Prof Stoeber H Boris

N Fourati
email: hassen.fourati@univ-reims.fr

L Manamanni

Afilal

Y Handrich

The proposed approach combines a quaternion-based nonlinear filter with the Levenberg Marquardt Algorithm (LMA). The algorithm has a complementary structure design that exploits measurements from a three-axis accelerometer, a three-axis magnetometer, and a three-axis gyroscope. Attitude information is necessary to calculate the animal's DBA in order to evaluate its energy expenditure. Some numerical simulations illustrate the nonlinear filter performance. Some quantitative assessments prove this efficiency such as the time constant of the filter ( = 2 s) and the rms magnitude of the quaternion error (rms = 0 0156).

I. INTRODUCTION

N OWADAYS, the attitude estimation of a moving rigid body is involved in several fields. For example, in aerial and marine vehicles [START_REF] Mahony | Nonlinear complementary filters on the special orthogonal group[END_REF]- [START_REF] Wang | Design a mini-type marine attitude measurement system for self-propelled model trials[END_REF], the attitude is used to achieve a stable controller. In rehabilitation and biomedical engineering [START_REF] Zhu | A real-time articulated human motion tracking using tri-axis inertial/magnetic sensors package[END_REF], the orientation is necessary to record patient movements and providing adequate feedback for the therapist.

Recently, the problem of attitude and orientation tracking are treated in bio-logging. This scientific field involves both bioengineering and animal behavior and aims to obtain new information on the natural world, which provides new insights into the hidden lives of animal species [START_REF] Rutz | New frontiers in biologging science[END_REF], [START_REF] Ropert-Coudert | Diving into the world of biologging[END_REF]. Generally, bio-logging involves attaching an electronic device to a free-ranging animal in order to record different aspects of the animal's biology [START_REF] Halsey | Fine-scale analyses of diving energetics in king penguins Aptenodytes patagonicus: How behaviour affect costs of a foraging dive[END_REF] (e.g., 3-D movements, behavior, physiology, and energy expenditure) and environment.

The main goal of this paper is to suggest an option concerning a body motion estimation method [body attitude and dynamic body acceleration (DBA)]. The latter is suitably used for marine mammals or birds. Marine animals are particularly hard to study during their long foraging trips at sea. However, their need to return periodically to the breeding colony gives us the opportunity to measure these different parameters using bio-logger devices. Note that the use of inertial and magnetic sensors is relatively recent, due to the difficulty to develop miniaturized technologies adapted to high rate record sampling (over 12-50 Hz). The obvious advantage of this new approach is that we gain access to the third dimension space, which is the key to a good understanding of the diving strategies observed in these predators [START_REF] Elkaim | Marine Mammal Marker (MAMMARK) dead reckoning sensor for In-Situ environmental monitoring[END_REF], [START_REF] Johnson | A digital acoustic recording tag for measuring the response of wild marine mammals to sound[END_REF]. The principle, proposed in those papers for tracking the animal's posture estimates in the aforementioned studies, consists of using a three-axis accelerometer and a three-axis magnetometer, with the reductive assumption that all movements are static or quasi-static. This assumption made, we can consider that the accelerometer outputs correspond to the gravity measurements in body coordinates, and it therefore becomes possible to estimate the attitude by the resolution of well known Wahba's problem [START_REF] Elkaim | Marine Mammal Marker (MAMMARK) dead reckoning sensor for In-Situ environmental monitoring[END_REF]. However, the assumption made above is not valid in all dynamic situations observed underwater in freeranging animals, e.g., the phases of prey pursuit. Thus, the performance of attitude estimation is significantly degraded. Other work as [START_REF] Watanabe | A new technique for monitoring the behaviour of terrestrial animals; A case study with the domestic cat[END_REF] is focused on the use of a low-pass filter to extract the gravity projection from accelerometer outputs. In our knowledge, the tune of this filter is not easy, which introduces some errors on the attitude in many cases. In [START_REF] Wilson | Prying into the intimate details of animal lives: Use of a daily diary on animals[END_REF], the authors assumed that the running means over a one second interval of total acceleration time is a good estimator of the gravity vector projection. However, this approximation may not be valid over time and depends on other parameters such as animal species and the type of movement it makes.

In this paper, we propose the addition of a three-axis gyroscope measurement to the sensors already used (accelerom-eter and magnetometer) in order to overcome the drawbacks of aforementioned papers in this area [START_REF] Elkaim | Marine Mammal Marker (MAMMARK) dead reckoning sensor for In-Situ environmental monitoring[END_REF]- [START_REF] Wilson | Prying into the intimate details of animal lives: Use of a daily diary on animals[END_REF]. Indeed, the estimation of attitude using fusion technologies with low-cost sensors such as gyroscopes, accelerometers, and magnetometers has already been used in other fields. Micro Air Vehicles [START_REF] Mahony | Nonlinear complementary filters on the special orthogonal group[END_REF], [START_REF] Wang | Design a mini-type marine attitude measurement system for self-propelled model trials[END_REF] and biomedical engineering [START_REF] Zhu | A real-time articulated human motion tracking using tri-axis inertial/magnetic sensors package[END_REF] are examples using either Kalman filters (KFs) [START_REF] Zhou | A linear fusion algorithm for attitude determination using low cost MEMS-based sensors[END_REF], extended Kalman filters (EKFs) [START_REF] Sabatini | Quaternion-based extended Kalman filter for determining orientation by inertial and magnetic sensing[END_REF], or nonlinear observers [START_REF] Thienel | A coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise[END_REF]. It is important to note that the rigid body attitude estimation using MEMS inertial and magnetic sensors is useful for calculating the DBA of the moving body, although the proposed methodology based on the use of these sensors in not a new one. Nevertheless, we propose another way to combine these measurements and the developed approach still the first applied work, in our knowledge, in bio-logging.

The main idea of the algorithm is to use a complementary nonlinear filter coupled with a Levenberg Marquardt Algorithm (LMA) to process the measurements from a three-axis gyroscope, a three-axis magnetometer, and a three-axis accelerometer. The proposed approach combines a strap-down system, based on the time integral of the angular velocity, with the LMA that uses the earth's magnetic field and gravity vector to compensate the attitude predicted by the gyroscope. It is important to note that the resulting structure is complementary: high-bandwidth-rate gyro measurements are combined with low-bandwidth vector observations (gravity and earth's magnetic field) to provide an accurate attitude estimate. Furthermore, the estimated attitude is used to calculate three components of DBA, which provides important information about the energy budgets of free-living animals.

This paper is organized as follows. Section II describes the prototype design for the bio-logging application. Moreover, the attitude parameterization, the rigid body kinematic model, and sensor measurement models are presented. Section III details the structure of the proposed complementary nonlinear filter for the attitude and DBA estimation. Sections IV and V are devoted to simulation and experimental results to illustrate the effectiveness of the proposed algorithm. Finally, Section VI summarizes the main conclusions of the paper.

II. MATERIALS AND METHODS

The prototype "logger," the attitude representation, and sensor models are described here.

A. Bio-Logging Application: Case Study and Logger Concept

Bio-logging refers to the use of electronic devices (logger) attached to the animal. The King penguin is one of the major models of diving birds studied at Strasbourg University thanks to the bio-logging technology [START_REF] Halsey | Fine-scale analyses of diving energetics in king penguins Aptenodytes patagonicus: How behaviour affect costs of a foraging dive[END_REF]. A future generation of loggers should be equipped with a kinematic sensing unit (ADIS16405 from Analog Device [15]) that integrates a three-axis accelerometer, a three-axis magnetometer, and a three-axis gyroscope. It is a micro-fabricated device that provides digital signals ( g, s, Gauss, and 23 mm 23 mm 23 mm). The core of the hardware would be an ATMEGA 644-QFN ultra low-power microcontroller [16] including several on-ship peripherals. It should read digital signals from the ADIS16405 via SPI bus. Other kinds of sensors need to be implemented in the prototype, such as a pressure transducer (Keller-3L from KELLER [17]) and a temperature sensor (TC77 from Microchip [18]) to obtain information about the animal's environment. These sensors have to be periodically sampled with different frequencies, filtered, corrected from errors and temperature effects, and stored for postprocessing. The main storage of data is provided by a flash memory (Micro SD from San Disk [19]). A prototype would be protected from the effect of low temperature and high-pressure data by a special material called resin [20] and powered by a 3.6-V lithium battery. To maximize battery life time, the system should be automatically started for periodically moments of the day. With these unit and sensors, the power consumption of the prototype is estimated around 350 mW of which most is consumed by the sensors. Once the bird returns from a foraging trip at sea, the logger should be recovered, and all recorded measurements on the memory are downloaded and used into the nonlinear filter. Note that the loggers are autonomous and do not use a satellite or RF transmission.

Before deploying such a logger, the goal in this paper is to be able to convert the raw data (acceleration, angular rate, and earth's magnetic field) into relevant information: attitude and energy expenditure (DBA). The algorithms that will exploit the measurements from this prototype are the main concerns of this work. It is important to note that during preliminary experimental tests in the paper, equivalent devices available on the market are used to evaluate the performance of the attitude estimation algorithm. Then, Inertial Measurement Units were used, namely, the MTi and MTi-G from Xsens technologies [21]. These units integrate the main sensors for attitude estimation such as a three-axis accelerometer, a three-axis magnetometer, and a three-axis gyroscope.

B. Rigid Body Attitude Description

The rigid body attitude in space is determined when the axis orientation of the body frame is specified with respect to the navigation frame . The navigation frame is attached to the earth and tangent on its surface (N-E-D:

: Magnetic North, : East, : Down) [START_REF] Grewal | Global Positioning Systems, Inertial Navigation, and Integration[END_REF]. In this paper, we consider the unit quaternion as the mathematical representation of rigid body attitude between the mentioned frames. The unit quaternion, denoted by , is a hypercomplex number of rank 4 such that [START_REF] Mahony | Nonlinear complementary filters on the special orthogonal group[END_REF] where is the scalar part and is the vector part of the quaternion. We invite the reader to refer to the Appendix and [START_REF] Kuipers | Quaternion and Rotation Sequences[END_REF] for more details about quaternion algebra.

C. Sensor Measurement Models

The sensors configuration consists of a three-axis gyroscope, a three-axis accelerometer, and a three-axis magnetometer containing MEMS technologies [START_REF] Beeby | MEMS Mechanical Sensors[END_REF], [START_REF] Fourati | Rigid body motions estimation using inertial sensors: Bio-logging application[END_REF]. Their theoretical outputs can be written, respectively, as follows:

(2)

(3) (4)
where is the unknown gyro-bias vector and , , and are assumed white Gaussian noises. represents the real angular velocity, is the gravity vector, denotes the DBA, and describes the direction of the earth's magnetic field.

is the rotation matrix defined in (18) (see the Appendix).

III. FILTER DESIGN FOR ATTITUDE AND DBA ESTIMATION

In this paper, the objective is to design a rigid body attitude estimation algorithm based on inertial and magnetic MEMS sensors. The proposed approach will subsequently be used to track the orientation and DBA during several motions in free-ranging animals. This work is proposed in order to improve the attitude estimation approaches developed in [START_REF] Elkaim | Marine Mammal Marker (MAMMARK) dead reckoning sensor for In-Situ environmental monitoring[END_REF]- [START_REF] Wilson | Prying into the intimate details of animal lives: Use of a daily diary on animals[END_REF] and limited to the static and quasi-static cases of animal's movements. We propose an approach based on a nonlinear filter to gain the most accurate attitude estimation using the advantages of accelerometers, magnetometers and gyroscopes. It is important to note that the resulting approach structure is complementary: high-bandwidth-rate gyro measurements are combined with low-bandwidth vector observations to provide an accurate attitude estimate [START_REF] Brown | Introduction to Random Signal and Applied Kalman Filterin[END_REF].

A. Attitude Filter Description

To achieve our goal, (20) is developed to form the nonlinear system ( ) given by ( 5), shown at the bottom of the page, where is the state vector, is the identity matrix, and is defined in (17) (see the Appendix). The output of this system is built by stacking the accelerometer and magnetometer measurements such as [START_REF] Ropert-Coudert | Diving into the world of biologging[END_REF] In order to estimate the attitude, the following nonlinear filter is proposed [based on ( 5) and ( 6)]: [START_REF] Halsey | Fine-scale analyses of diving energetics in king penguins Aptenodytes patagonicus: How behaviour affect costs of a foraging dive[END_REF] where and denote the estimated states (attitude) and the filter gain, respectively. Note that is the modeling error which represents the difference between real measurements , as defined in ( 6), and estimated values , given by ( 8)

The components of are calculated using ( 21) and ( 22) (see the Appendix). The filter gain in ( 7) is used to correct the modeling error . This can be done if we are able to locate the minimum of the scalar squared error criterion function defined such as [START_REF] Johnson | A digital acoustic recording tag for measuring the response of wild marine mammals to sound[END_REF] In this paper, the LMA is used to minimize the nonlinear function [START_REF] Dennis | Numerical Methods for Unconstrained Optimization and Nonlinear Equations[END_REF]. Then, the unique minimum can be written in the following form [START_REF] Marquardt | An algorithm for the least-squares estimation of nonlinear parameters[END_REF], [START_REF] Fourati | A rigid body attitude estimation for Bio-logging application: A quaternion-based nonlinear filter approach[END_REF]: [START_REF] Watanabe | A new technique for monitoring the behaviour of terrestrial animals; A case study with the domestic cat[END_REF] where guarantees that the inverted term will be nonsingular.

is the Jacobian matrix defined in [START_REF] Kuipers | Quaternion and Rotation Sequences[END_REF] (see the Appendix).

B. Nonlinear Filter Performance Analysis

The scheme of the attitude estimation algorithm is summarized in Fig. 1. A frequency analysis of accelerometer signals [see [START_REF] Wang | Design a mini-type marine attitude measurement system for self-propelled model trials[END_REF]] shows that gravity components ( ) tend to lie towards the low end of the frequency spectrum while the DBA ( ) has higher frequencies components [START_REF] Brown | Introduction to Random Signal and Applied Kalman Filterin[END_REF]. The frequency analysis of magnetometer measurements shows also that the magnetic field tends to lie towards the low end of the frequency spectrum [see [START_REF] Zhu | A real-time articulated human motion tracking using tri-axis inertial/magnetic sensors package[END_REF]] [START_REF] Brown | Introduction to Random Signal and Applied Kalman Filterin[END_REF]. The gyroscope signals analysis shows that the drift , which is a slow-moving process [see [START_REF] Zhu | A small low-cost hybrid orientation system and its error analysis[END_REF]], tends to lie towards [START_REF] Rutz | New frontiers in biologging science[END_REF] the low end of the spectrum, and the angular velocity tends to have higher frequency components [START_REF] Brown | Introduction to Random Signal and Applied Kalman Filterin[END_REF]. Therefore, signals coming from the accelerometer-magnetometer pair and signals from the gyroscope have a complementary frequency spectrum [START_REF] Brown | Introduction to Random Signal and Applied Kalman Filterin[END_REF]. The resulting structure of the nonlinear filter is complementary: it blends the low-frequency region (low bandwidth) of the accelerometer and magnetometer data, where the attitude is typically more accurate, with the high-frequency region (high bandwidth) of the gyroscope data, where the integration of the angular velocity yields better attitude estimates. Indeed, the filtering approach can perform a low-pass filtering on the signals from the accelerometer-magnetometer pair and a high-pass filtering on the signals from the rate gyroscope. By filtering the high-frequency components of the signals from the accelerometer (DBA) and the low-frequency components of the gyroscope signal (slow-moving drift), the nonlinear filter produces an accurate estimate of the attitude. More details on the filter performance are given in the Appendix.

IV. SIMULATION RESULTS

This section aims to illustrate the performance and accuracy of the designed complementary nonlinear filter. Some numerical simulations were carried out under MATLAB to estimate a rigid body attitude based on theoretical inertial and magnetic measurements. We consider an attitude variation example taken from angular velocity data over 50 s. The following angular rate values are simulated: for s for s s (11) Then, the kinematic equation ( 5) is solved to obtain the continuous time motion in quaternion representation using the angular velocity measurements in [START_REF] Wilson | Prying into the intimate details of animal lives: Use of a daily diary on animals[END_REF]. The obtained quaternion is used as a reference to compare it with the estimated quaternion from the proposed nonlinear filter. Moreover, the specific force and magnetic field measurements are created using (3) and ( 4). The rotation matrix in (18) (see the Appendix) is computed using the theoretical quaternion (reference). To represent the sensor imperfections, an additive random zero-mean white Gaussian noise and disturbance was considered for all measurements, with a large standard deviation (see Table I). The angular velocity measurements are also assumed to be corrupted by theoretical bias given by rad s [START_REF] Zhou | A linear fusion algorithm for attitude determination using low cost MEMS-based sensors[END_REF] Note that the bias is very common and is an undesirable characteristic of low-cost rate gyros. The sampling rate was chosen as 100 Hz for all measurements.

The theoretical components of the quaternion (reference motion) as well as those of the nonlinear filter are initialized with different random values. These conditions are summarized in Table II. Notice that this choice allows us to illustrate the convergence of the filter even though it was initialized far from the actual states. At each integration step, ( 5) is resolved to obtain the theoretical quaternion . Thereafter, accelerometer and magne- tometer measurements are used in the LMA to calculate the correction term (see [START_REF] Beeby | MEMS Mechanical Sensors[END_REF] and ( 25) in the Appendix). Finally, ( 7) is solved to obtain the estimated quaternion . The filter gain , from [START_REF] Watanabe | A new technique for monitoring the behaviour of terrestrial animals; A case study with the domestic cat[END_REF], guarantees convergent estimates and is set according to the considered sensor noise levels as . The constant in ( 10) is chosen as . Notice that the integration loop of the algorithm has the length . Fig. 2 shows the flowchart representation of the estimation algorithm. It gives a step-by-step solution to the given problem (attitude estimation) using a simulation experiment.

The time histories of the quaternion (reference) as well as filter estimates are shown in Fig. 3. Despite the fact that the nonlinear filter and theoretical model of quaternion were initialized with different initial conditions, one can note that the estimated quaternion rapidly converges towards the theoretical values. The filter converges quite fast with a time constant . It is important to note that the rate gyro bias and noises in the accelerometer and magnetometer measurements are rejected by the filter. In order to evaluate the overall performance of the attitude estimation, we plotted the time history evolution of the quaternion estimation error. This error is computed using (19) (see the Appendix). Fig. 4 depicts the convergence of this error towards zero during the simulated motion. Moreover, the filter performance is shown quantitatively using the root mean square error (rmse) in [START_REF] Brodie | Dynamic accuracy of inertial measurement units during simple pendulum motion[END_REF] (see the Appendix). So, the orientation estimates are accurate within 0.0156 RMS on the estimation error (19). In conclusion, the obtained results show that the complementary nonlinear filter is an efficient method to improve the quality of sensor measurements and attitude estimation.

V. EXPERIMENTAL RESULTS AND DISCUSSION

A. Experimental Setup

In order to evaluate the efficiency of the proposed approach in real-world applications, an experimental setup was developed. Two inertial measurement units were used, namely the MTi and MTi-G from Xsens Technologies [21], which output data at a rate of 100 Hz. The Xsens devices are miniature, lightweight, and 3-D digital output sensors. They output a 3-D acceleration from accelerometer, a 3-D angular rate from gyroscope, and a 3-D magnetic field data from magnetometer with a built-in bias, sensitivity, and temperature compensation. Note that the MTi-G is a GPS-enhanced attitude and heading reference system (AHRS). The Xsens MT devices can track the attitude in Euler angles, quaternion, and rotation matrix representations. Table III summarizes the sensor specifications of the Xsens MT devices (see the Appendix). It is important to note that MT devices serve as tools for the evaluation of the nonlinear filter efficiency and cannot be suitable for use in free-ranging animals due to their dependences on an energy source as well as their heavy weights. In the following set of experiments, the calibrated data from the MTi and MTi-G are used as inputs to the proposed filter.

B. Attitude Estimation by the MTi and MTi-G

The orientation determined by the MTi is computed using a KF. It uses the gravity and earth's magnetic field in order to compensate the slowly increasing drift from the time integral of angular velocity. The KF uses the assumption that on average the acceleration due to the movement (linear acceleration) is zero [21]. Then, during the time of this assumption, the gravity direction can be estimated and used to stabilize the attitude obtained from the gyroscope (roll and pitch angles). The heading angle is stabilized by using the local magnetic field from the magnetometer.

The attitude given by the MTi-G is computed using an EKF. In the prediction step, the gyroscope data are integrated over time to obtain an orientation estimates (by a dead-reckoning technique). The accelerometer calculates attitude estimations by using the gravity vector projection. So, it leads to improve the orientation estimates from the gyroscope. Due to small errors in the gyroscope and transient accelerations (linear accelerations coming from the movement), the attitude estimates suffer from some drifts. In the correction step, these drifts are corrected by removing transient accelerations (using a mathematical derivation of velocity measurements coming from GPS receiver) and then the estimation of gravity vector projection is enhanced [21]. The heading is estimated based on GPS and earth's magnetic field measurements.

C. First Experiment: Improvement of the MTi Shortcomings

In this first experiment, the performance of the nonlinear filter in estimating the 3-D orientation during a simple motion is compared to the accuracy of the Xsens MT devices. Before starting the motion, the MTi and MTi-G are attached to a plate using an adhesive tape and their frame axes are aligned (see Fig. 5). The experiments were chosen to cover a wide part of the 3-D motions. The subject was asked to perform the four motions outlined in Fig. 6. Thus, the whole system is manipulated to move with up-and-down moving acceleration as follows. [Fig. 6(d)]. The nonlinear filter is fed with calibrated data from the MTi-G to estimate the quaternion describing the orientation in each motion. We chose to express the orientation in Euler angles representation because it is more intuitive than quaternion for the reader. The mathematical transformation between the quaternion and Euler angles is given in [START_REF] Phillips | Review of attitude representations used for aircraft kinematics[END_REF]. Fig. 7 shows the time history evolution of the Euler angles (roll, pitch and yaw) calculated by the MTi, MTi-G and nonlinear filter. It is important to note that during the intervals (17-34 s), (37-42 s), and (76-95 s), the motion is performed with high linear acceleration (DBA). One of these intervals (76-95 s) is zoomed in Fig. 8 for more clarity. This figure shows the effectiveness of the nonlinear filter in estimating the 3-D orientation during the motion since the differences between the MTi-G and nonlinear filter outputs are small (about 2 on roll, 2 on pitch and 3 on yaw). In conclusion, this experiment shows the drawbacks of the MTi Kalman Filter in estimating orientation and confirms data already published about its shortcoming [START_REF] Brodie | Dynamic accuracy of inertial measurement units during simple pendulum motion[END_REF].

D. Second Experiment: Human Segment Motion Evaluation

An experimental trial was designed to evaluate the body motion of one subject. The MTi-G is attached to the hand using an adhesive tape (see Fig. 9). Some efforts are carried out to align the axes frame of the MTi-G with those of the hand to represent the ongoing orientation of the subject. To show the nonlinear filter effectiveness, the experiments were chosen to cover a wide part of 3D hand motion. Then, the subject was asked to perform 3) Clockwise and anti-clockwise rotation around the dashed line axis defined along the forearm segment [Fig. 10(c)]. While the number of subjects used was limited to one, the range of movement over which the technique is evaluated is large and comprehensive. The quaternion errors are shown in Fig. 11 to provide an overview of the overall performance of the nonlinear filter. These errors are computed as the difference between quaternion estimates produced by the nonlinear filter and the MTi-G. The performance level consistency of the complementary filter may be observed in this figure, even in dynamic situations. These estimation errors are very small and the convergence rate is very fast (around 2 s). This performance is also proved by using the rmse criterion. So, the orientation estimates of are accurate within 0.008 rms on , 0.0065 rms on , 0.009 rms on , and 0.0104 rms on . It is important to note although our approach did not exploit GPS data, it is able to reconstruct the orientation of the hand (given by the MTi-G) with a small error.

E. Third Experiment: Free Movement of Animal

In this set of experiments, the accuracy of the nonlinear filter is evaluated during the free motion of a domestic animal [START_REF] Fourati | A rigid body attitude estimation for Bio-logging application: A quaternion-based nonlinear filter approach[END_REF]. Then we chose to carry out tests on a horse. Before starting experiments, the MTi-G is attached to the head of the horse [see Fig. 12(a)]. The choice of this body limb stems from our biologist colleagues' need to evaluate the head motions and the DBA of the horse during its course. So, this information helps them to study its behavior during the high level competitions.

During the movement, the circular course of the horse (see Fig. 12(b)) is composed of natural gaits such as a walk, trot, and gallop. The experiment is started by a walk where the horse moves its head and neck in a slight up and down motion. During the trot, the horse made small balancing motions with its head. The movement of the horse is finished by a gallop. In this gait, the horse moves its head faster in up/down and right/left motions. The three gaits and head motions of the horse are depicted in Fig. 13. Fig. 14 plots the evolution of the difference between quaternion estimates produced by the nonlinear filter and the MTi-G. This figure illustrates the performance of the complementary nonlinear filter in estimating the orientation of the horse's head even in dynamic situations (gallop). The orientation estimates of the quaternion are accurate within 0.0201 rms on , 0.0074 rms on , 0.0062 rms on , and 0.0203 rms on . Fig. 15 shows the time history evolution of the Euler angles calculated by the nonlinear filter and the MTi-G, plotted on different scales. The Euler angles estimation errors are shown in Fig. 16 to provide an overview of the overall performance of the nonlinear filter. These errors are computed as the difference between Euler angles estimates produced by the nonlinear filter and the MTi-G. Note that this difference is always smaller than 3 on roll, 3 on pitch and 5 on yaw, even during fast motions. In static and quasi-static situations (walk and trot), the difference is smaller than 2 . The rms differences between filter and MTi-G outputs during the free movement experiments are within 0.392 rms on roll, 0.577 rms on pitch, and 2.521 rms on yaw. This mismatch is still acceptable in our application and allows implementing the filter in the final prototype.

In our opinion, the differences between the quaternion outputs of the MTi-G Kalman filter and the nonlinear filter are due to different reasons. First, the MTi-G technical documentation [21] indicates that the orientation errors committed by the Xsens Kalman filter in dynamic situations are as follows: pitch/roll: 1 and yaw: 3 . Second, the internal algorithm of the MTi-G uses GPS measurements such as the velocity to calculate transient accelerations (moving accelerations). This information is used to correct the attitude estimated by the gyroscope. The GPS signal is not always good during the motion, because of the lack of satellites in view, presence of obstructions to line-of-sight to the satellites and atmospheric conditions. Then the transient acceleration calculation is degraded and consequently the precision of the KF is less accurate for short time.

To represent more accurately the ongoing orientation of the subject (the head of the horse or body limb), some efforts are made to align the three axes of the body frame (the subject) with respect to the sensor frame so that the estimated orientation by the filter or by the MTi-G represents the real attitude of the subject. To be sure that the algorithm represents the ongoing orientation of the subject, a manual step can be done after the attitude estimation of by the filter or the MTi-G. This step is given in the Appendix.

The DBA performed by the head of the horse can be determined using the following equation: [START_REF] Sabatini | Quaternion-based extended Kalman filter for determining orientation by inertial and magnetic sensing[END_REF] where is the rotation matrix defined in (18), is the gravity vector, and is the accelerometer readings.

Fig. 17 plots the evolution of the difference between the DBA estimates produced by the nonlinear filter and the MTi-G. The performance level consistency of the complementary filter may be observed in this figure since the difference is very small. Indeed, the corresponding rms difference is within 0.18 m s on , 0.19 m s on , and 0.03 m s on . Such information on the DBA will be necessary in the future to assess energy expenditure of the studied animal species.

VI. CONCLUSION

This paper proposes a quaternion-based nonlinear filter for the rigid body motion tracking. The filter was designed with the goal of being able to produce highly accurate orientation and DBA estimates over wide motion range in free-ranging animals (bio-logging). We have proposed a state estimation algorithm that adds to the data from a three-axis accelerometer and a three-axis magnetometer the one provided by a three-axis gyroscope. The main idea is to combine a strap-down system, based on the time integral of angular velocity, with the LMA. The estimated attitude is then exploited to calculate the DBA of the moving body (the animal in our case). The performance of the filter is demonstrated by some simulations and quantitative assessments. Moreover, the efficiency of the proposed approach is shown with a set of experiments on a domestic animal through sensor measurements provided by an Inertial Measurement Unit (MTi-G). The experimental results point out that the algorithm is able to track the accurate orientation and DBA under several motions. Indeed, the rms difference on DBA is within 0.18 m s on , 0.19 m s on , and 0.03 m s on .

APPENDIX

Quaternion: The unit quaternion, denoted by , is expressed as [START_REF] Thienel | A coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise[END_REF] where represents the imaginary vector, is the scalar element, and can be written such as (15) The quaternion product of and is defined such as (16) where is the identity matrix and the operator represents the standard vector cross product which is defined as (17) The rotation matrix in term of quaternion can be written as (18)

The estimation error on the quaternion is denoted by Rigid Body Kinematic Model: The kinematic differential equation, in terms of the unit quaternion, that describes the re-lationship between the rigid body attitude variation and the angular velocity in the body frame is represented by [START_REF] Shuster | A survey of attitude representations[END_REF] (20) where is the equivalent update quaternion relative to the angular velocity of the rigid body measured in and relative to . denotes the quaternion multiplication (16).

Sensor Measurement Notations: We give some notations of the sensor measurements.

Real angular velocity.

Linear acceleration (DBA).

Gravity vector.

Earth's magnetic field .

The theoretical model of the magnetic field nearest to reality is given in [33] and considers a magnetic field vector with a dip angle and a norm vector Gauss, so that: . Usually, the magnetometer error is described by taking into account the effects of external magnetic interferences. These errors can be detected and minimized using an automatic procedure of in-line calibration [START_REF] Roetenberg | Compensation of magnetic disturbances improves inertial and magnetic sensing of human body segment orientation[END_REF].

Bias in the gyroscope which is function of environmental conditions.

Levenberg Marquardt Algorithm

where is the quaternion representation of the gravity vector and is the quaternion representation of the earth's magnetic field.

The Jacobian matrix is defined in [START_REF] Fourati | Nonlinear attitude estimation based on fusion of inertial and magnetic sensors: Bio-logging application[END_REF] as [START_REF] Kuipers | Quaternion and Rotation Sequences[END_REF] The gain factor in ( 10) is used to tune the balance between measurement noises suppression and response time of the filter. Generally, it combines low bandwidth accelerometer/ magnetometer readings with high bandwidth gyroscope measurements. Notice that the nonlinear filter has a better convergence when is chosen somewhere between 0.1 and 1 [START_REF] Mahony | Nonlinear complementary filters on the special orthogonal group[END_REF].

The gain takes the following form:

(24) The correction term is computed using the gain such as [START_REF] Fourati | Rigid body motions estimation using inertial sensors: Bio-logging application[END_REF] Complementary Filter Performance: Complementary filtering is a theory that is based upon the use and availability of multiple independent noisy data of the same signal. This technique has been deeply rooted in many papers and books such as [START_REF] Brown | Introduction to Random Signal and Applied Kalman Filterin[END_REF], [START_REF] Wiener | Extrapolation, Interpolation and Smoothing of Stationary Time Series[END_REF]. An unknown signal can be estimated using corrupted measurements from one or more sensors whose information naturally stands in distinct and complementary frequency bands [START_REF] Higgins | A comparison of complementary and Kalman filtering[END_REF]. Complementary filtering explores the sensor redundancy to successfully reject measurements disturbances in complementary frequency regions without distorting the signal. If the measurements have complementary spectral characteristics, transfer functions may be chosen in such a way as to minimize estimation error. The general requirement is that one of the transfer functions complements the sum of the others. Thus for measurements of a signal [START_REF] Brown | Introduction to Random Signal and Applied Kalman Filterin[END_REF] where is the time integration Laplace operator.

The simplest complementary filter involves two noise contaminated measurements of a signal. This situation is shown in Fig. 18.

If is predominantly high-frequency noise and is low-frequency noise, the two noise sources have complementary spectral characteristics. If is chosen as a low-pass filter, then is a high-pass filter and both noise signals are attenuated. Since both high and low frequency data are utilized, the filter output will not suffer from any delay in dynamic response due to low-pass filtering.

Indeed, the complementary filtering theory is used in the paper. To study the performance of the nonlinear filter let us show the transform domain block diagram of the linearized quaternion estimation filter. This diagram (see Fig. 19) is obtained from Fig. 1.

The transform of the quaternion from accelerometer and magnetometer readings is , while is the quaternion obtained by integrating gyroscope signals.

is the Laplace transform of [see [START_REF] Rutz | New frontiers in biologging science[END_REF]]. In our case, the accelerometer is noisy and sensed both gravitational acceleration and DBA. The gyroscope suffers from bias. Therefore, and are both perturbed.

From Fig. 18, the filter transfer function based on accelerometer and magnetometer inputs is given by ( 27) Equation ( 27) has the form of a first-order low-pass filter. This proves that the LMA optimization is integrated into the quaternion estimation filter to help obtaining a low-pass filter on with respect to the output . So, the perturbations due to high-frequency components of accelerometer signal (DBA) are filtered from . Similarly, from Fig. 18, the filter transfer function based on gyroscope input is given by ( 28) Equation ( 28) has the form of first-order high-pass filter. The gyroscope measurements are high-pass filtered with respect to the output . So, the perturbations due to low frequency components of gyroscope signal (the biases) are filtered from . Finally, (26) can be verified as [START_REF] Fourati | A rigid body attitude estimation for Bio-logging application: A quaternion-based nonlinear filter approach[END_REF] This equation confirms the complementary aspect of the filter. The global transfer function of the filter is written as [START_REF] Phillips | Review of attitude representations used for aircraft kinematics[END_REF] One can conclude from (30) that the complementary filter blends low-pass filtering on the signals from the accelerometer-magnetometer pair, and high-pass filtering on the signals from rate gyroscope. RMSE: [START_REF] Brodie | Dynamic accuracy of inertial measurement units during simple pendulum motion[END_REF] where is the time interval and is the computed error. Arbitrary Alignment: Since the attitude is estimated by the nonlinear filter or the MTi-G, one can calculate the rotation matrix . After that, one considers the coordinate system of the subject (the body limb or the head of the horse) and the rotation matrix can be measured directly based on the difference between two frames and . Finally, the 3-D orientation that expresses the ongoing attitude of the subject is given by the rotation matrix . Some methods are presented in literature to calculate the rotation matrix that describes the orientation of the subject with respect to the sensor frame attached to it. For example, in the case of the horse, we can apply the method proposed in [START_REF] Johnson | A digital acoustic recording tag for measuring the response of wild marine mammals to sound[END_REF]. In the case of human limb motion, we can apply the method proposed in [START_REF] O'donovan | A inertial and magnetic sensor based technique for joint angle measurement[END_REF].

A

  Nonlinear Filtering Approach for the Attitude and Dynamic Body Acceleration Estimation Based on Inertial and Magnetic Sensors: Bio-Logging Application Hassen Fourati, Student Member, IEEE, Noureddine Manamanni, Lissan Afilal, and Yves Handrich Abstract-This paper addresses the problem of rigid body orientation and Dynamic Body Acceleration (DBA) estimation. This work is applied in bio-logging, an interdisciplinary research area at the intersection of animal behavior and bioengineering.

  Moreover, the effectiveness of the algorithm is experimentally demonstrated. In the experiments a domestic animal is equipped with an Inertial Measurement Unit (MTi-G), which provides a truth attitude for comparison with the complementary nonlinear filter. The rms difference between the filter and MTi-G outputs in the free movement experiments is within 0.392 rms on roll, 0.577 rms on pitch, and 2.521 rms on yaw. Index Terms-Attitude, bio-logging, complementary nonlinear filter, dynamic body acceleration (DBA), inertial measurement unit (MTi-G), MEMS inertial/magnetic sensors.

Fig. 1 .

 1 Fig. 1. Block diagram for the attitude estimation algorithm.

Fig. 2 .

 2 Fig. 2. Flowchart representing the estimation algorithm.

Fig. 3 .

 3 Fig. 3. Four components of the quaternion: Theoretical model (solid line) and estimated quaternion from the nonlinear (dashed line).

Fig. 4 .

 4 Fig.4. Estimation error E = k(q q) 0 1k.

Fig. 5 .

 5 Fig. 5. Whole system. The frame axes of the plate, MTi, and MTi-G are aligned.

Fig. 6 .

 6 Fig. 6. Motions performed during the evaluation test; the dashed arrow describes the direction of the motion.

1 )

 1 Move along the -axis of the plate in the level plane [Fig. 6(a)]. 2) Move along the -axis of the plate [Fig. 6(b)]. 3) Move along the -axis of the plate in the level plane [Fig. 6(c)]. 4) Move around -, -, and -axes of the plate (free motion)

Fig. 7 .

 7 Fig. 7. Attitude estimation with the MTi, MTi-G, and nonlinear filter.

Fig. 8 .

 8 Fig. 8. Zoom of the interval (76-95 s).

Fig. 9 .

 9 Fig. 9. Subject with the MTi-G attached to the hand.

Fig. 10 .

 10 Fig. 10. Exercises performed during the hand motion. (a) Clockwise and anticlockwise shoulder rotation. (b) Elbow extension. (c) Clockwise and anti-clockwise rotation around the dashed line axis defined along the forearm segment.

Fig. 11 .

 11 Fig. 11. Differences between quaternion estimates produced by the complementary nonlinear filter and the MTi-G.

Fig. 12 .

 12 Fig. 12. (a) MTi-G attached to the head of the horse. (b) Schematic diagram of how the horse performed its motion.

Fig. 13 .

 13 Fig. 13. Gaits of the horse and the movement performed by the head. (a) Walk. (b) Trot. (c) Gallop.

Fig. 14 .

 14 Fig. 14. Differences between quaternion estimates produced by the complementary nonlinear filter and the MTi-G.

Fig. 15 .

 15 Fig. 15. Euler angles estimates calculated by the nonlinear filter (dashed line) and the MTi-G (solid line).

Fig. 16 .

 16 Fig. 16. Differences between Euler angles calculated by the nonlinear filter and the MTi-G.

Fig. 17 .

 17 Fig. 17. Differences between the DBA calculated using the complementary nonlinear filter and the MTi-G.

  Details: The components of are calculated by (21)

Fig. 18 .

 18 Fig. 18. Complementary filter.

TABLE I CHARACTERISTICS

 I OF THE VARIOUS NOISES FOR SENSOR MEASUREMENTS

	TABLE II
	INITIAL CONDITIONS

ACKNOWLEDGMENT

The authors would like to thank the Ph.D. students of AUTO team (CReSTIC), especially Dr. T. Bouarar and D. Jabri for their help during the horse experiments. The authors would also like to thank the Centre Equestre Reims Université Champagne for allowing the use of the equipments during the experiments.

part by the Alsace and Champagne-Ardenne regions through Project NaviMeles.

Lissan Afilal received the "Matrise de Physique" degree and the Ph.D. degree in automatic control and computer science from the University of Reims Champagne-Ardenne (URCA), in 1984 and 1990, respectively.

He runs the theme of embedded systems in the Automatic Control and Hybrid System Research Group, CReSTIC Centre de Recherche en Sciences et Technologies de l'Information et de la Communication, Reims, France. His research interests include the automatic biomedical engineering, the control of complex systems, computer engineering and the design of embedded systems. His application areas range from the interoperability of systems on vehicles to bio-logging. In 1989, he became a Researcher with the Centre National de la Recherche Scientific (CNRS), Reims, France. Since 1989, he has been with CNRS (which became the Institut Pluridisciplinaire Hubert Curien (IPHC-CNRS) in 2007), Strasbourg, France. His research field is focused on the behavioral energetic and ecophysiology of wild animal living in extreme environmental conditions. He is a specialist of fasting, thermoregulation, and diving in marine birds. One key of his approach is the use of the bio-logging technology.

Yves Handrich was born in


Patricia Margaritte-Jeannin

Bernard Biot

Audrey Labalme

Jean-Claude Bernard

Joëlle Chastang

Behrouz Kassai

Marie- Helene Plays

Florina Moldovan

Francoise Clerget-Darpoux

P Edery

New disease gene location and high genetic heterogeneity in idiopathic scoliosis

Keywords: Idiopathic scoliosis, complex disorder, high density genome-wide linkage, genetic mapping, chromosomes 5q13-q14, 3q11.1-q13.2

Idiopathic scoliosis is a spine disorder of unknown origin with 1.5-3% prevalence in the general population. Besides the large multifactorial form sample of idiopathic scoliosis, there is a good evidence for the existence of a monogenic subgroup in which the disease is inherited in a dominant manner. However, results from literature suggest a strong heterogeneity in the locations of the mutated genes. Using a high resolution genome-wide scan, we performed linkage analyses in three large multigenerational idiopathic scoliosis families compatible with dominant inheritance including 9 to 12 affected members or obligate carriers. In two of these families, our results suggested intra-familial genetic heterogeneity whereas, in the other, we observed a perfect marker disease co-segregation in two regions at 3q12.1 and 5q13.3. We can state that one of these two locations is a novel idiopathic scoliosis disease gene locus, since the probability of having this perfect co-segregation twice by chance in the genome is very low (p=0.001). Lastly, in all three families studied, linkage to the previously mapped dominant idiopathic scoliosis loci on chromosomes 19p13.3, 17p11.2, 9q34, 17q25 and 18q is unlikely, confirming that there is a high genetic heterogeneity within the subgroup of dominant forms of idiopathic scoliosis.

Introduction

Idiopathic scoliosis (IS) is a spine deformity of unknown cause commonly defined by a rotation of vertebral bodies leading to a structurally fixed lateral curvature of the spine with a Cobb's angle of at least 10°. Idiopathic scoliosis affects up to 3% of the population [START_REF] Rogala | Scoliosis: incidence and natural history. A prospective epidemiological study[END_REF] , girls being more commonly affected than boys 2 3 . Adolescent idiopathic scoliosis (AIS, MIM 181800) occurs in otherwise healthy adolescents and accounts for approximately 85% of IS cases, other cases being infantile or juvenile (15%). About 3 per 1,000 children present with severe AIS forms over 30° which may require braces or surgical treatment and may have severe functional and aesthetic consequences.

Aside from idiopathic forms, scoliosis may also occur secondary to other hereditary disorders including defects in extracellular matrix components (e.g. Marfan syndrome, Ehlers-Danlos syndromes), neuromuscular disorders (e.g. Friedreich ataxia, muscular dystrophies), osteochondrodystrophies (e.g. osteogenesis imperfecta, neurofibromatosis), vertebral segmentation defects or other malformation syndromes including CHARGE syndrome and Smith-Magenis (del17p11.2) syndrome.

Several population studies established that up to 40% of IS are familial [START_REF] Bonaiti | Genetics of idiopathic scoliosis][END_REF] and that there is a high concordance rate of this disease in monozygotic twins [START_REF] Kesling | Scoliosis in twins. A meta-analysis of the literature and report of six cases[END_REF] . Many studies concluded that the most likely mode of inheritance was multifactorial, postulating that predisposing alleles were required together with environmental factors for phenotypic expression [START_REF] Riseborough | A genetic survey of idiopathic scoliosis in Boston, Massachusetts[END_REF] . Although the exact mode of inheritance remains unknown in many IS cases, there is now evidence for a monogenic subgroup. A segregation analysis using a model with age and gender effects established a significant contribution of a major causal gene transmitted according to an autosomal dominant inheritance with sex-dependent incomplete penetrances [START_REF] Axenovich | Segregation analysis of idiopathic scoliosis: demonstration of a major gene effect[END_REF] . A small number of model-dependent linkage analyses using an autosomal dominant model in large multigenerational IS families have been performed successfully to date. Indeed such large multiplex IS families compatible with dominant inheritance are very rare. A genome-wide scan of seven unrelated multiplex families of Chinese descent with AIS concluded that there was a linkage to chromosome 19p13.3 (IS1, MIM 181800) [START_REF] Chan | A genetic locus for adolescent idiopathic scoliosis linked to chromosome 19p13.3[END_REF] . This finding was replicated in several IS multiplex families in Alden et al's study [START_REF] Alden | Idiopathic scoliosis: identification of candidate regions on chromosome 19p13[END_REF] . A similar study in an Italian family mapped a second locus to 17p11.2 (IS2, MIM 607354) [START_REF] Salehi | Assignment of a locus for autosomal dominant idiopathic scoliosis (IS) to human chromosome 17p11[END_REF] . Another study reported on the mapping of two loci for IS on chromosome 9q31.2-q34.2 (IS4, MIM 612238) and 17q25.3qtel (IS5, MIM 612 239), the latter by combining information obtained from two unrelated IS families [START_REF] Ocaka | Assignment of two loci for autosomal dominant adolescent idiopathic scoliosis to chromosomes 9q31.2-q34.2 and 17q25.3-qtel[END_REF] . Recently, a locus for IS and pectus excavatum was found in a single large family on chromosome 18q [START_REF] Gurnett | Genetic linkage localizes an adolescent idiopathic scoliosis and pectus excavatum gene to chromosome 18 q[END_REF] .

In the present study, we performed model-dependent linkage analyses in three large IS families compatible with autosomal dominant inheritance of the disease trait with a high number of informative meioses. We compared our results to published linkage data.

Families and Methods

Patient and Family Inclusion

We selected large multiplex IS families and we genotyped the largest families, i.e. those including the highest number of affected individuals or obligate carriers. A positive diagnosis for IS was reached by combining clinical examination of the spine, including the forward bending test (Adams test), and the Cobb's angle measurement on radiographic evaluation.

Both a Cobb's angle above 15° (in order to minimize the risk of phenocopies) and vertebral rotation were required for the positive diagnosis of IS. Patients harbouring a spine curvature but no rotational component were assigned an "unknown status", as were those with any other associated anomaly of the spine or, more generally, with atypical IS. Patients with normal clinical examination and strictly normal spine on radiograph were considered "unaffected".

All individuals underwent clinical examination by at least one clinical geneticist (PE) and one orthopaedist (BB and/or JCB). Each radiograph was carefully and often repeatedly checked, by the clinical geneticist and one or both orthopaedists.

Pedigrees of the three selected IS families are shown in the figure. All families are threegeneration pedigrees suggestive of the transmission of an autosomal dominant mutation with incomplete penetrance. Clinical data on patients belonging to the families studied are provided in table 1.

DNA sample collection

We selected for genotyping 57 individuals including 27 affected patients or obligate carriers belonging to the three families presented here. Blood samples were collected from each participant (IS patients and their first-degree affected or unaffected relatives) after providing them with an information sheet describing the research study and obtaining their informed written consent. This research study was approved by the local Ethical Committee and was promoted by the Hospices Civils de Lyon, France. Genomic DNA was extracted from blood samples according to standard procedures. EBV-transformation of lymphocytes was performed in at least one affected individual per family. After the cell-lines were established, transformed lymphocytes were stored frozen in liquid nitrogen. Normal standard chromosomal studies on blood lymphocytes were obtained from at least one affected individual in each family ruling out gross chromosomal abnormalities.

Genotyping

Considering the large number of possible IS candidate genes with regard to their function and the wide genetic heterogeneity of IS, we decided to favour genome-wide approaches rather than candidate loci mapping strategies. We genotyped the three large multiplex IS families using a high density 1,000 microsatellite DNA marker set, with an average spacing of 3.7 cM between each marker, where genetic locations are based on the deCODE map [START_REF] Kong | A high-resolution recombination map of the human genome[END_REF] .

Statistical analyses

The classical method used for mapping a disease gene under monogenic transmission is the Morton Lod Score analysis (1955). A lod score value of 3.00 (for autosomal loci) means that the hypothesis of linkage between the disease trait locus and the marker locus is 1,000 times more likely than the hypothesis of no linkage. This method is very powerful when the disease is due to a mutation of the same gene in all families with full penetrance for mutation carrier.

In the present situation, it is realistic to assume that the disease is due to segregation of a dominant mutation in the families selected for this study. However, the results from literature suggest a strong heterogeneity in the mutated genes. In addition, penetrance values may differ according to the genes and mutations involved. Clerget-Darpoux et al [START_REF] Clerget-Darpoux | Power and robustness of the linkage homogeneity test in genetic analysis of common disorders[END_REF] showed that a misspecification of the penetrance values results in a lod score decrease and hence in a loss of the power to detect linkage.

To overcome the interfamilial genetic heterogeneity, we performed a lod score analysis separately in each family. Analyses were performed assuming the segregation of a dominant mutation, with and without the possibility of phenocopies. In linkage analysis, marker segregation was considered only in affected individuals. The status of other family members, including unaffected individuals, was assumed to be unknown. In such an analysis, the results do not depend on the absolute penetrance values, but only on the relative ones (probability to be affected for a non mutation carrier versus this probability for a mutation carrier).

In a first step, we measured in each family the lod score value that could be reached at the disease gene locus in the situation of no phenocopy and full informativity. This theoretical maximum lod score value Z imax can be simply inferred from the number Ti of disease allele transmissions to affected individuals in each family. If we denote Mi the number of meioses between a common ancestor and all affecteds, ni the number of common ancestors and Ti = Mi-ni, in each family Fi : Z imax = log 10 [ L( =0)/L( =0.5)] Ti = -Ti* log 10 (1/2) where L( =0) is the likelihood of H1 (hypothesis of linkage at =0, i.e. no recombination event between the marker and the disease locus) and L( =0.5) is the likelihood of H0 (hypothesis of no linkage).

In a second step, we estimated the probabilities under H0 to reach the maximum lod score on each arm of autosomes 0, 1 or more than once. This maximum occurs according to a Poisson distribution with a parameter equal to the probability of reaching Z imax if hypothesis H0 is true = (0.5) Ti . On the whole genome (44 chromosome arms), this parameter is equal to 44*(0.5) Ti .

In a third step, we computed maximum Morton lod scores using Merlin software [START_REF] Abecasis | Merlin--rapid analysis of dense genetic maps using sparse gene flow trees[END_REF] separately in each of the selected families.

In a final step, we tested in each of the three families studied the compatibility with linkage to the five loci previously considered to be IS loci (19p13.3, 17p11.2, 9q34, 17q25, 18q), using both the genetic model without phenocopy described here and a more flexible model allowing for various rates of phenocopies (0.01, 0.03, 0.10 and 0.020).

Results

Table 2 gives the values Mi, ni , Ti and the expected maximum lod score Z imax in each family Fi. It also gives the probabilities to reach it 0, 1 or more than once under the H0 hypothesis in whole genome analysis. Expected maximum lod scores are 1.51, 3.01 and 2.71 in families F1, F2 and F3 respectively. The probability of never reaching Z imax under H0, and thus only once (under H1), in whole genome analysis, is 25% for F1, 96% for F2, and 92% for F3. Table 3 gives, for each family F1, F2 and F3, the maximum lod score (max Z) and marker informativity at this location. In families F1 and F3, the maximum lod scores obtained after whole genome scan were significantly lower than the expected Zmax previously estimated.

Affected individuals never shared a same transmitted allele in any chromosome arm.

Consequently, the hypothesis that the disease is due to the transmission of a dominant mutation has to be reconsidered in these families. The variable severity in these two families suggests a possible intra familial genetic heterogeneity. In family F1, the severity of IS in affected individuals is highly variable since, in several patients of this family, Cobb's angles are between 16° and 24 ° and over 50° in four patients. Similarly, patient III1 of family F3 is affected with a severe form of IS (Cobb's angle of 65°), whereas angles for the five other patients are between 15° and 24°. When patient III1 of family F3 is removed from the linkage study, the lod score reaches the theoretical maximum value of 2.11 at the D15S979 locus.

In family F2, the lod score Zimax of 3.01 is reached both on chromosome 3 (3q12.1) and on chromosome 5 (5q13.3). As noted in table 2, the probability of having twice obtained this value randomly (under H0) is very small (p=0.001). It means that there is strong evidence that a disease gene lies in one of these two regions. The lengths of chromosome intervals shared by all patients at the two possible disease gene locations are 7cM and 12cM respectively (table 3). The boundaries of these intervals, ]D3S3690-D3S3045[ and ]D5S2851-D5S1397[ respectively, were deduced from recombination events in affected individuals as observed on haplotypes. To test whether the study of unaffected members of family F2 may help in the discrimination between these two locations, we performed a mod score analysis [START_REF] Clerget-Darpoux | A new method to test genetic models in HLA associated diseases: the MASC method[END_REF] with all genotyped family members (affected and non affected) using the software Genehunter-Modscore 2.0.1 [START_REF] Strauch | How to model a complex trait. 2. Analysis with two disease loci[END_REF] . The results from four unaffected individuals in this family, II7, II8, III6, III7 (see the figure) proved to be informative. The mod scores obtained at chromosome 3 and 5 loci respectively were identical and did not allow us to discriminate between the two possible locations.

Lastly, we considered lod scores obtained in families F1, F2 and F3 on the 5 regions claimed to be idiopathic scoliosis loci. All lod scores were negative when assuming no phenocopy and a maximum value of 0.00 was reached when the model allowed for phenocopies.

Discussion

Using a set of 1000 polymorphic markers (decode set), we performed a genome-wide linkage analysis under a dominant mode of transmission in three large multiplex IS families. Several pitfalls such as inter-familial genetic heterogeneity, possible phenocopies or unknown penetrance for mutation carrier were taken into account. First, lod score analyses were computed for each family independently. Second, in order to decrease the risk of phenocopies, individuals with a Cobb's angle of at least 15°, instead of the widely accepted threshold of 10°, were considered as affected. Indeed, mild scoliosis or even mild curvature and/or rotation of the spine are likely to be caused by a combination of environmental and minor gene effects rather than by a major gene effect [START_REF] Axenovich | Segregation analysis of idiopathic scoliosis: demonstration of a major gene effect[END_REF] . In addition, previous reports and our own observations showed that there are important variations in Cobb's angle measurements of up to 5° among different physicians [START_REF] Kuklo | Reliability analysis for manual adolescent idiopathic scoliosis measurements[END_REF][START_REF] Dang | Intra-observer reproducibility and interobserver reliability of the radiographic parameters in the Spinal Deformity Study Group's AIS Radiographic Measurement Manual[END_REF] .

This study shows the mapping of a new locus for dominant IS in a large family (F2) either on 5q13-q14 or on 3q11-q13. Indeed, in this family, the observed Zmax reached the expected Zmax value of 3.01, on both genomic locations. Since the probability of obtaining a Zmax value in the genome twice by chance is very low (p=0.001), we can conclude that there is strong evidence that a novel IS disease gene lies in one of these two locations. This IS gene still remains to be identified. Among the genes in the two chromosomal regions, many are involved in cellular signalling, some are coding for hypothetical/uncharacterized proteins yet to be discovered, and others were cloned but their function is not yet fully documented.

Mutation screening using either exome sequencing or targeted high throughput sequencing, gene expression studies, and in vivo functional analysis of candidate genes within these loci need to be performed.

In this study, we also provide further evidence for a high heterogeneity among monogenic forms of IS. In each of the three families studied, the lod scores are either negative or inconclusive, depending on the phenocopy rate assumption, at the five previously suggested locations for IS genes, namely 19p13.3, 17p11.2, 9q34, 17q25 and 18q. Such high genetic heterogeneity favours the view that computing IS families independently, when performing linkage analyses based on a monogenic model hypothesis, is critical to obtaining accurate results.

We also show that the selection of multiplex IS families diagnosed with strict clinical criteria does not warranty that the apparently dominant inheritance is simply explained by the segregation of a unique gene mutation. We have to consider the difficulty of selecting families with a high number of informative meioses without introducing intra-familial heterogeneity or phenocopies, to be able to detect linkage at a family level. It seems that the criteria of a minimum angle of 15° is not sufficient to ensure intra-familial genetic homogeneity, and we hypothesize that clinical variability between family members also has to be minimized. Indeed, in both families F1 and F3, the Zmax values obtained were far beyond the values expected under the assumption of a single causative mutation segregating in all affected members. In family F1, IS patients have a wide range of Cobb's angle values extending from 16° to 130°, and in family F3, one patient is affected with a severe form of IS (Cobb's angle of 65°), whereas angles for the five other patients are between 15° and 24°.

When the patient with the Cobb's angle of 65° from family F3 is removed from the analysis, we observe a perfect co-segregation of the disease and the D15S979 marker. Therefore, in such studies, we suggest paying careful attention to the clinical aspects of scoliosis, possibly including Cobb's angle values, even within single families. Such stratification by the severity of the phenotype or by other discrete clinical aspects of scoliosis may minimize the risk of intra-familial heterogeneity. The results provided in our study, supporting a high degree of inter-and intra-familial genetic heterogeneity in IS, raise questions regarding several previously reported mapping studies based on linkage data obtained by pooling together independent IS families. It should be noted that a posteriori selection of families compatible with linkage to a specific locus may lead to false primary linkage and/or fine mapping. Thus, in genetically heterogeneous disorders in which a different mutated gene in each family may be dealt with, pooling linkage data obtained from unrelated families should be considered with caution. Accurate linkage mapping studies are required to allow the subsequent identification of dominant IS genes. To date, no dominant IS gene has been identified, and we hope that the present study will prove to be an important step toward this goal. Also, although dominant IS genes, yet to be identified, may account for a restricted number of IS cases, they may provide insights into the genetic bases of multifactorial IS, which remain largely unknown [START_REF] Gao | CHD7 gene polymorphisms are associated with susceptibility to idiopathic scoliosis[END_REF] . Future genetic and functional analyses of candidate genes located within these defined regions will hopefully reveal gene defects accounting for IS, which will allow, in turn, the identification of secondary molecular factors that may point towards yet unrecognized IS pathophysiological pathways.

Titles to tables

Figure

  Figure Titles and Legends

Figure

  Figure Legend: Simplified pedigrees are drawn for families 1-3. All pedigrees are compatible

Table 1 .

 1 Clinical data for patients included in this study

Table 2 .

 2 Informativity studies performed in each family. Only affected individuals have been taken into account.

Table 3 .

 3 Maximum lod score values obtained in each family using only affected individuals.

	Table 1
	Family	Patient	Age at	Cobb's angle on	Cobb's angle	Apical	Therapy
			diagnosis	radiograph		vertebrae
			(years old)	(age: years old)
	F1	IV-2	13	76° (65)	right thoracic	D9	physiotherapy
		IV-3	adolescence	55° a (61)	right thoraco-lumbar	D12	spinal surgery
		IV-4	fortuitous	18° (50)	right lumbar	L2	none
		IV-6	12	35°/28°(64)	right thoracic/left lumbar b D8-D9/L2-L3	bracing
		IV-7	6	90°/93° (20)	right thoracic/left lumbar b	D7/L2	unknown
		IV-8	9	21° a (26)	right thoracic/left lumbar b	D8/L3	spinal surgery
		V-1	15	16° (15)	left lumbar	L3	none
	F2	II-1	fortuitous	27° (73)	right thoraco-lumbar	D12-L1	none
		II-3	fortuitous	19° (74)	right lumbar	L1	none
		II-4	fortuitous	23° (71)	right lumbar	L2	none
		II-6	adolescence	41° (67)	right thoraco-lumbar	L1	none
		III-3	adolescence	15° (49)	right thoraco-lumbar	L1	bracing
		III-4	fortuitous	29° (46)	right thoraco-lumbar	D12	none
		III-8	fortuitous	15° (42)	right thoraco-lumbar	L1	none
		IV-1	adolescence	29°/29° (20)	right thoracic/left lumbar b	D8/L2	physiotherapy
		IV-2	fortuitous	15° (16)	right thoracic	D7	none
		IV-3	fortuitous	14°/16° (19)	right thoracic/left lumbar b	D8/L1	none
		IV-4	12	18° (15)	left thoraco-lumbar	L1	physiotherapy
	F3	II-3	fortuitous	14°/16°(58)	right thoracic/left lumbar b	D8/L1	none
		II-8	fortuitous	24°(66)	right thoracic	D8	none
		III-1	adolescence	65°(18)	right thoracic-left lumbar	D8	spinal surgery
		III-2	fortuitous	15°(32)	right lumbar	L2	none
		IV-1	9	22°(15)	right thoracic	D8	physiotherapy
		IV-2	10	20° (15)	right thoraco-lumbar	L1	bracing
	a Cobb's angle after arthrodesis
	b Double major curve

Table 2

 2

	Family Number of	Number	Number of	Expected Maximum	Probabilities to reach Z imax
		meioses	of	disease allele	lod score	under H0, 0, 1 or more
		between a	common	transmissions		than once in whole
		common	ancestors	to affected		genome analysis
		ancestor and		individuals
		all affected
		individuals
					Z imax = -Ti*log 10 (0.5)
	Fi	Mi	ni	Ti=Mi-ni		0	1	>1
	F1	9	4	5	1.51	0.253	0.346	0.401
	F2	12	2	10	3.01	0.958	0.041	0.001
	F3	10	1	9	2.71	0.918	0.079	0.003

Acknowledgements

We warmly thank the patients who participated to this study. We also thank Heather Yampolsky for her editorial assistance. This work was supported by the Fondation Yves Cotrel-Institut de France, the Hospices Civils de Lyon (PHRC regional 2001), the Fondation pour la Recherche Médicale, the Rhône-Alpes region (contract Emergence), the University Lyon 1 and the Association GEN'HOM.

Competing interests: none

Web resources

Online Mendelian (OMIM), http://www.ncbi.nlm.nih.gov/sites/entrez?db=omim National Center for Biotechnology Information, http://www.ncbi.nlm.nih.gov/ deCODE Genetics, http://www.decode.com/