Xiucong Sun

Pei Chen

Christophe Macabiau

Chao Han
email: hanchao@buaa.edu.cn

Low-Earth Orbit Determination from Gravity Gradient Measurements

Keywords: Gravity gradient, Gravity gradiometry, LEO orbit determination, Adaptive hybrid least squares, GOCE

An innovative orbit determination method which makes use of gravity gradients for Low-Earth-Orbit satellites is proposed. The measurement principle of gravity gradiometry is briefly reviewed and the sources of error are analyzed. An adaptive hybrid least squares batch filter based on linearization of the orbital equation and unscented transformation of the measurement equation is developed to estimate the orbital states and the measurement biases. The algorithm is tested with real flight data from the European Space Agency's Gravity field and steady-state Ocean Circulation Explorer (GOCE). The orbit determination results are compared with the GPSderived orbits. The radial and cross-track position errors are on the order of tens of meters, whereas the along-track position error is over one order of magnitude larger. The gravity gradient based orbit determination method is promising for potential use in GPS-denied navigation and in outer space planetary exploration.

Introduction

Satellite orbit determination usually relies on geometric measurements. Two typical examples are the groundbased radar tracking and the Global Positioning System (GPS), both of which utilize electromagnetic wave propagation to measure relative distance and direction. Current GPS technology achieves centimeter-level accuracy for LEO satellites with dual-frequency carrier phases [START_REF] Montenbruck | Reduced dynamic orbit determination using GPS code and carrier measurements[END_REF][START_REF] Bock | GPSderived orbits for the GOCE satellite[END_REF]. An alternative method to the geometry-based orbit determination is the geophysical navigation, which derives position from local geophysical data. One representative is the magnetometer-based autonomous navigation. Orbital position errors ranging from a few to a hundred kilometers have been achieved with real flight data from several LEO satellites [START_REF] Psiaki | Ground tests of magnetometer-based autonomous navigation (MAGNAV) for low-Earth-orbiting spacecraft[END_REF][START_REF] Thienel | Results of the magnetometer navigation (MAGNAV) inflight experiment[END_REF][START_REF] Wu | Particle filter using a new resampling approach applied to LEO satellite autonomous orbit determination with a magnetometer[END_REF][START_REF] Juang | Design and verification of a magnetometer-based orbit determination and sensor calibration algorithm[END_REF]. Despite the low accuracy, the geophysical navigation does not need supports from ground stations or any other satellites and is thus suitable for autonomous spacecraft operation in GPS-denied environments.

Besides the magnetic field, the gravity is another kind of geophysical information that can be exploited for orbit determination. The Earth has a relatively stable gravity field. Its effects on satellites include two aspects. Firstly, the gravity force (or gravitational acceleration) is the main driving force of the orbital motion. The modeling of the gravity field is of crucial importance to precise orbit prediction. Secondly, the gravity gradients (or gravitational gradients) vary as functions of position and orientation and can be measured by a spaceborne gradiometer [START_REF] Difrancesco | Gravity gradiometer systems-advances and challenges[END_REF][START_REF] Rummel | GOCE gravitational gradiometry[END_REF].

With a known gravity model as well as the satellite orientation information, the orbit could be estimated from the gravity gradient observations. The application of gravity gradiometry for navigation has been studied since the 1960s. One of the main research interests is to incorporate a gradiometer into an airborne or shipborne inertial navigation system (INS) for real-time compensation of gravity model uncertainties [START_REF] Levine | Geodetic and geophysical uncertainties -Fundamental limitations on terrestrial inertial navigation[END_REF][START_REF] Siouris | Gravity modeling in aerospace applications[END_REF][START_REF] Wells | Real-time gravity gradiometer utilization to improve INS accuracy[END_REF][START_REF] Jekeli | Precision free-inertial navigation with gravity compensation by an onboard gradiometer[END_REF]. Metzger and Jircitano [START_REF] Metzger | Inertial navigation performance improvement using gravity gradient matching techniques[END_REF] presented an early form of mapmatching technique by cross-correlating sensed gravity gradients with previously mapped values. The premise was to let a vehicle travel a course twice and to compute the state lag from gravity gradient measurements on both passes.

The idea developed into a passive gravity gradiometer navigation system in which the gravity gradient map was not provided by a first flight but generated from the terrain elevation data base [START_REF] Affleck | Passive gravity gradiometer navigation system[END_REF]. An optimal filter was designed to update positions and to correct instrument errors. During the following twenty years further contributions were made on this topic, including the Fast Fourier Transformation based rapid map generation [START_REF] Gleason | Passive airborne navigation and terrain avoidance using gravity gradiometry[END_REF], extended application to a hypersonic cruise [START_REF] Richeson | Gravity gradiometer aided inertial navigation within non-GNSS environments[END_REF], and feasibility investigation using a modern gradiometer [START_REF] Rogers | An investigation into the feasibility of using a modern gravity gradient instrument[END_REF]. By contrast, little research has been conducted on the applications of gradiometry for spacecraft navigation. The major difference between the inertial navigation aiding and the application for spacecraft navigation lies in the fact that high-precision attitude information could be easily obtained onboard a satellite by star trackers. This decouples position estimation from attitude estimation. In addition, the higher frequency terrain contributions are dramatically attenuated at the height of a spacecraft. A truncated spherical harmonic gravity model will be accurate enough for space users.

In 2009, ESA's GOCE satellite was launched into a sun-synchronous LEO orbit to determine the Earth's gravity field and geoid heights. The satellite carried an Electrostatic Gravity Gradiometer (EGG) and measured gravity gradients from an unprecedented low altitude of about 260 km in space. Post flight analysis showed that a noise density level of 0.01 E/√Hz was achieved within the measurement band (MB) from 5×10 -3 to 0.1 Hz [START_REF] Rummel | GOCE gravitational gradiometry[END_REF]. The satellite was also equipped with three advanced star trackers and two dual-frequency GPS receivers. These conditions make GOCE an ideal testbed for the research of gravity gradient based space navigation. X. Sun [START_REF] Sun | Test of GOCE EGG data for spacecraft positioning[END_REF] introduced an idea of using full-tensor gravity gradients combined with high-precision attitudes to determine a spacecraft's position. A least squares searching algorithm was developed and a mean positioning error of 620 m was achieved with real GOCE data. An eigendecomposition method using the J2 gravity model was presented in P. Chen [START_REF] Chen | Gravity gradient tensor eigendecomposition for spacecraft positioning[END_REF] and position errors ranging from 421 to 2690 m were achieved.

In the previous studies mentioned above, the gravity gradient observation errors are modeled as low-level white noise only. In fact, the gradiometer measurements contain significant biases and low-frequency noises. The present work deals with the biases and the drifts in the actual measurements. The noise characteristics are investigated and a simplified error model is formulated. An adaptive hybrid least squares batch filter is developed to estimate the orbital states and the biases. The filter combines the advantages of the linear approximation of the orbital equation and the unscented transformation of gravity gradient observations to achieve fast and accurate orbit determination.

The measurement time span at each iteration step is adaptively adjusted to restrict the linearization errors and thus to guarantee convergence. An augmented state iterated least squares filter is also implemented to further estimate the drifts. The algorithms are tested with real GOCE data and the orbit determination results are compared with the Precise Science Orbit (PSO) solutions derived from the GPS system. This contribution is organized as follows. Section 2 briefly reviews the measurement principle of GOCE gravity gradiometry and investigates the sources of error in gravity gradient retrieval. Section 3 presents the orbital dynamic model, the gravity gradient observation model, and the measurement error model for orbit determination. Section 4 summarizes the iterated least squares filter and the unscented least squares filter for nonlinear estimation and gives the algorithm of the adaptive hybrid least squares filter. Section 5 presents the orbit determination results obtained with real GOCE data. Conclusions are drawn in Section 6.

GOCE gravity gradiometry

A differential accelerometry technique was employed by GOCE to measure gravity gradients. The gradiometer was placed close to the spacecraft's center of mass (CoM) and consisted of three orthogonal pairs of capacitive accelerometers. Each accelerometer had two ultra-sensitive axes and one less-sensitive axis. The three pairs were mounted at the ends of three baselines having an approximate length of 0.5 m. The gradiometer reference frame (GRF) is materialized by the three orthogonal baselines with the X axis in the flight direction, the Y axis normal to the orbit plane, and the Z axis radially downwards, as depicted in Fig. 1. Inside each accelerometer, a platinumrhodium proof mass was electrostatically levitated at the center of a cage, leading to control voltages that were representative of the sum of the non-gravitational accelerations at the location of the proof mass [START_REF] Cesare | Performance requirements and budgets for the gradiometric mission, 4th edn[END_REF]. The gravity gradients were contained in the accelerometer differences.

X GRF Y GRF Z GRF Y 5 Y 6 X 6 Z 6 Y 1 X 1 Z 1 Y 4 X 4 Z 4 Y 2 X 2 Z 2 Y 5 X 5 Z 5 Y 3 X 3 Z 3 orbit trajectory

Earth direction

Measurement principle

To describe the measurement process, an ideal gradiometer is considered by assuming that: (i). The centers of the three baselines are coincident;

(ii). The baselines are mutually perpendicular and perfectly aligned with the three axes of GRF;

(iii). The accelerometers occupy their nominal positions and their axes are also aligned with GRF;

(iv). The accelerometers can sense the local non-gravitational accelerations exactly, which means that there are no errors in the conversion of control voltages.

Under these ideal conditions, the output of each accelerometer has the following expression

  2 i i      a a Ω Ω V r d ( 1
)
where i is the identifier of the accelerometer, i r is the vector from CoM to the position of the ith accelerometer,

2 i
Ω r is the centrifugal acceleration induced by the spacecraft's angular rotation, i  Ωr is the Euler acceleration, V is the gravity gradient tensor (GGT), and a d is the non-gravitational acceleration at CoM. The accelerations due to the higher order (  3rd) derivatives of the gravitational potential, relative motion of the cage (e.g., Coriolis effects), selfgravity, and the coupling with the external magnetic field are not considered. The matrices Ω, ,  Ω and 2 Ω are defined as

0 0 0 z y z x y x                     Ω (2) 0 0 0 z y z x y x                            Ω (3)       2 2 2 2 2 2 2 y z x y x z x y x z y z x z y z x y                                      Ω (4)
where , The non-gravitational acceleration a d is first isolated by forming the common-mode (CM) and the differentialmode (DM) accelerations

    2 , 1 2 c ij i j        a a a a Ω Ω V c d (5)     2 , 1 1 2 2 d ij i j ij      a a a Ω Ω V L ( 6
)
where   14, 25, 36 ij  represents the index of the accelerometer pairs, c is the vector from CoM to the center of the baselines, and ij L is the vector from the jth to the ith accelerometer.

Then combine the three DM accelerations to form a matrix equation

  2 1 2 d     A Ω Ω V L (7) with ,14 ,25 ,36 d d d d      A a a a (8)
 

14 25 36 0 0 0 0 0 0 x y z L L L             L L L L (9)
where ,

x L , y L and z L are the lengths of the three baselines. The right-hand side of Eq. ( 7) contains the GGT, the centrifugal accelerations, and the Euler accelerations. Based on the symmetry of 2 Ω and V and the skew-symmetry of ,  Ω the Euler accelerations can be isolated

T d d    A A ΩL (10)   2 d d T    A A Ω V L (11)
The gravity gradient tensor is finally retrieved as follows

  1 2 T d d          V A A L Ω (12)

Error sources

The ideal GGT retrieval procedure is subject to several sources of error. For the most part, the errors are attributed to the accelerometers. First, accelerations are measured by voltage signals. The transformation from voltage to acceleration requires the accurate knowledge of the electrostatic gains and the read-out gain. Uncertainties of these gains result in scale factors in the outputs [START_REF] Cesare | Performance requirements and budgets for the gradiometric mission, 4th edn[END_REF]. Second, non-linearity exists in the electronic components and the transfer functions of the control loop, leading to an additional quadratic term, which is proportional to the square of the input acceleration. Last but not least, the accelerometer outputs contain unknown biases and noises due to the intrinsic imperfection. In addition to the accelerometer-related errors, the gravity gradiometer also has geocentric imperfections. For example, the misalignments of the six accelerometers induce small rotation angles with respect to the nominal GRF frame and non-perfect orthogonality between the accelerometer axes causes cross coupling errors.

Taking all of the factors above into consideration, the actual output of each accelerometer should be written as

  2 i i i i i i i i i d d        a K R S a K2a b n ( 13
)
where i K and i K2 are the scale factor and the quadratic factor, i dR is the rotation matrix due to misalignments, i dS is the accelerometer inter-axis coupling matrix, and i b and i n are the accelerometer bias and noise.

The quadratic factor of each accelerometer has been identified and calibrated in flight and the calibration error is negligible [START_REF] Cesare | Gradiometer on-orbit calibration procedure analysis[END_REF]. Let i M denote the sum of , i K , i dR and .

i dS According to Eq. ( 6), the actual obtained CM and

DM accelerations are , , , , , , , ,
c ij c ij c ij c ij ij d ij d ij d ij d ij                              a a b n M a a b n (14) with 1 2 i j i j ij i j i j            M M M M M M M M M (15) , , , , 1 1 , 2 2
c ij i j c ij i j d ij i j d ij i j                               b b b n n n b b b n n n (16) where
ij c ij c ij c ij c ij c ij ij ij ij d ij d ij d ij d ij d ij d ij                                           a a a a b n M M M a a a a b n ( , , , 1 1 1 , , , , , , ˆˆˆĉ
) 17
where , ˆc ij a and , ˆd ij a are called the calibrated CM and DM accelerations. The d A matrix can now be formed using , ˆc ij a and , ˆd ij a according to Eq. ( 8) and the error consists of a 1 ˆij  M related term and linear combination of the accelerometer biases and noises.

In the final step of GGT retrieval, the accurate knowledge of centrifugal accelerations is required, as seen in Eq. [START_REF] Jekeli | Precision free-inertial navigation with gravity compensation by an onboard gradiometer[END_REF]. The angular velocities have been derived from an optimized combination of angular accelerations from the gradiometer, as seen in Eq. ( 10), and attitude quaternions from the star trackers [START_REF] Frommknecht | GOCE level 1b data processing[END_REF][START_REF] Stummer | Upgrade of the GOCE Level 1b gradiometer processor[END_REF]. The angular velocity estimation error will definitely affect the accuracy of the GGT measurements.

Based on the analysis of error sources, the GGT observation error can be decomposed as

ICM a a       V N B N N (18) with           ,14 ,14 ,14 ,14 ,25 ,25 ,25 ,25 , 1 1 1 1 ,14 ,14 ,14 ,14 1 1 1 1 ,25 ,25 ,25 ,25 1
d c T T c d c d T T c d c d T c d c d d ICM c d c d c d c d                                                  a a a a N a a a a a a M M M M M M M M M M M a M a   1 36 T T                                  L (19)             1 1 1 1 ,14
,14 ,14 ,14

1 1 1 1 ,25 ,25 ,25 ,25 1 1 1 1 ,36 , ,14 ,14 ,14 ,14 , 2 5 , 2 5 , 2 5 , 2 5 ,36 ,36 ,36 , 36 ,36 , 3 36 6
ˆˆˆT ˆˆˆT

c d c d a c d c d c T c d c d T T c d c d T T c d c d c d d                                          b b b b B b b b b b b b b M M M M M M M M M M M M 1 T                       L (20)             1 1 1 1 ,14 ,14 ,14 ,14 1 1 1 1 ,25 ,25 ,25 ,25 1 1 1 1 ,36 , ,14 ,14 ,14 ,14 ,25 ,25 ,25 ,25
c d c d a c d c d c T c d c d T T c d c d T T c d c d c d d                                         M n M n M n M n N M n M n M n M n M n M n M n M n 1 T                       L (21)       2 2 2 z z y z z y z z y y y z z x y y x x x x y y x x x z z x x x x y z z y y                                                                                 N ( 22
)
Where ICM N is the noise due to the ICM calibration error, a B and a N are the bias and noise due to biases and noises in the six accelerometers, respectively,  N is the noise due to angular velocity estimation error,

Orbit determination models

This section describes the orbital dynamic model and the gravity gradient observation model used for GOCE orbit determination. In addition, the characteristics of the sources of error are analyzed and a simplified measurement error model is formulated.

Orbital dynamic model

The spacecraft's orbital motion is described by the following first-order differential equation

 

, ,

d t dt              v r a r v v ( 23
)
where r and v are the inertial position and velocity vectors and a (t, r, v) is the inertial acceleration. In this study, the International Celestial Reference Frame (ICRF) is used as the inertial coordinate system. The definition of ICRF refers to Petit et al. [START_REF] Petit | IERS Technical note no.36[END_REF]. Given initial values and accurate force models, position and velocity over time can be obtained by numerical integration of Eq. ( 23).

The forces on LEO satellites usually include the Earth's gravitational attraction, third-body attractions from the Sun and the Moon, atmospheric drag, solar radiation pressure and thruster forces. In the case of GOCE, the nongravitational forces in the flight direction were continuously compensated by electric propulsion. The remaining perturbation acceleration is on the order of 10 -7 m/s 2 . In this study, a 70×70 subset of the Earth Gravitational Model 2008 (EGM2008) [START_REF] Pavlis | The development and evaluation of the earth gravitational model 2008 (EGM2008)[END_REF] is used to compute the acceleration due to the Earth's static gravity field. The tidal effects such as solid Earth tides, polar tides and ocean tides are not considered. The gravitational attractions of the Sun and Moon are modeled by using analytical series expansions of luni-solar coordinates [START_REF] Montenbruck | Satellite Orbits[END_REF].

The linearization of orbital equation requires the state transmission matrix, which refers to the partial derivative of the orbital state (position and velocity) at arbitrary time t with respect to the initial state. The state transmission matrix is obtained by integration of the following differential equation

        3 3 3 3 0 0 , , , , , , d t t t t t t dt                  0 I Φ Φ a r v a r v r v ( 24
)
where   0 , t t Φ denotes the state transmission matrix from 0 t to t. The initial value of the equation above is

  0 0 6 6 , t t   Φ I ( 25
)
The accuracy requirement for   0 , t t Φ is not as stringent as that for the trajectory integration. A simplification of the force model is used, where only the contribution of the Earth's gravitation up to degree 2 and order 0 is considered.

Gravity gradient observation model

The gravity gradients are second-order derivatives of the gravitational potential with respect to position and form a 3×3 matrix. The gravity gradients measured by GOCE are expressed in the GRF frame, whereas most of the Earth gravity models utilize the Earth-Centered Earth-Fixed (ECEF) frame. The GGT in GRF and the GGT in ECEF have the following relationship

T    V C Γ C (26) with , xx x y x z x x x y x z xy y y y z x y y y y z xz y z z z x z y z z z V V V V V V V V V                   Γ Γ Γ Γ ( 27
)
where V is the GGT in GRF, Γ is the GGT in ECEF, and C is the rotation matrix from ECEF to GRF. The coordinate transformation is implemented via the ECI frame. The star trackers provide accurate attitude information which can be used to compute the rotation matrix from ECI to GRF. The IERS models provide high-precision transformation from ECI and ECEF [START_REF] Petit | IERS Technical note no.36[END_REF].

The GGT is a symmetric matrix and contains 9 elements. As mentioned in Section 2, the symmetric property of GGT is used to isolate Euler accelerations. Thus the GOCE gradiometer outputs only 6 components at each epoch.

The measurement equation is usually given in a vector form. Rewrite V and Γ into column vectors

, xx x x yy yy zz z z xy x y xz x z yz yz V V V V V V                                       Γ Γ Γ Γ ( 28
)
where the array symbol denotes the vector form of GGT. The relationship between  V and  Γ is

    V Π Γ ( 29
)
                        (30)
where the coefficient matrix Π comprises elements from the rotation matrix C and ij c represents the ith row and jth column element in C.

The gravitational potential is expressed as a series of spherical harmonics

  0 0 n nm nm nm nm n m GM U V W C S R        (31) with   1 sin cos n nm nm R V P m r           (32)   1 sin sin n nm nm R W P m r           ( 33
)
where GM is the Earth's geocentric gravitational constant, R is the reference equatorial radius of the Earth, n and m are degree and order, nm C and nm S are normalized spherical harmonic coefficients, nm V and nm W are the associated normalized terms, nm P is the normalized Legendre function, and r, ϕ, and λ are the geocentric distance, latitude, and longitude of position in ECEF. GM, R, , nm C and nm S are constants and their values are provided in Earth gravity model files. The recursive computation of nm V and nm W are given in Montenbruck et al. [START_REF] Montenbruck | Satellite Orbits[END_REF].

The expression of the GGT in ECEF can be obtained by evaluating the second-order derivatives of U

2 2 0 0 n nm nm ij nm nm n m W GM C S R i ij V j                    Γ ( 34
)
where  

, , , , , ij xx yy zz xy xz yz

 represents the index of the gravity gradient components. In this study, a 120×120 subset of the EGM2008 gravity model is used to compute gravity gradients. The contributions of tidal effects are on the order of 0.1 mE and are thus not considered [START_REF] Bouman | Preprocessing of gravity gradients at the GOCE high-level processing facility[END_REF].

The partial derivatives of the GGT in ECEF with respect to position are calculated by T is also not stringent and only the Earth's gravitation up to degree 2 and order 0 is involved.

n nm nm ij k nm nm n m W GM T C S R ijk V i j k                     (

Measurement error model

As stated in Section 2, the GOCE gravity gradient measurements contain complicated errors. An appropriate error model is essential to the formulation of the measurement equation. The following is an analysis of the characteristics of the errors in Eq. ( 18), from perspectives of frequency spectrum and noise level.

First consider the ICM calibration-induced noise . The bias a B and the noise a N are determined by the characteristic of the accelerometers. According to Rummel et al. [START_REF] Rummel | GOCE gravitational gradiometry[END_REF], the accelerometers achieve high sensitivity only in frequencies between 5×10 -3 and 0.1 Hz. Inside the measurement band, the accelerometers show white noise behavior. Along the ultra-sensitive axes, the noise density level is about 10 -12 m/s 2 /√Hz, whereas along the less sensitive axes, the noise density level is about 3×10 -10 m/s 2 /√Hz [START_REF] Cesare | Performance requirements and budgets for the gradiometric mission, 4th edn[END_REF]. Below the measurement band, the noise is proportional to the inverse of the frequency, and shows a typical drift behavior in the time domain. As seen in Eqs. ( 20) and ( 21), a B and a N are linear functions of the accelerometer biases and noises. Therefore, the sum of the two errors can be modeled as a drifting bias plus white noise.

The characteristic of the noise  N is determined by the angular velocities and their estimation errors. Similar to the differential-mode accelerations, the angular velocities of GOCE are also nearly constant and contain periodical variations having a magnitude of about 10 -4 rad/s. Due to the spectral combination, the angular velocity errors show a f behavior in lower frequencies and show a 1/f 2 behavior in higher frequencies. The maximum value of the angular velocity errors is about 10 -6 rad/s [START_REF] Cesare | Performance requirements and budgets for the gradiometric mission, 4th edn[END_REF]. Therefore, according to Eq. ( 22), the periodical variations of  N are on the order of 0.1 E (10 -10 s -2 ).

By summing up all the errors above, the noise characteristic of the total observation error ∆V can be obtained.

Inside the measurement band, the noise shows white noise behavior. For the , , xx y y z z V V V and xz V components, the noise density levels are on the order of 10 mE√Hz, whereas for the xy V and yz V components, the noise density levels are much higher, 350 mE√Hz and 500 mE√Hz, respectively. Below the measurement band, the noise increases inversely with frequency and is superimposed by cyclic distortions (due to the variations of ICM N and  N ). Fig. 2 plots the measurement error of the yy V component as an example. Therefore, the observation error of each gravity gradient component in this study is modeled as a drifting bias plus small periodical variations as well as low-level white noise

    0 ij ij ij ij ij V b d n t t t v       ( 36
)
where ij denotes the index of the gravity gradient component, ij b is the bias at the reference epoch 0 t , ij d is the constant drift of the bias, ij n is the low-frequency noise, and ij v is the white noise. The bias ij b represents the sum of a B and the constant parts of ICM N and .

 N The drift ij d is due to the 1/f behavior of the accelerometer noises.

The noise ij n represents the sum of periodical variations of ICM N and  N and has a magnitude of about 0.1 E and ij v represents the white noise part of a N .

Let the vector z denote the gravity gradient measurements. The observation equation can be written as

    0 t tt             z V V Π Γ b d n v (37)
where b is the bias vector, d is the constant drift vector, n is the low-frequency noise, and v is the white noise.

Batch filter design

The orbital motion is close to linear over a small range, whereas the gravity gradient observations are highly nonlinear functions. A hybrid least squares (HLS) batch filter based on linearization of the orbital equation and unscented transformation of the measurement equation is developed to deal with different degrees of nonlinearity in the system. In addition, to restrict orbit linearization errors, the filter adaptively adjusts the time span of the measurement data arc at each iteration step. This section starts with an overview of the iterated least squares and the unscented least squares, and a detailed description of the adaptive HLS filter design is given thereafter.

The iterated least squares and the unscented least squares

The principle of least squares (LS) batch filter is to determine a set of states that minimizes the sum of the squares of measurement residuals. For nonlinear measurement equations, the iterated least squares (ILS) filter iteratively improves state estimation using the first-order partial derivatives of the linearized system [START_REF] Bar-Shalom | Estimation with applications to tracking and navigation: theory algorithms and software[END_REF]. For highly nonlinear functions, sigma-point transformation is introduced in Park et al. [START_REF] Park | Satellite orbit determination using a batch filter based on the unscented transformation[END_REF], and a non-recursive unscented least squares (ULS) filter is developed. To illustrate the methods of ILS and ULS, the measurement equation is redefined in the following general mathematical form

        1 1 1 2 2 2 n k n z w h z w h z w h                                               x x z h x w x ( 38
)
where x is the state vector, z is the stacked vector of measurements, h is the stacked measurement function, and w is the additive zero-mean measurement noise vector.

Assume the initial state estimate and its covariance as

ILS ILS 0 , 0 ˆ, , 0 j j j    xx x x P P ( 39
)
The ILS filter updates the estimate ILS ˆj x as follows

      1 ILS ILS 1 1 1 ILS 1 ILS 1 0 0 0 ˆˆˆˆT T j j j j j j j                     x x P H R H P x x H R z h x (40)
where

  ILS ˆj j     x x h x H x ( 41
)
is the measurement Jacobian matrix and R is the covariance matrix of the noise w. The convergence criterion for the iteration is usually given by

ILS ILS 1 ILS ˆĵ j j     x x x ( 42
)
where η is a predefined relative error tolerance. The covariance of the ILS final state estimate is

  1 ILS 1 1 0 ˆˆT j j      xx P P H R H (43)
The ILS filter performs well for weakly nonlinear equations. For highly nonlinear systems, however, the linear approximation will induce significant errors and make it difficult to achieve convergence. The ULS filter is an extension of the sequential unscented Kalman filter and deals with the nonlinearity problem using a set of selected sigma points. The mean and the covariance of the measurements are calculated and are used to correct the state estimate.

Assume the initial state estimate and its covariance as

ULS ULS 0 , 0 ˆ, , 0 j j j    xx x x P P ( 44
)
where j is the iteration number. The sigma points are selected as follows

        ULS 0, ULS ULS , , ULS ULS , , ˆˆ1, , ˆ1, , 2 j
j i j j j i i j j j i L i L L i L L               , , xx xx χ x χ x P χ x P ( 45
)
where L is the dimension of x, Each sigma point is propagated using the nonlinear measurement function

  2 L L       is a scaling parameter, α is a constant

 

, , , 0, ,2

i j i j i L    γ h χ (46)
The mean and the covariance of the measurement vector are calculated as follows

  2 , 0 L m j i i j i W    z γ (47)      2 , , , 0 ˆL T c j i i j j i j j i W       zz P γ z γ z R ( 48
)
and the cross-correlation matrix of x and z is

     2 ULS , , , 0 ˆL T c j i i j j i j j i W      xz P χ x γ z ( 49
)
where   m i W and   c i W are the weighting factors and are defined as follows

            0 2 0 1 1 , 1, ,2 2 m c m c i i W L W L W W i L L                    (50)
where β is the third scaling parameter and is used to incorporate prior knowledge of the distribution of x.

The ULS filter updates the estimate ULS ˆj x as follows

  ULS ULS 1 ˆĵ j j j     x x K z z (51) with 1 , , ˆĵ j j 
 xz zz K P P (52)
where j K is the optimal filter gain.

The convergence criterion can be set to be the same as that of the ILS. The covariance of the final estimate is given by

ULS 0 1 , 1 1 ˆˆˆT   xx zz P P K P K (53)
The covariance matrices ILS ˆxx P and ULS ˆxx P are consistent with the state estimation if and only if the measurement noise w is white and Gaussian. Otherwise, a fudge factor has to be added to guarantee the filter consistency

  1 ILS ILS 1 ˆT j j     xx P H R H (54)   ULS ULS 0 1 , 1 1 ˆˆˆT    xx zz P P K P K (55)

Adaptive hybrid least squares batch filter design

The batch orbit determination problem is to estimate unknown orbital elements from a set of measurements. The measurement function h actually consists of not only the observation equation but also the orbital equation. The adaptive HLS filter exploits the different degrees of nonlinearity in the two equations and autonomously adjusts the measurement time span to bound orbit linearization errors. In this study, the adaptive HLS filter is first implemented to estimate the initial position and velocity as well as the initial biases. An additional augmented state ILS filter is then carried out to obtain the bias drifts and to correct the adaptive HLS filter's results. The low-frequency noise n is not estimated and is included with v into the measurement noise vector w as follows

0 0 1 1 1 1 N N                    n v n v w n v ( 56
)
where N is the total measurement epochs. The standard deviations of the six components of the noise (n + v) are set to be 100 mE, 100 mE, 100 mE, 350 mE, 100 mE, and 500 mE, respectively.

The algorithm of the adaptive HLS for GOCE gravity gradient based orbit determination is proposed as follows.

The state vector HLS x comprises the 6-dimensional orbital state vector y (the initial position 0 r and the initial velocity 0 v ) and the 6-dimensional bias vector b. The initial state and its covariance are assumed as HLS HLS 0 , 0 ˆ, , 0

j j j    xx x x P P ( 57
)
where j is the iteration number. The orbital state y and the bias vector b and their covariances can be extracted as follows HLS HLS , , 1:6 1:6,1:6 ˆ, In the case of GOCE orbit determination, 6.

j j j j          

L 

The sigma points are propagated using the orbital integrator and the state transmission matrices to obtain the position at each measurement epoch   0, , 1

k j t k N          0, , 0, 1:3 , , 0, , 0 , 0, 1
:3,1:6 , , , 1 , , 2
k j j k i k j k j k i j j t t t i L               ψ f χ ψ ψ Φ χ χ (
A A A     A (64)
The sigma points of positions are further propagated to obtain the stacked gravity gradient measurements as follows

      ,0, ,1, , , , ˆ, 0, ,2 ĵ i j j i j j i j i N j j i L                           ψ b ψ b V ψ b V γ V ( 65
)
The mean and covariance of the gravity gradient measurements are

  2 , 0 L m j i i j i W    z γ (66)      2 , , , , 0 ˆL T c T j i i j j i j j j i W        zz b bb b P γ z γ z H P H R ( 67
)
where b H is the partial derivative matrix of the stacked vector of measurements z with respect to the bias vector b

  6 6 6 6 6 6 6 6 j N                   b I I H I (68)
and R is the covariance of the measurement noise vector w.

The cross-correlation matrix of y and z is The HLS filter updates the estimate HLS ˆj x and its covariance

     2 , , , 0 ˆL T c j i i j j i j j i W      yz P χ y γ z ( 69
ˆˆĵ j j j           bb xx b x P P 0, j χ 2 , L j χ 1, j χ          ,0, ,1, , , , ˆ, 0, ,2 ĵ i j j i j j i j i N j j i L                           ψ b ψ b V ψ b V γ V   , , max j N j p N N     ψψ P           0, , 0, 1:3
, , 0, , 0 , 0, 1:3,1:6 , , 0 , 0 1:3,1:6 1:6,1:3 , , 1, ,2 ; 0, , 1 ˆ, , After the adaptive HLS filter, an augmented state ILS filter is used to estimate the drifts and to improve the orbit determination results. The augmented state vector is defined as

k j j k i k j k j k i j j j T k j k j k t t t i Lk N t t t t                           ψψ yy ψ f χ ψ ψ Φ χ χ P Φ P Φ        2 , 0 2 , , , 0 , ˆˆL m j i i j i L T c j i i j j i j j i T j W W           zz b bb b z γ P γ z γ z H P H R      2 , , , 0 , , , ,
a            y x b d ( 75
                      d 0 I H I (77)
The state update and the final covariance are the same as Eq. ( 40) and Eq.(54), respectively.

Results

The orbit determination algorithm has been tested with real GOCE data. The Level 1b product EGG_NOM_1b which contains the raw GGT measurements (EGG_GGT dataset) and the gradiometer inertial attitude quaternions (EGG_IAQ dataset) are used as inputs. The precise orbit solutions provided in the Level 2 product SST_PSO_2 (reduced-dynamic orbits from GPS) are used to evaluate the accuracy of the GGT derived orbits. The test covers 12hour arc data starting 8 September 2013, 00:00:00.0 (GPS Time). The data are reported to have good quality and no special events (data anomaly or calibration) occurred. The data are resampled at intervals of 30s. Thus the gravity gradients accumulate a total number of 8640 measurements.

The initial position and velocity as well as the initial biases are estimated using the ILS filter, the ULS filter, and the adaptive HLS filter, respectively. The initial errors added in position and velocity are set to [10 4 (10 E) 2 , (10 E) 2 , (10 E) 2 , (10 E) 2 ]. The bias drifts are all assumed to be zero. The relative error tolerance for the convergence criterion is set to 10 -5 and the maximum iteration number is set to 10. The fudge factors of the covariance computation are all set to 10. For the ULS filter and the adaptive HLS filter, the three scaling parameters of the unscented transformation, i.e., α, β, and κ are set to 1, 2, and 0, respectively. The tolerance of the position uncertainty in the adaptive HLS filter is set to 5×10 4 m.

Figs. 4(a) and 4(b) show the histories of iterations and the variations of the position and velocity errors. The ULS filter and the adaptive HLS filter are successfully converged, whereas the ILS filter fails to converge due to the large initial errors. The adaptive HLS filter shows better convergence performance and better accuracy than the ULS filter since the system's nonlinearity are kept much lower by the adaptive adjustment of measurement time span and the iterative updating of the covariance matrix. Another advantage of the adaptive HLS filter is the short computation time. For each iteration loop, the execution time of the ULS filter is 8107 s and the execution time of the adaptive HLS filter is 2382 s. The reason is that the ULS filter propagates 13 (= 2L + 1) sigma points of orbits at each time, whereas the adaptive HLS filter propagates one orbit only.

Fig. 1 .

 1 Fig. 1. The arrangement of the six accelerometers inside GOCE gradiometer and the orientation of the GRF reference frame. The solid arrows at each accelerometer show the ultra-sensitive axes and the dashed arrows show the less sensitive axes.



   and z   are the angular accelerations. All the vectors and matrices are expressed in the GRF frame.

anM

  are true CM and DM accelerations, ij M is a scaling calibration matrix related to i M and , are CM and DM biases and noises, respectively. To obtain , the calibration matrix (ICM) is needed. The ICMs of the three baselines were directly determined from an in-flight satellite shaking procedure, which was achieved by the ion thruster and the gradiometer calibration device. Details of the calibration are found in[START_REF] Cesare | Gradiometer on-orbit calibration procedure analysis[END_REF], respectively. The CM and DM accelerations after ICM calibration will be , , ,



  are the angular velocity errors. The characteristics of these errors will be discussed in the next section.

  of the position components. The 18 partial derivatives form a 6×3 matrix called the gravity gradient Jacobian matrix. The accuracy requirement for the computation of , ij k

3 .

 3 GOCE data show that the differential-mode accelerations , d ij a are nearly constant and are disturbed by small periodical variations. The variations are concentrated near the orbit revolution frequency and their magnitudes are between 10 and 100 E. Therefore, the noise ICM N can be modeled as a constant bias plus small periodical variations (between 0.03 and 0.3 E).

Fig. 2 .

 2 Fig. 2. Measurement error of the GOCE yyV component.

  and is usually set to a small positive value, κ is a secondary scaling parameter which is usually set to 0 or 3-L, and  column of the matrix square root.

  refer to the elements of the vector and the matrix. The sigma points of the orbital state are selected by

  ) and the cross-correlation matrix of b and z is

Fig. 3 .

 3 Fig. 3. Flowchart of the adaptive hybrid least squares.

  m, 10 4 m, 10 4 m, 10 m/s, 10 m/s, 10 m/s] and the diagonal elements of the initial covariance of position and velocity are set to [(10 4 m) 2 , (10 4 m) 2 , (10 4 m) 2 , (10 m/s) 2 , (10 m/s) 2 , (10 m/s) 2 ]. The initial errors added in the 6 biases are set to [10 E, 10 E, 10 E, 10 E, 10 E, 10 E] and the diagonal elements of the initial covariance of the biases are set to [(10 E) 2 , (10 E) 2 ,

Fig. 4 .

 4 Fig. 4. Histories of iterations and the variations of the position and velocity errors of the ILS, the ULS, and the adaptive HLS.

Fig. 5 .

 5 Fig. 5. Evolution of the radial, along-track, and cross-track position and velocity errors of the adaptive HLS filter.

    V Γ Γ Γ Γ Γ Γ

      V Γ Γ Γ

Acknowledgements

This research was supported by the National Natural Science Foundation of China (No. 11002008). The European Space Agency is acknowledged for providing the GOCE data. The authors acknowledge the valuable remarks of the reviewers and the editor.

Table 1 lists the reference values of the biases and the drifts, which are determined from the differences between the actual measurements and the gravity gradients computed using the GPS-derived orbits and a 300×300 subset of the EGM2008 gravity model. The estimation errors of the 6 biases are -3.7 mE, 64.3 mE, 0.4 mE, -542 mE, -641.9 mE, and -314.0 mE, respectively. The poor observability of the xz b component results in the biases in the orbit estimation. The post-fit measurement residuals of the adaptive HLS filter are also investigated and are illustrated in Fig. 6. The obvious drifts in the residuals (yy, xy, and yz components) are due to the underfitting of the filter. The bias drifts are further estimated using the augmented state ILS filter. The initial values of the orbital states and the biases are set to the estimates obtained from the adaptive HLS filter. The initial covariance of these parameters is set to be the same as that for the adaptive HLS filter. The initial values of the drifts are all set to be zero. The diagonal elements of the initial covariance of the drifts are set to [(1 mE/h) 2 , (10 mE/h)

Conclusions

In this contribution, an orbit determination method using gravity gradient measurements has been described and the GOCE satellite has been used as a case study. Within the strategy, satellite orbits are estimated from a combined usage of the spaceborne gravity gradiometer and star trackers. Actually, the function of the star trackers is two-fold: to provide estimations of angular rates and angular accelerations for gravity gradients retrieval and to provide precise attitude quaternions in order to isolate the orientation contributions.

The orbit determination is implemented by an adaptive hybrid least squares batch filter. The performance of the algorithm is evaluated using real GOCE data, and a position accuracy of tens of meters has been achieved for the radial and cross-track position components. The large along-track position error is due to the poor observation of one of the measurement biases. The low-frequency noises are remained in the measurement residuals and need to be dealt with in the future study to further improve the accuracy. Nevertheless, the present work demonstrates the feasibility of orbit determination from gravity gradients containing drifting biases and provides an alternative autonomous navigation method for satellites in near-Earth orbits.


domain: info shs

text:
Mathieu Mangeot-Nagata
email: mathieu.mangeot@imag.fr

Construction of an open-source multilingual lexical system targeted on French and Japanese through contributive and automatic methods Research project in Japan

Introduction

This research is located in the natural language processing (NLP) domain, at the intersection of computer science and linguistics, more specifically on multilingual lexicography and lexicology.

In a first long stay in Japan from November 2001 to March 2004, we made the observation that the French-Japanese lexical resources available on the Web were almost nonexistent. Which gave birth to the Papillon project of building a multilingual lexical database with a pivot structure [START_REF] Sérasset | Papillon Lexical Database Project: Monolingual Dictionaries & Interlingual Links[END_REF]. Since then, progress has been made in several areas (technical, academic, social) [START_REF] Mangeot | Papillon project : Retrospective and perspectives[END_REF], but the production of real data has made very little progress. On the other hand, reuse of lexical resources is trendy (WSD, use of open-source resources like Wiktionary or Dbpedia, fusion with ontologies, etc.). Even if they can consolidate and expand the coverage of existing resources, these experiences always start from data created manually by lexicographers.

Based on this observation, we defined the following project which involves building a multilingual lexical system with focus on the French-Japanese language pair. Construction will be based on the reuse of existing resources (Franco-Japanese lexicons, Wiktionary) and automatic operations (reification of translation links, word senses disambiguation) and also on a community of volunteer contributors working on the Web. They will be asked to contribute either via serious lexical games, or directly on dictionary entries according to their level of expertise and knowledge in the field of lexicography or bilingual translation.

Resources generated will be royalty free and designed to be used both by humans via bilingual dictionaries and tools for automatic language processing (analysis, machine translation, etc.).

We will begin with a brief inventory of bilingual lexicography in general and focus on French-Japanese in particular. We then present recent advances in the field of construction of lexical resources online. Then, we describe in more detail the lexical system that we plan to build. We conclude with a description of the steps involved in this construction.

State of the art of bilingual lexicography

Lexicography is now at a turning point that began in the end of the twentieth century with the rise of the computer. Paper dictionaries are increasingly difficult to find an audience and their electronic versions are far from being commercial successes.

The main difficulty of the current bilingual lexicography lies in the prohibitive costs of building large amounts of data. For example, the Electronic Dictionary Research project [START_REF] Edr | EDR Electronic Dictionary Technical Guide[END_REF] whose aim was to build a Japanese-English dictionary took more than 1,200 person/years of work. The selling price of about € 84,000, is much lower than the actual costs of construction, costs that will probably never be recouped.

Anyway, these costs are too high for an individual. Therefore, only institutions can acquireit. On the other hand, the data provided at this price can only be used by some machine translation systems based on specific techniques.

In front of these unmanageable costs, publishers end up living on their laurels and offer mainly new editions of existing dictionaries. Few publishers have the courage to embark on the construction of a new high quality bilingual dictionary from scratch.

On the other hand, even in the most comprehensive dictionaries, there is almost always a lack of information especially on collocations. Scarce resources that take them into account do not do it systematically.

Despite the advent of the Internet, there are now few good quality lexical resources freely available online. Most are in fact bilingual lexicons made by volunteers non-specialist in lexicography.

Multilingual lexicography as such is in fact its infancy. Indeed, there is no real way to print a true "multilingual dictionary." On the contrary, it is quite possible to find multilingual terminology databases (as IATE) or small lexicons and multilingual phrasebooks.

State of the art of Japanese bilingual lexical resources

Although French and Japanese languages are considered well resourced in terms of tools and linguistic resources, the French-Japanese couple is considered an under-resourced language pair. There are indeed no royalty free high quality bilingual lexicons. French-Japanese aligned bilingual corpora and machine translation systems are logically also rare.

For historical as well as practical reasons, the Japanese society quickly put emphasis on English. The English-Japanese couple is one of the best equipped at present with very substantial resources like the EDR dictionary (1993) and machine translation systems are among the most efficient.

French-Japanese published dictionaries

Existing high quality Japanese-French dictionaries are published dictionaries that exist only on paper or in compiled electronic dictionaries (denshi-Jishou). There is no online interface for consultation.

French→Japanese dictionaries

Le Dico (Hakusuisha, 1993) contains 34,000 entries.

The Crown (Sanseido) contains 47,000 entries.

Figure 1: Screen shot of the Crown dictionary in electronic version

Japanese→French dictionaries

The Royal (Obunsha, 1992) contains 42,000 entries.

The Concise (Sanseido) contains 38,000 entries. Disadvantages: not free, these dictionaries are protected by copyright. They are not transferable and non-reusable in other projects. The latest edited versions date of twenty years ago. Furthermore, many of these dictionaries have been designed for Japanese speakers (Le Dico, The Royal, etc). Therefore, there is no latin transcription of Japanese words. Their reading requires mastery of kanji, which greatly reduces their usability.

Wiktionary projects

The French Wiktionary has currently 2.2 Mo of entries of which 1.2 Mo are French and a little less than 7,000 are Japanese translations of which half of them are a proper names (this is often a simple transcription of the headword in Japanese syllabary). There are also some translations included from the website dictionnaire-japonais.com (see below). Translations are specified in the entry level and not word sense level. There is no description of the translation context (gloss, examples), or information on the Japanese translations (part-of-speech, etc).

The Japanese Wiktionary has 83,000 entries of which 26,000 are Japanese and 2,800 are French entries translated into Japanese (there are inflected forms or verbal forms, eg 32 entries for the verb "aimer"). The coverage is very poor.

Wiktionary projects are interesting and trendy but they have several limitations:

• The entry structure is free. It is not possible to use the same microstructure for all entries.

• Although it is possible to describe in a Wiktionary of language A, a word sense of an entry in language B, the interface is not designed to prepare bilingual dictionaries. For example, the description of the reverse link language A → language B must be done by hand in the Wiktionary of language B.

• It is also not possible to automatically add existing data from other sources to construct a draft for later refinement.

• Contributions are anonymous. It is not possible to use a high quality data or reviewing / validation system.

Online Japanese bilingual resources

2.2.1 Japanese→English dictionary: JMdict

The JMdict1 (Japanese-Multilingual Dictionary) [START_REF] Breen | JMDict: a Japanese-multilingual dictionary[END_REF]) is a project led by Jim Breen. It contains 165,000 entries Japanese-English entries with some translations in other languages : German (from WaDokuJiten), 31,000 French equivalents (from dico FJ), Russian, etc.

Figure 3: entry 食べる of the JMdict dictionary

Advantages: resource coverage, royalty-free and available for free download. It is regularly revised and completed.

Disadvantages: unidirectional Japanese→English. There is no reverse dictionary English→ Japanese. The microstructure is limited: translation contexts are not described. It also lacks of a definition and examples.

Japanese→German dictionary: WaDokuJiten

The WaDokuJiTen2 from Ulrich Apel [START_REF] Apel | WaDokuJT -A Japanese-German Dictionary Database[END_REF] consists of more than 280,000 entries. Its coverage and its microstructure are more developed than the JMdict.

Figure 4: entry 食べる of the WaDokuJiTen dictionary

Advantages: JMdict the most complete in terms of coverage and information free of charge and available for free download.

Disadvantages: like JMdict, the dictionary is unidirectional. It does not use examples to illustrate the translation contexts. This dictionary is to date the most comprehensive Japanese-other language lexical resource available for free download. It is a goal for our resource in terms of coverage.

French-Japanese online resources

Projet Dico FJ

The dico FJ project, pioneer in the field, was launched in early 2000 by Jean-Marc Desperrier [START_REF] Desperrier | Analyze [sic] of the results of a collaborative project for the creation of a Japanese-French dictionary[END_REF]. It contains a little more than 10,000 entries from translation of Japanese-English JMdict dictionary. There has been no change since 2003.

Advantages: free of charge and available for free download.

Disadvantages: more thant the same disadvantages of JMdict, there are translation errors due to the fact that some contributors unfamiliar with the Japanese language directly translates English translations instead of Japanese headwords, which increases the number of misinterpretations.

dictionnaire-japonais.com

The dictionnaire-japonais.com 3 project contains a little over 28,000 entries. It is a significant improvement compared to other French-Japanese dictionary online projects. Each user can directly contribute by adding entries. The community of contributors seems quite active as evidenced by the project forum. Information available for each entry is relatively limited to a "grammatical type" a "category" (domain), a language level, and sometimes an "origin of the word" (etymology). Disadvantages: more than the FJ disadvantages , it seems that the a atesource files are not available for download.

Lexicons from machine translation systems

The UNL4 project defines a semantic pivot language called UNL. For each natural language N, it is then possible to program language N→ UNL enconverters and UNL → language N deconverters. French and Japanese partners of the project have each produced a lexicon for their conversion systems. These lexicons are available online5 .

The Japanese-UNL lexicon contains 1,266,694 entries. Here is the Japanese entry 食べる (to eat):

[食べる]{} "eat(icl>food)";

The French-UNL lexicon contains 520,305 entries. Here are the entries corresponding to the verb "manger":

[manger]{AUX(AVOIR),CAT(CATV),GP1(DE),VAL1(GN)}"consume(icl>event)";

[manger]{AUX(AVOIR),CAT(CATV),VAL1(GN)}"eat";

Advantages: these lexicons have a large coverage large and can be used to build a skeleton dictionary.

Disadvantages: the microstructure is very small: a lemma in French or Japanese and its equivalent in UNL, sometimes with information from systems such as part-of-speech (CAT). These lexicons are not available for free download.

Conclusion

The Japanese-French lexicons available online have a small size and are all oriented from Japanese to French.

Most French-Japanese dictionaries also lacks specific information concerning Japanese. There is no example to our knowledge of a dictionary with both kanji (ideograms), kana (syllabaries) and romaji (roman alphabet transcription). Dictionaries without romaji are designed for Japanese speakers. Dictionaries without kanji are designed for French beginners. It is also difficult to find some important information such as counters (objects are not counted the same way: one car = ichi dai =, one dog = ippiki, etc.) or language levels.

In conclusion, for personal use, it is possible to find good quality printed (or electronic) dictionaries printed if one knows how to read kanji, but when there is a need for a free online dictionary, there is often no other choice but to use an English-Japanese dictionary, which, as we know, can only increase errors of understanding and translation.

Advances in the construction of online resources

Our Ph.D. thesis [START_REF] Mangeot | Environnements centralisés et distribués pour lexicographes et lexicologues en contexte multilingue[END_REF] defines a number of theoretical foundations in the area. The first concrete steps have been made with the Papillon project: requirements definition, structures and types of desired information, implementation of the Jibiki platform for handling lexical resources online. Followed a reflection on how to contribute online indirectly through serious games with the JeuxDeMots project. Several resources construction projects were also conducted using the Jibiki platform but none was based on the French-Japanese language pair. The ongoing GDEF project [START_REF] Mangeot | Dictionary building with the Jibiki platform : the GDEF case[END_REF] aims to build a high quality Estonian-French dictionary for translators. The LexALP project [START_REF] Sérasset | Multilingual Legal Terminology on the Jibiki Platform: The LexALP Project[END_REF] has produced a multilingual terminology (German, French, Italian, Slovenian) with a pivot structure for the vocabulary of the Alpine Convention. The MotÀMot project [START_REF] Mangeot | MotÀMot project: building a multilingual lexical system via bilingual dictionaries[END_REF] focused on the recovery of an existing French-Khmer dictionary.

On contributing aspects: Wikipedia and Wiktionary projects

The Wikipedia online contributive encyclopedia met a big resounding success. Although the Wiktionary projects are developing, the success is not yet there for some languages (26,000 entries for Japanese) and they are rarely used to build bilingual dictionaries. The WiktionaryZ project, who claimed to ward off Wiktionary defects does not have the desired effect.

One hypothesis to explain this problem is the motivation. Indeed, when a person contributes to a Wikipedia article, it is rewarded with fame. It will then be recognized as an expert in her field. This is not possible with a dictionary. The contributions focus on small parts of highly targeted information and are therefore anonymous. On the other hand, there is a technical aspect related to the structure. An encyclopedia article is more or less free while a dictionary entry must follow a very precise microstructure (headword, grammatical information, semantic blocks, translation blocks, examples, etc.). It is therefore not possible to reuse a wiki platform to build a dictionary with a well defined structure.

Once accepted the idea that writing dictionary entries is not as much fun as working on a Wikipedia article, one must find ways to motivate a community of volunteers to contribute to a dictionary. L Serious lexical games are a first track. It is also possible to highlight the contributors, for example through a table of top contributors of the month. And finally, the social networks such as Facebook should also bring grist to the mill.

On technical aspects: the Jibiki platform

Jibiki [START_REF] Mangeot | Construction collaborative d'une base lexicale multilingue[END_REF]) is a generic platform for managing online lexical resources with users and groups handling, heterogeneous resources lookup and generic dictionaries entries editing. This is a community website originally developed for the Papillon project. The platform is programmed entirely in Java, based on the "Enhydra" framework. All data is stored in XML format in a relational database (Postgres). This website offers two main services: a unified interface for simultaneous access to many heterogeneous resources (monolingual, bilingual dictionaries; multilingual databases, etc.) and an editing interface in order to contribute directly to specific dictionaries available on the platform.

The editor is based on a HTML interface model instantiated with the entry to edit. The model can be automatically generated from a description of the structure of the entry with an XML schema. It can then be modified to improve the screen rendering. The only information required is the XML schema representing the structure of this entry. In addition, it is possible to edit any type of dictionary entry as soon as it is encoded in XML.

Several lexical resources construction projects used or still use this platform successfully. This is the case for example of the GDEF Estonian-French bilingual dictionary project6 . The code for this platform is open source and downloadable from the LIG Laboratory forge7 .

On the data collecting through serious games: the JeuxDeMots projects

JeuxDeMots8 [START_REF] Lafourcade | JeuxDeMots : un prototype ludique pour l'émergence de relations entre termes[END_REF] is an attempt to answer to the anonymous contributing problem. This project aims to build a lexical network rich and scalable, which can be compared to some degree to the famous WordNet base [START_REF] Miller | Introduction to WordNet : an on-line lexical database[END_REF]. The principle is as follows: one game requires 2 players. Player A starts a game. The type of relation to build (synonyms, opposites, domain, etc.) is displayed, and a word M randomly selected in a database of words. Player A then has a limited time to respond with proposals corresponding to, according to him, the word M and the type of game. This same word with the is then asynchronously offered to another player B, the process is identical.

The two half-games, the one of player A and of player B are not simultaneous but asynchronous. For each common response to proposals A and B, both players earn a number of points. The structure of the lexical network is based on the concepts of nodes and relations between nodes to build a network type of [START_REF] Polguère | Structural properties of lexical systems : Monolingual and multilingual perspectives[END_REF]. Each node consists of a lexical unit (word or phrase) involving all its lexical items and the relationships between nodes reflect the lexical functions. The first version of the game for French was launched in July 2007. There are also English, Arabic, Japanese, Vietnamese, Thai and more recently Portuguese versions [START_REF] Mangeot | A Serious Lexical Game for Building a Portuguese Lexical-Semantic Network[END_REF]. They are available on the Web.

4 Description of the lexical system to build

Microstructure of entries based on the Meaning-Text Theory

The microstructure of the entries gathered into monolingual volumes is a simplification of the Papillon project one. Each entry is based on the vocable. A word is either a group of lexies (a word sense), or an idiom.

The lexie consist of a name, grammatical properties, a semantic formula that can be seen as a formal definition -in the case of a predicative lexie, the formula describes the predicate and its arguments and the syntactic realization of the arguments -and a list of lexical semantic featuresthere are 56 basic lexical functions applicable to any language that can be combined together -a list of examples and finally to a list of idioms.

To cope with different contributor skill levels, the editing interface can adapt itself and display appropriate information. For example, a beginner contributor will be prompted for a simple gloss to characterize a lexie, while an expert linguist will describe a complete semantic formula. Similarly, some contributors only have access to the list of lexical functions to fulfill.

Pivot macrostructure pivot via bilingual interfaces

The macrostructure is derived from the Papillon project with a monolingual volume for each language and a center pivot volume. This macrostructure has been tested and validated in the LexAlp9 project [START_REF] Sérasset | Multilingual Legal Terminology on the Jibiki Platform: The LexALP Project[END_REF] that aimed to build a multilingual terminology for the vocabulary of the Alpine Convention. This project also uses the Jibiki platform [START_REF] Mangeot | Construction collaborative d'une base lexicale multilingue[END_REF] as well as for its development and online consultation.

When a new entry in a language A is added, it must be connected to interlingual volume. These links are created either by reusing existing bilingual dictionaries language A → language B, or by adding them manually from a translation. The link language A → language B then becomes language A → pivot → language B. If the entry of language B is already connected to another entry of C language, then the entry of language A also benefit from these links. However, in order not to confuse users, they contribute through an interface with a classic bilingual dictionary view. Each bilingual link language A → B language added via this interface will actually be translated in the background by the creation of two interlingual links as well as a pivot link representing the original translation link in order to obtain finally language A → pivot axie → language B. This idea has been used for the MotÀMot 10 project [START_REF] Mangeot | MotÀMot project: building a multilingual lexical system via bilingual dictionaries[END_REF] which led to the construction of a bilingual French-Khmer dictionary.

Establishing bilingual and interlingual links

If a contributor wants to add a translation link between a vocable in language A Va and a vocable of language B Vb, s/he can establish that link at different levels.

The ideal solution is to connect a word sense Sa vocable Va to another word sense Sb of vocable Vb. In this case, the link is bijective and Sb is also connected to Sa.

If the vocable Vb has not yet any specific word sense or if the contributor can not choose a word sense, s/he can connect directly Sa to Vb. In this case, a new word sense Sb' of the vocable Vb is created and tagged with a draft quality and the link as well as the word sense Sb' are marked for further refinement.

In the case of existing data recovery, it is often impossible to attach information to a specific word sense. In this case, at the end of the vocable Va is added an information to indicate that one of the word senses of vocable Va is connected to one of the word senses of vocable Vb, but this information will not be added to Vb. It will of course be marked for emergency refinement! Thanks to the pivot macrostructure, if two links language A → language B and language B → language C exist, then a link language A → language C will be automatically created and its level will be marked as draft and to be revised.

Data and contributors quality levels

Each piece of information for each entry is assigned a level of quality. The levels range from 1 star for a draft (recovered data whose quality is not known) to 5 stars, for an entry certified by an expert (eg, a translation link validated by a sworn translator).

Likewise, the contributors will be assigned a skill level (1 to 5 stars as well). 1 star is a beginner level unknown in the community and 5 stars is the level of a recognized expert.

Then, when a level 3 contributor revises a level 2 entry, the entry automatically rises to level 3. Similarly, if the work of a contributor is systematically validated without corrections by other contributors of a better level, s/he can automatically be switched to the next level after a certain threshold (eg 10 contributions).

To go further, we plan to analyze the work of contributors. If a person contributes massively eg on a particular domain, the system will automatically send her regular contribution proposals in its domain.

Working plan

Gathering existing resources

The first stage of the project is to collect existing resources that we are allowed to reuse (primarily

Figure 2 :

 2 Figure 2: Screen shot of the Concise dictionary in electronic version

Figure 5 :

 5 Figure 5: entry 食べる of the dictionnaire-japonais.com Advantages: available online, covering a little more vocabulary than the dico FJ dictionary,

http://www.csse.monash.edu.au/~jwb/jmdict.html

http://www.wadoku.de

http://www.undl.org/

http://www.undl.org/uwgate/

http://www.estfra.ee/

http://jibiki.ligforge-imag.fr/

http://jeuxdemots.org/

http://217.199.4.152:8080/termbank/LexALP.po

royalty free). These resources are lexical (dictionaries, lexicons) and corporal (aligned bilingual corpus or similar). The latter will be used for extracting bilingual examples.

Dictionaries available on the Internet

• JMdict dictionary with its 31,000 French equivalents;

• French and Japanese Wiktionary translations.

Resources of NLP projects

• Data of the ongoing sakura-survitra project with Kyoto university;

• French-UNL and Japanese-UNL lexicons from the UNL project with United Nations Universoty and Tokyo Soft company;

• Pre-terminological data from Mohammad Daoud's co-supervised with Pr. Kyo Kageura from University of Tokyo [START_REF] Daoud | Multilingual Lexical Network from the Archives of the Digital Silk Road[END_REF], [START_REF] Daoud | Utilisation de ressources non conventionnelles et de méthodes contributives pour combler le fossé terminologique entre les langues en développant des "préterminologies" multilingues[END_REF];

• Any data from other Japanese partners (Hosei University, National Institute of Informatics, Laboratory of Prof. Fuji Ren at University of Tokushima, Laboratory of Prof. Yoshinori Sagisaka at Hokkaido University, Laboratory of Prof. Yves Lepage at Waseda University, etc.).

Personal lexicons from translators and interpreters

In the absence of a reference dictionary with broad coverage or specialized terminology databases, it is common for translators and interpreters to build their own lexicons most often using Excel files. These resources could be reused in the project.

Production of a skeleton dictionary to be revised

Once collected, the resources must be converted into a single format and then automatically merged. Two entries with the same headword and the same part-of-speech are merged into a single entry with several word senses. The translation links must be reified: from a French article with a Japanese translation Fj, a Japanese article J and a pivot entry A connecting the two entries A are generated (Fj ⇒ F← A→ J).

The second step consists in merging the different word senses of an entry resulting from a previous fusion. Word sense disambiguation techniques [START_REF] Navigli | Word Sense Disambiguation: a Survey[END_REF] such as ant colony algorithms [START_REF] Schwab | Désambiguïsation lexicale par propagation de mesures sémantiques locales par algorithmes à colonies de fourmis. 18ème conférence sur le Traitement Automatique des Langues Naturelles[END_REF] can then be applied.

When the first version of the resource is ready, it is then put online to be corrected and enriched semi-automatically by other programs and also manually by a community of volunteer contributors.

Semi-automatic enrichment of the resource

We plan to enrich the resource with the JeuxDeMots data via two main methods:

1. Lexical elicitation: recent research has proved that "serious lexical games", such as JeuxDeMots, are quite useful to gather monolingual lexical resources. Promising research is currently done about ways to develop bilingual or multilingual games on top of JeuxDeMots games. While JeuxDeMots has already produced a large lexical network for French, an important objective of the project is to considerably expand the audience of the (existing) Japanese JeuxDeMots 11 .

2. Collocation discovery and translation: an important part of the needed resources is made of collocations, and in particular of multiword terms. Statistical processing and analogous processing have been successfully used to find such terms in specialized English corpora (e.g. in the Genia corpus), and to propose translations of them in other languages.

Animating a community of contributors around the project

A first conclusion of our experiments is that JeuxDeMots projects based on the work of voluntary contributors must always be animated constantly otherwise, they can be abandoned rapidly past the early stages of discovery. One of our main tasks will be to lead a community of contributors through project advertising, forums moderating, social networks, etc. It will also be necessary to recruit experts to supervise contributors and lead also the community. Tools can also be set up to reward contributors (best contributor of the month, etc.).

Finally, we plan to start again the task of scientific communication around the project by organizing a new Papillon seminar in Japan, giving presentations in partner laboratories and participating in conferences in the domain.