as it was made possible in [START_REF] Cully | Robots that can adapt like animals[END_REF].

Walking behaviour applied to hexapod robots is a well mastered issue. We can thus benefit from such knowledge to directly customize an already designed hexapod robot. Since customization step will increase the payload of the robot, it is important to have one with good payload capacity without disturbance over speed performance. Moreover, stability of the robot should not be affected by carrying a payload when walking. As we estimate global embedded-to-be electronics and sensors weight to be around 450 g, we rapidly came to hesitate between two hexapod robots.

The hexapod robots: Hexabot versus PhantomX

The first robot is PhantomX AX 18 (see fig. 1, top), designed by InterbotiX Labs (Trossen Robotics, Downers Grove, IL, USA). It has six legs, each using three degrees of freedom made using ultra fast DYNAMIXEL AX-18A series robot servos, providing an maximum speed of 80 cm/s when tested in optimal conditions: on a flat terrain, without sliding effect and with tripod gait. It should be noted that PhantomX maximum speed is presently the best performance. Another advantage of this robot is that it can carry heavy payloads (up to 2 kg, but with reduced speed).

The second robot is Hexabot (see fig. 1, bottom), a fully 3D-printed, open source walking robot based on the Metabot1 concept developed by [START_REF] Passault | Optimizing morphology and locomotion on a corpus of parametric legged robots[END_REF] at LaBRI Lab in Bordeaux, France. Since legs are printed with PLA filament (polylactic acid, a biodegradable thermoplastic aliphatic polyester derived from renewable resources), they require less effort to be moved during the transfert phase of walking movement. Indeed, each part of the legs are printed with low infill (20%) and 0.2 mm layer height. In those conditions, Hexabot reaches an optimum between maximum strength (the maximum stress the specimen can take before breaking) and overall weight.

Another consequence of a reduced weight is that joints can be operated by less energy-demanding servos. Each leg has three degrees of freedom made with DYNAMIXEL XL-320 series robot servos, which are smaller and lighter than those used by PhantomX. The maximum speed of Hexabot is only of 35 cm/s in optimal conditions, but it will be enough to complete our navigation tasks. Finally, Hexabot can carry payloads up to 450 g without disturbing its walking behaviour (stability and speed).

Stability analysis

Geometric analysis provides interesting evidence that Hexabot would be a starting point for our experiments. The Fig. 2. The effect of locomotion over a flat terrain on Hexabot's roll stability. From top to bottom, graphs show roll behaviour when speed is 6.2 cm/s (resp. 13.3 cm/s and 22.6 cm/s). In this case, the maximum peak-to-peak roll disturbance is about 4.5 • (resp. 6.7 • and 9.0 • ). Blue curves correspond to average roll angle while red curves correspond to maximum and minimum roll angles over the ten iterations of the walking task.

geometric parameters of both robots are given in table 1.

In particular, [START_REF] Chu | Comparison between different model of hexapod robot in faulttolerant gait[END_REF] demonstrated that the hexagonal model shows better stride length (in specific conditions), better turning ability and better stability margins than the rectangular model. Furthermore, the rectangular shape of PhantomX's body implies the use of crab walking or turn. On the contrary, the hexagonal shape of Hexabot provides a holonomic property that enhances direction changing aptitudes. In order to make the analogy with ants, table 1 includes Cataglyphis desert ants caracteristics. Considering the maximum body length, Hexabot is 10 times bigger than ants, and PhantomX is 20 times bigger. The ants' maximum speed is often in the range of 70 cm/s, which matches with PhantomX's speed performance, but walking speed is generally measured at 30 cm/s and coincides to Hexabot's speed regime. In order to analyze orientation disturbances caused by the walking behaviour of both Hexabot and PhantomX, we used the Flying Arena of the Mediterranean 2 (6m x 8mx 6m-height) equipped with 17 motion-capture cameras (VICON) covering a uniquely reconfigurable flight space. Therefore, orientation angles were measured over time during straight-forward walking tasks of the robots.

Based on first observations, it seems PhantomX has heavy walk that involves lots of vibrations over its structure 2 http://flying-arena.eu/ which would cause major visual artefacts and thus affect navigation performances. A contrario, Hexabot has a smoother walking behaviour due to its low weight.

The effect of walking tripod gait on the orientation of Hexabot was observed and figure 2 shows its roll stability over time at different speeds. For each speed, walking task was repeated ten times. We can notice that the faster the robot walks, the more important the roll disturbances.

The wave shape is due to time intervals of tripod gait: the maximum of orientation disturbances occur during transfert phase of legs. The entire results of orientation analysis are given in table 2. PhantomX shows greater roll and pitch turmoils than Hexabot, be the speed set to 50%, 75% or 100% of maximum speed. Consequently, Hexabot tends to be less prone to visual artefacts caused by locomotion. Interestingly, when walking at maximum speed, Hexabot shows the same roll displacement than ants. This observation is no longer true when considering pitch displacements. Indeed, there is a great similarity between Hexabot's roll and pitch displacements which can be explained by the strict symmetry of the hexagonal body of the robot, while ants' visual system is located on the head which is not directly connected to the legs. In other words, Hexabot's roll and pitch displacements are both caused by tripod gait locomotion, like ants' roll displacement, whereas ants' pitch displacement is greatly affected by carried food and roughness of terrain. Furthermore, it should be noted that yaw angles are considerably greater than roll or pitch angles. The reason is that both Hexabot and PhantomX use tripod gait, that is a gait where three legs (one on the left, and the two others on the right, forming a triangle shape, and vice-versa) are in contact with a ground at any moment.

CONCLUSION AND FUTURE WORK

Hexapods are well mastered and can be used to test biological theories about insect locomotion and navigation abilities. Hexapod robots with hexagonal shape show better performances in key tasks, especially in body orientation stability. We compared two walking robots, Hexabot and PhantomX, using our motion-capture system, and came to the conclusion that Hexabot, a 3D-printed, light and small robot, entirely met the requirements and will be a more reliable candidate for navigation tasks based on visual cues. Our robotic platform Hexabot has shown interesting walking skills but remains to be customized with sensors in order to complete navigation tasks such as foraging and homing, as desert ants like Cataglyphis and Melophorus do.

Hexabot will be equiped with sensors that will reproduce the same visual cues as desert ants in order to fulfill homing tasks in both indoor and outdoor conditions [START_REF] Dupeyroux | A novel insect-inspired optical compass sensor for a hexapod walking robot[END_REF]. Ceteris paribus roll and pitch behaviours will afford the opportunity to adapt Hexabot to an ant-inspired walking robot, considering it provides the same range of roll displacements as due to its tripod gait locomotion.

  This work was supported by the French Direction Générale de l'Armement (DGA), CNRS, Aix-Marseille University, the Provence-Alpes-Côte d'Azur region and the French National Research Agency for Research (ANR) with the Equipex/Robotex project.

Fig. 1 .

 1 Fig. 1. The two hexapod robots PhantomX (top) from InterbotiX Labs, USA, and Hexabot (bottom) from LaBRI, France.

Table 1 .

 1 Hexabot and PhantomX specifications. Overall weight does not include sensors, nor batteries. Equivalent values are given for desert ants Cataglyphis.

			Hexabot	PhantomX	Ants
	Body	Max length	115 mm	210 mm	10mm
	Leg		215 mm	290 mm	8mm
	Robot Max length	525 mm	870 mm
		Max height	145 mm	185 mm
		Max speed	35 cm/s	80 cm/s	70 cm/s
		Overall weight	675 g	1975 g	10 mg

Table 2 .

 2 The effect of walking on orientation stability

		Speed	Hexabot PhantomX	Ants
		50%	4.5 •	7.4 •
	Roll	75%	6.7 •	9.3 •	10 •
		100%	9.0 •	11.3 •
		50%	4.4 •	7.0 •
	Pitch 75%	5.4 •	9.7 •	60 •
		100%	9.9 •	17.1 •
		50%	19.8 •	9.7 •
	Yaw	75%	19.8 •	17.5 •
		100%	19.8 •	28.4 •
	of both Hexabot and PhantomX, and the correspond-
	ing average value measured in ants. Angle values of roll,
	pitch and yaw are given in degrees and correspond to
	the peak-to-peak maximum amplitude observed during
	a straight-forward walking task, over 7 seconds. Speed
	values are percentages of maximum speed.

http://metabot.cc/

ACKNOWLEDGEMENTS

We sincerely thank Grégoire Passault and Olivier Ly from LaBRI for their technical support to develop and manage Hexabot.


Heading direction error from 0.3° to 2.9° under various weather conditions: High Reliability.

Sensory modalities very similar to those of insects : robotic insect embodiment.

Even under poor weather conditions, these results highlights good precision to make the optical compass suitable for field robotics.

A bio-inspired celestial compass for a hexapod walking robot in outdoor environment

Julien Dupeyroux, Julien Diperi, Marc Boyron, Stéphane Viollet, Julien Serres. Aix-Marseille Université, CNRS, ISM, Biorobotics Team, Marseille, France

Variable angular step resolution of the sensor (max res is 0.95°)

42-seconds Acquisition Time

Most insects like desert ants and bees use the polarization pattern of skylight to get their orientation [START_REF] Labhart | Polarization-opponent interneurons in the insect visual system[END_REF].

The skylight is linearly polarized and the direction of polarization remains slightly constant within short times.

The compound eye

The DRA

The solar meridian e-Vector

The polarized light detection in insects

The celestial compass UV0

Conclusion

Top of the Hexabot robot equipped with the UV-polarized light compass [START_REF] Dupeyroux | Hexabot: a small 3D printed six-legged walking robot designed for ant-like navigation tasks[END_REF].

cm

The signal processing unit

Example of signals acquired by the celestial compass

Results

Heading lock over a straight-forward walking task

Heading recovery performances

Performances of the celestial compass under various weather conditions [3]

Celestial Compass Magnetometer

where is the solar meridian direction angle [0, ]. We then define p(x) as the log ratio of both normalized and corrected UV0 and UV1 POL-signals:

POL-sensor raw responses depending on the gear rotation angle x in [0, 2 ]: Finally, is computed as follows:


Julien Dupeyroux

Stéphane Viollet

Julien Serres

Julien Serres Providing

Providing an autonomous hexapod walking robot with the ability to reorientate: application of a novel ant-inspired celestial compass

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

Providing an autonomous hexapod walking robot with the ability to reorientate: application of a novel ant-inspired celestial compass Julien Dupeyroux, Stéphane Viollet and Julien Serres Aix Marseille Univ, CNRS, ISM, Marseille, France.

Abstract-Insects like desert ants have photoreceptors in their Dorsal Rim Area sensitive to the ultraviolet skylight polarization pattern. Taking inspiration from insects allows us to develop smart sensors. A bio-inspired celestial compass has been developed in the aim of stabilizing the heading of a hexapod robot, which can estimate its heading direction from the linearly polarized skylight in the ultraviolet (UV) range. Field results will be presented and will show their reliability and accuracy with respect to various weather conditions. Finally, our celestial compass can be used as an alternative to both magnetometers failure in highly disturbed environment, and GPS limited precision.

I. INTRODUCTION

The classical methods to provide the heading direction of mobile robots mainly rely on the use of inertial measurement units (IMUs) and global positionning systems (GPS). Unfortunately, IMUs remain highly sensitive to any electromagnetic disruptions produced by ferrous materials. In addition, GPS-based solutions only provide metric precision which does not fit small scale mobile robotics tasks.

The study of the insects' navigational toolkit showed that many of them such as desert ants, bees, and locusts make use of the skylight polarization as a visual cue to estimate their heading direction [START_REF] Wehner | Visual navigation in insects: coupling of egocentric and geocentric information[END_REF], mostly in the UV-range. The sunlight is indeed linearly polarized when crossing the Earth's atmosphere [START_REF] Coulson | Polarization and Intensity of Light in the Atmosphere[END_REF]. The angle of polarization keeps stable over short-term so that insects can rely on that information to complete their navigation tasks like homing.

II. THE CELESTIAL COMPASS

A new insect inspired celestial compass has been developed and embedded onto a six-legged walking robot ([3], [START_REF] Dupeyroux | A novel insect-inspired optical compass sensor for a hexapod walking robot, A bio-inspired celestial compass applied to an ant-inspired robot for autonomous navigation[END_REF], [START_REF] Dupeyroux | Hexabot: a small 3D-printed six-legged walking robot designed for desert ant-like navigation tasks[END_REF]) (fig. 1). This sensor is made of two UV-light sensors (SG01D18, SgLux) topped with rotating UV linear sheet polarizers (HNP'B replacement). The angular directions of the polarizers are phase-shiffted by 90 • . The full prototype is 3D-printed with polyactic acid (PLA) filament.

According to both scanning and simultaneous models introduced by Lambrinos et al. [START_REF] Lambrinos | An autonomous agent navigating with a polarized light compass[END_REF], we propose to merge these models into one, so that the measurement of the polarization direction is performed by rotating the polarizers by 360 • at fixed angular resolution (up to 0.64 • ). The signals measured by the two units are cosine waves with 180 • phase shift. The log-ratio of the two signals is then computed, where the polarization direction corresponds to the angle with the minimum log-ratio value.

Outdoor tests under various weather conditions (clear, variable and overcast sky) were performed to describe qualitatively and quantitatively the ability of the sensor to provide reliable heading direction to the robot during heading lock navigational tasks. Tests were conducted during winter (February 2017) with a very low UV-index (up to 2). The results were highly precise and reliable under clear-sky conditions, with an average heading steady state error as small as 0.4 • . Results under cloudy-sky conditions exhibits good performances as well, from 0.8 • under variable weather, to 1.9 • under overcast sky, but slightly less reliable due to the high variability of meteorological conditions.

Fig. 1 .

 1 Fig. 1. Top view of the Hexabot robot equipped with the UV-polarized light compass.


Yves Laprie
email: yves.laprie@loria.fr

Benjamin Elie

Pierre-André Vuissoz

Anastasiia Tsukanova

Articulatory model of the epiglottis

Keywords:

Introduction

The position of the epiglottis has an acoustic influence in the lower part of the vocal tract by decreasing the transverse area when it is contacting the tongue root, or when it reduces the volume of the laryngeal vestibule. Conversely, it is assumed to have a negligible acoustic impact when the pharyngeal cavity is wider (as /i/ for instance) and the epiglottis is well detached from the tongue. Epiglottis is a cartilage connected to the hyoid bone via the thyroid cartilage. Unlike tongue and lips whose intrinsic muscles are at the origin of a substantial part of their deformations, those of the epiglottis are mainly due to the influence of other articulators, i.e. the tongue, larynx, and indirectly mandible. A global articulatory model of the vocal tract should take these properties into account.

Description of the model

Similarly to the model of [START_REF] Maeda | Un modèle articulatoire de la langue avec des composantes linéaires[END_REF] the deformation modes of our articulatory model (Y. [START_REF] Laprie | Construction and evaluation of an articulatory model of the vocal tract[END_REF][START_REF] Laprie | 2D Articulatory Velum Modeling Applied to Copy Synthesis of Sentences Containing Nasal Phonemes[END_REF] are obtained by applying Principal Component Analysis (PCA) on the articulator contours (mandible, tongue, lips. . . ) by using prior knowledge of the interactions between articulators. For instance, the tongue is attached to the mandible whose movement is subtracted from the tongue contours before applying PCA. So far our epiglottis articulatory model only incorporated the indirect influence of the mandible and the influence of the tongue via a collision model. Its deformation modes thus integrated movements due to the tongue and larynx.

Other deformation modes were due to the errors of delineation. In the case of the tongue, these errors are marginal, or at least give rise to deformation modes coming after the genuine deformations whose amplitude is bigger. On the other hand, the width of epiglottis is small on the image, and the errors of delineation, whether manual or automatic, are of the same order of magnitude as genuine deformations. Consequently, a PCA applied without precaution will mix both types of deformation.

To prevent the apparition of these spurious deformation components the epiglottis was approximated as a thick curve, and only the centerline of the epiglottis was analyzed. As a matter of fact, the centerline was determined after delineation of all the epiglottis contours, and the width was set as the average width of all these contours in the upper part where the two epiglottis edges are clearly visible (see Fig. 1). The height of the upper part (where both contours are visible) is adjusted by hand to fit the contours extracted from images. The centerline is approximated as a B-spline and represented by its control points P l (0 ≤ l < M where M is the number of control points) in the form of a two-coordinate vector, and the reconstruction of the epiglottis from the centerline amounts to draw a line at a distance of half the width from the centerline.

In order to take into account the influences of the jaw, larynx and tongue, we applied multiple linear regression Rao, R. Toutenburg, and Heumann 2008 on the control points P l (0 ≤ l < M ) of the epiglottis contour:

P l = jaw 0,l B0 + j=T -1 j=0 tg j,l Cj + lx 0,l D0 + E l
where jaw 0,l is the control factor of the first linear component of the jaw in the global articulatory model and B0 the twocoordinate regression vector for the jaw, tg j,l the control factors of the T first linear components of the tongue and Cj the corresponding regression vector, lx 0,l the first control factor of the larynx component and D0 the corresponding regression vector, and E l is the residue vector.

Actually, since the centerline is represented by the M control points, each centerline occurrence is represented by a 2×M vector formed by the coordinates of the M points. Each of the N occurrences of the epiglottis is represented by a 2×M vector named Pi with 0 ≤ i < N . These contour vectors Pi are the observations, and the control factors of the jaw jaw0,i, tongue tgj,i and larynx lx0,i are the input explanatory variables. For the jaw and larynx we kept only the first factor (jaw0, resp. lx0), which is in both cases the most informative one. For the tongue, which is in front of the epiglottis, we kept the first six deformation factors tg0..5. All the variables are centralized and therefore the intercept can be ignored.

By grouping all the explanatory factors jaw0, tgj, and lx0, in a vector of K = T + 2 coordinates which are named Xj, the previous equation is

Pi = j=K-1 j=0 bjXi + ei or in a matrix form P = XB + E
, where P is the N × 2M matrix of the observations, i.e. the control points of the centerlines, X is the N × K matrix of the explaining articulatory factors, B is the K × 2M regression matrix and E is the N × 2M matrix of residue not explained by jaw, tongue and larynx. B is given by

B = (X t X) -1 X t Y .
Then, the contributions of the jaw, tongue and larynx can be subtracted from the observations of the epiglottis, i.e. P -XB, and PCA can be applied to the residue E. Given the nature of epiglottis, i.e. a cartilage, the number of relevant linear components should be small.

We applied this analysis scheme to two sets of data: midsagittal slice of 3D MRI images recorded at IADI of 90 static syllables covering vowels and blocked CV articulations made by a male speaker and a corpus of 1021 mid-sagittal images from an X-ray film (from the DOCVACIM database) made of short sentences recorded by a female speaker. most of the variance is explained by the articulators influencing the epiglottis and that two linear components are enough to approximate the epiglottis fairly accurately.

Concluding remarks

The regression stage enables most of the extrinsic influences exerted on the epiglottis to be removed before PCA. It should be noted that these influences explain as much variability as the intrinsic components. This new model is currently used with our articulatory copy synthesis approach. Fig. 2 and 3 show that the first factor corresponds to a vertical movement, although the second corresponds to a horizontal movement in the lower and mid part of the epiglottis which contributes to the constriction of the whole laryngeal vestibule. This retraction is not explicitly used in French which is the

Figure 1 :

 1 Figure 1: The epiglottis contour and the centerline (left), and the complete articulatory model in red approximating the vocal tract for the articulation /sø/ (right)

Figure 2 :

 2 Figure 2: First two deformation modes of the epiglottis (male speaker). The green line is the neutral position (middle)

Table 1 :

 1 Tab. 1 shows that Standard deviations in millimeter of the epiglottis: total, after subtracting the influence of other articulators, after subtracting 1 st , 2 nd and 3 rd linear components.

	σ in mm	male (static) female (dynamic)
	total	8.44	14.32
	regression 5.08	6.51
	PCA 1st	2.66	3.17
	PCA 2nd	1.33	1.73
	PCA 3rd	0.78	0.93

Acknowledgements

This work benefited from the support of the project ArtSpeech of ANR (Agence Nationale de la Recherche).