Marc Khalifé
email: marc.khalife@aphp.fr

Claudio Vergari

Emmanuelle Ferrero

Valérie Attali

Cécile Heidsieck

Ayman Assi

Wafa Skalli

The rib cage: a new element in the spinopelvic chain

Keywords: Thoracic cage, Rib cage anatomy, Aging variations, Sagittal alignment, Pelvic incidence

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

Introduction

The spine and pelvis form a biomechanical chain of balance that allows to maintain an erect posture and a horizontal gaze. This results in a sagittal spinopelvic profile that can largely vary in different subjects: it depends on the morphology of the subject's pelvis, and in particular on his pelvic incidence, but also on other factors such as age, gender, pain, and spine deformity [START_REF] Jgjc | Pelvic incidence: A fundamental pelvic parameter for three-dimensional regulation of spinal sagittal curves[END_REF][START_REF] Iyer | Variations in sagittal alignment parameters based on age: a prospective study of asymptomatic volunteers using full-body radiographs[END_REF]. Alterations of the sagittal balance can be compensated by postural adjustments aiming at keeping the erect posture with a minimal energy expenditure [START_REF] Barrey | Compensatory mechanisms contributing to keep the sagittal balance of the spine[END_REF]. Hence, thoracic kyphosis and lumbar lordosis can increase or decrease, and pelvic tilt can change within a physiological range in order to compensate [START_REF] Beyer | Pelvic compensation in sagittal malalignment: how much retroversion can the pelvis accommodate?[END_REF]. In this sagittal chain of balance, the rib cage is often overlooked. However, ribs help stabilize the thoracic spine while also significantly reducing its range of motion [START_REF] Liebsch | The rib cage stabilizes the human thoracic spine: an in vitro study using stepwise reduction of rib cage structures[END_REF][START_REF] Ignasiak | Thoracolumbar spine model with articulated ribcage for the prediction of dynamic spinal loading[END_REF]. Therefore, the rib cage geometry and kinematics can play a role in the compensating mechanisms which are deployed to keep the balance.

Alterations of balance can be permanent or semi-permanent, such as in spinal deformity, or dynamic, such as in breathing. While quiet breathing can be accompanied by a minor postural sway [START_REF] Clavel | Decreased respiratoryrelated postural perturbations at the cervical level under cognitive load[END_REF][START_REF] Hodges | Coexistence of stability and mobility in postural control: evidence from postural compensation for respiration[END_REF], it has been shown that attaining maximal or minimal lung volume requires significant changes of the spinopelvic alignment [START_REF] Dally | An inquiry into the physiological mechanism of respiration, with especial reference to the movements of the vertebral column and diaphragm[END_REF][START_REF] Attali | Compensation of respiratory-related postural perturbation is achieved by maintenance of head-to-pelvis alignment in healthy humans[END_REF]. Thus, breathing function is correlated to thoracic cage anatomy and on its dynamic behavior, but also on the spinopelvic alignment and its dynamic changes. Furthermore, age and gender can have an impact on sagittal alignment [START_REF] Iyer | Variations in sagittal alignment parameters based on age: a prospective study of asymptomatic volunteers using full-body radiographs[END_REF][START_REF] Roussouly | Age-and sex-related variations in sagittal sacropelvic morphology and balance in asymptomatic adults[END_REF], as well as respiratory or musculoskeletal pathology [START_REF] Turner | Elasticity of human lungs in relation to age[END_REF]. Breathing is a basic vital function, and its study is therefore of interest for a wide range of physicians from pneumologists to spine surgeons, to understand more precisely the aging phenomena as well as gender variations of the rib cage. For instance, thoracic cage should be taken into account when addressing spinal malalignment. It has been demonstrated that restoring kyphosis in spinal deformity corrections helps avoiding respiratory repercussion [START_REF] Assi | A Novel Classification of 3D Rib Cage Deformity in Subjects With Adolescent Idiopathic Scoliosis[END_REF], but the impact of this surgery on the rib cage is not well studied. Indeed, little literature is available about thoracic cage anatomical variations, or its relationship with spinopelvic alignment. There are a few studies that analyzed rib cage anatomy variations using CT-scanner, however, to our knowledge, there are no articles in the literature describing age-, gender-and spinopelvic-related variations in upright position [START_REF] Holcombe | The effect of age and demographics on rib shape[END_REF][START_REF] Weaver | Morphometric analysis of variation in the ribs with age and sex[END_REF]. Bi-planar radiography allows skeletal 3D reconstructions in erect position, allowing a more accurate sagittal alignment analysis than supine position [START_REF] Yeung | Accuracy on the preoperative assessment of patients with adolescent idiopathic scoliosis using biplanar low-dose stereoradiography: a comparison with computed tomography[END_REF]. In addition, radiation exposure is decreased for the subject, up to 30 times lower compared to CT-scan [START_REF] Delin | Ionizing radiation doses during lower limb torsion and anteversion measurements by EOS stereoradiography and computed tomography[END_REF]. The 3D thoracic cage reconstruction method from bi-planar radiographs was validated in several articles analyzing rib cage deformities, brace treatment impact and predict pulmonary function in scoliotic adolescents [START_REF] Assi | A Novel Classification of 3D Rib Cage Deformity in Subjects With Adolescent Idiopathic Scoliosis[END_REF][START_REF] Courvoisier | EOS 3D imaging: assessing the impact of brace treatment in adolescent idiopathic scoliosis[END_REF][START_REF] Bouloussa | Biplanar stereoradiography predicts pulmonary function tests in adolescent idiopathic scoliosis: a cross-sectional study[END_REF].

The hypothesis of this study is that rib cage geometry is related to spinopelvic alignment. Hence, the aim of this study was to analyze the anatomical variations of the thoracic cage (TC) according to spinopelvic parameters using bi-planar X-Rays in erect position. Secondary objectives were to determine age-and gender-related variations of the TC.

Material and methods

Population

This multicentric study retrospectively included, apart from any medical consultation, healthy volunteer subjects from previous retrospective studies, with no reported pain. All subjects had free-standing position EOS® bi-planar radiographs (upright position, fingertips on the zygomatic bones, one foot slightly forward) [START_REF] Janssen | Differences in male and female spino-pelvic alignment in asymptomatic young adults: a three-dimensional analysis using upright low-dose digital biplanar X-rays[END_REF]. Subjects were excluded if they presented thoracic scoliosis (with Cobb angle greater than 10°) or any musculoskeletal deformity, and history of respiratory disease. This study was approved by the regional ethics committee (approval N° 6001, C.P.P Ile-de France VI) and FM 312 ethical committee at the Saint-Joseph University, Beirut. All subjects provided their informed written consent (or parents' if minor subject).

Parameters

Spinal and thoracic cage three-dimension reconstructions were performed according to previously validated methods [START_REF] Humbert | 3D reconstruction of the spine from biplanar X-rays using parametric models based on transversal and longitudinal inferences[END_REF][START_REF] Vergari | A novel method of anatomical landmark selection for rib cage 3D reconstruction from biplanar radiography[END_REF], by two specifically trained doctors (Fig. 1). Briefly, the user digitized the spinal line in the frontal and lateral X-rays, from T1 to L5. The method then proposed a first 3D reconstruction of the spine, and retro-projected the 3D models of the vertebra on the radiographs. The user could modify the models to make them fit the contours visible on the radiographs. In a second phase, the user digitized the sternum, the midlines of ribs 2, 5, 8 and 10 in the frontal radiograph, as well as the most posterior points of the ribs in the lateral view. Again, a 3D model was retro-projected on the radiographs for the user to adjust.

Apart from age and sex, all data were collected from 3D reconstructions:

• Pelvic parameters (Fig. 2): Pelvic incidence (PI), pelvic tilt (PT) and sacral slope (SS) • Type of sagittal profile according to updated Roussouly's classification [START_REF] Laouissat | Classification of normal sagittal spine alignment: refounding the Roussouly classification[END_REF] • Spinal parameters (Fig. 2): T1-T12 kyphosis (TK), T4-T12 kyphosis, L1-S1 lordosis (LL), curvilinear spinal length was measured from T1 to L5 • Global thoracic cage parameters (Fig. 3a): maximum thickness (anteroposterior diameter-mm) maximum width (lateral diameter-mm) T1-T10 rib cage volume (cm 3 ) mean Spinal Penetration Index (SPI: ratio between rib cage volume and volume occupied by the spine-%)

• Rib parameters, computed for each rib from 1 to 10 (Fig. 3b): absolute sagittal angle: measured between horizontal line and rib sagittal axis.

relative sagittal angle: measured between the upper endplate of the corresponding vertebra and sagittal rib axis.

"umbrella angle": the coronal angle between two ribs.

Statistical analysis

All variables were tested for normality using Shapiro-Wilk's. Left and right ribs parameters were averaged, after statistically checking for symmetry. Rib groups have then been constituted: upper (ribs 2 to 5), middle (ribs 6 and 7) and lower (ribs 8 to 10). Correlations were searched between spinopelvic and thoracic parameters, using Pearson's coefficients. Using ANOVAs, rib and thoracic cage parameters have been compared according to sagittal profile as described in updated Roussouly's classification [START_REF] Laouissat | Classification of normal sagittal spine alignment: refounding the Roussouly classification[END_REF].

In order to distinguish more accurately variations between age intervals, and to separate growing from aging phenomena, the cohort has been divided in two subgroups for gender analysis (growing [START_REF] Hodges | Coexistence of stability and mobility in postural control: evidence from postural compensation for respiration[END_REF][START_REF] Dally | An inquiry into the physiological mechanism of respiration, with especial reference to the movements of the vertebral column and diaphragm[END_REF][START_REF] Attali | Compensation of respiratory-related postural perturbation is achieved by maintenance of head-to-pelvis alignment in healthy humans[END_REF][START_REF] Roussouly | Age-and sex-related variations in sagittal sacropelvic morphology and balance in asymptomatic adults[END_REF][START_REF] Turner | Elasticity of human lungs in relation to age[END_REF][START_REF] Assi | A Novel Classification of 3D Rib Cage Deformity in Subjects With Adolescent Idiopathic Scoliosis[END_REF][START_REF] Holcombe | The effect of age and demographics on rib shape[END_REF][START_REF] Weaver | Morphometric analysis of variation in the ribs with age and sex[END_REF][START_REF] Yeung | Accuracy on the preoperative assessment of patients with adolescent idiopathic scoliosis using biplanar low-dose stereoradiography: a comparison with computed tomography[END_REF][START_REF] Delin | Ionizing radiation doses during lower limb torsion and anteversion measurements by EOS stereoradiography and computed tomography[END_REF][START_REF] Courvoisier | EOS 3D imaging: assessing the impact of brace treatment in adolescent idiopathic scoliosis[END_REF][START_REF] Bouloussa | Biplanar stereoradiography predicts pulmonary function tests in adolescent idiopathic scoliosis: a cross-sectional study[END_REF] versus mature skeleton (20 +)) and into five groups for age analysis: Children (8-12 years), Adolescents [START_REF] Assi | A Novel Classification of 3D Rib Cage Deformity in Subjects With Adolescent Idiopathic Scoliosis[END_REF][START_REF] Holcombe | The effect of age and demographics on rib shape[END_REF][START_REF] Weaver | Morphometric analysis of variation in the ribs with age and sex[END_REF][START_REF] Yeung | Accuracy on the preoperative assessment of patients with adolescent idiopathic scoliosis using biplanar low-dose stereoradiography: a comparison with computed tomography[END_REF][START_REF] Delin | Ionizing radiation doses during lower limb torsion and anteversion measurements by EOS stereoradiography and computed tomography[END_REF][START_REF] Courvoisier | EOS 3D imaging: assessing the impact of brace treatment in adolescent idiopathic scoliosis[END_REF][START_REF] Bouloussa | Biplanar stereoradiography predicts pulmonary function tests in adolescent idiopathic scoliosis: a cross-sectional study[END_REF], Young (20-39), Middle-aged (40-59) and Seniors (60 +).

Comparisons between gender or age categories were conducted using t-tests and ANOVAs for normally distributed variables, whereas Wilcoxon and Kruskal-Wallis tests were used for non-normally distributed ones. Correlations have then been tested between age and all the other numeric variables. A multivariate linear model was built to assess the relative effects of gender and subject height on rib cage volume. The model thus included sex, T1-L5 curvilinear spinal length and rib cage volume as variables, as well as their linear interactions. The statistical analyses have been carried out using RStudio, with p values lower than 0.05 considered significant.

Results

Population

A total of 256 healthy subjects were included for analysis, with 140 females (55%) and 116 males (Table 1). Mean age was 34 years (SD = 20.1, range: 8-83) with no significant difference between genders. Mean pelvic incidence was 50° (SD = 11, range: 24-87).

Relationship between rib cage and spinopelvic parameters

There were significant correlations between rib cage and spinopelvic parameters (Table 2, Fig. 4). Pelvic tilt and spinal length were correlated both to global rib cage parameters (volume, thickness, width, SPI) and to rib orientations (sagittal relative and absolute angles, umbrella angles, Table 2). T1-T12 and T4-T12 kyphosis were mostly correlated to global rib cage parameters (thickness and volume), but also to upper and middle ribs sagittal relative angles (Table 2). A higher thoracic kyphosis was associated to a lower SPI. Lumbar lordosis showed correlations with lower rib sagittal angles (Table 2). Figure 4 shows the relationships between pelvic incidence, T4-T12 kyphosis, SPI and rib cage volume. The association between PI and TC volume varied in opposite manners before and after skeletal maturation, with a -0.2 correlation among growing group and 0.5 in the mature group. Analysis according to Roussouly's classification revealed significantly larger rib cage volume and thickness for types 1 and 3, along with lower SPI (Table 3). Relative sagittal angle of lower ribs was significantly greater in type 4 compared to type 1 (p = 0.02).

Analysis according to age

Cohort characteristics by age are reported in Table 1. Age was significantly correlated to pelvic tilt, sacral slope, LL, TK and spinal length (Table 4). PI, PT and spinal length increased with age among growing subjects while SS, LL and spinal length decreased in mature group, along with PT and TK augmentation. No significant differences were exhibited between Young, Middle-aged and Senior groups regarding TC volume and width, whereas children and adolescents had significantly lower volume and width values than older subjects (p < 0.02, not shown). Thickness increased significantly with age between all subgroups except for young compared to adolescents (p = 0.12) and middle-aged compared to seniors (p = 0.26, not shown). SPI significantly decreased after 20 years old (p < 0.001) with no variation between children and adolescents. During growth, TC thickness, volume and width correlated with age (respectively, 0.47, 0.73, 0.73, p < 0.001). However, only thickness and SPI significantly changed after skeletal maturity (respectively, 0.39 and -0.52, p < 0.001). None of these parameters significantly varied between middle-aged and seniors (Fig. 5).

Absolute sagittal angles did not vary with age; however, relative sagittal angle showed negative correlation with age in skeletally mature subjects, with coefficients at, respectively, -0.52, -0.35, and -0.23 for upper, middle and lower ribs (p < 0.001). A negative correlation was found between TC Thickness and relative sagittal angles (-0.71, p < 0.001). Umbrella angles showed no correlation to age in the whole cohort, whereas children had significantly higher angles than the three older groups considering middle and lower ribs (not shown). All rib parameters are represented in Fig. 6.

Analysis according to gender

Male subjects presented larger TC in terms of thickness, width and volume (p < 0.001, Table 5). Female subjects had higher SPI than males (p < 0.001). In accordance, they had lower T1-T12 kyphosis than males with 45.3 ± 10.7° versus 49.9 ± 10.5° (p = 0.0002). Relative sagittal angles at all

Discussion

This study gives an insight on the correlation between rib cage anatomy and spinal and pelvic parameters. For instance, higher thoracic kyphosis and lumbar lordosis were both associated with lower SPI and higher TC thickness, TC volume and rib absolute sagittal angles. TK was correlated with upper and middle ribs while lumbar lordosis presented correlations with lower ribs. Pelvic tilt showed correlations with several thoracic parameters: a higher pelvic tilt was correlated with higher TC thickness, higher rib relative sagittal angles and lower SPI. Pelvic incidence and sacral slope were negatively correlated with TC volume. This confirms the hypothesis of the study that the thoracic cage and spine anatomy are linked to each other, but also that the rib cage is an integral part of the spinopelvic chain of balance. Hence, spine, pelvis and rib cage should not be considered separately. Expectedly, thoracic kyphosis showed stronger relationships with rib cage anatomy than lumbar lordosis. Indeed, thoracic spine presents anatomical joints with the ribs, and an increase in the first, leads to rising TC thickness and rib sagittal angles, particularly in the upper cage. Thoracic kyphosis was more strongly correlated to rib cage than to pelvic parameters. Lumbar lordosis was correlated with lower ribs, probably as it is directly influencing lower thoracic spine. Its correlations were stronger with pelvic parameters than rib cage ones. Pelvic incidence and sacral slope showed weak correlations with rib cage anatomy parameters. Among pelvic parameters, pelvic tilt was the most strongly associated with rib cage anatomy, especially TC thickness. This could be a compensation mechanism to compensate the forward shift of the gravity line due to an increase in TC thickness, with a posterior displacement driven by the increase in pelvic tilt.

This study describes for the first time the correlation between rib cage and spinopelvic anatomy in upright position, with respect to age and gender. Bi-planar radiography has been more largely used to analyze rib cage in adolescent idiopathic scoliosis patients [START_REF] Ilharreborde | Use of EOS imaging for the assessment of scoliosis deformities: application to postoperative 3D quantitative analysis of the trunk[END_REF]. This technique offers several advantages such as radiation reduction for the subject-compared to CT-scan-and analysis in standing position [START_REF] Melhem | EOS(®) biplanar X-ray imaging: concept, developments, benefits, and limitations[END_REF]. Indeed, it has been proven in spine care that prone position can significantly alter spine morphology compared to standing position, thus bias sagittal alignment assessment [START_REF] Yeung | Accuracy on the preoperative assessment of patients with adolescent idiopathic scoliosis using biplanar low-dose stereoradiography: a comparison with computed tomography[END_REF]. More, in lying position, lung volumes-functional residual capacity and forced vital capacity-are smaller due to cephalic displacement of the diaphragm, a consequence of increased abdominal pressure, as well as increased pulmonary blood volume. This reduction in the lung volume in the recumbent position is correlated to a reduction in elastic recoil of the lungs and the chest wall and a modification of ribs orientations, which could both modify the forces applied to the thoracic cage and spinal curvatures [START_REF] Katz | The effect of body position on pulmonary function: a systematic review[END_REF]. Corroborating further the association between rib cage and spine, Bouloussa et al. exhibited correlations between rib cage volume measured using bi-planar radiography and all pulmonary capacities (total lung, slow and forced vital capacities) in an adolescent idiopathic scoliosis population [START_REF] Bouloussa | Biplanar stereoradiography predicts pulmonary function tests in adolescent idiopathic scoliosis: a cross-sectional study[END_REF]. Adult spinal deformity patients present spine and thoracic cage deformity. However, surgical treatment of these patients usually focuses on the spine, including the restoration of thoracic kyphosis. This has an indirect effect on pelvic tilt through compensatory mechanism to maintain sagittal balance [START_REF] Barrey | Compensatory mechanisms contributing to keep the sagittal balance of the spine[END_REF]. Given the significant anatomical relationships between pelvis, spine and rib cage, the latter should not be overlooked. However, the potential changes induced by spinal surgery on the rib cage are not well studied. Correction-fusion surgeries, through spinal deformity and kyphosis correction might have consequences on TC anatomy and pulmonary function. Therefore, a failure in restoring spinal physiological curvatures could lead to pulmonary disfunction as it is known that respiratory function and spinal anatomy are linked.

Aging has been proven to modify spinal alignment [START_REF] Diebo | Recruitment of compensatory mechanisms in sagittal spinal malalignment is age and regional deformity dependent: a full-standing axis analysis of key radiographical parameters[END_REF], increase rib fracture risk [START_REF] Kent | Fatality risk and the presence of rib fractures[END_REF], and alter respiratory function. One CT-scan-based study have analyzed changes in aging rib cage morphology. In that study, Weaver et al. reported significant changes in aging thoracic cage [START_REF] Weaver | Morphometric analysis of variation in the ribs with age and sex[END_REF], which appear consistent with the present results. Both studies observed an increase in ribs sagittal angle during growth, then a decrease until adulthood and a further increase during aging (Fig. 6). However, relative rib orientations in the previous study were computed in supine position, according to the rib itself, whereas angles were computed relative to the vertebra in the present study. Hence, this relative angle represents the orientation of the costovertebral junction. This relative angle in the sagittal plane decreased with age, with a sharp decrease during adulthood (Fig. 6). Thoracic kyphosis increases with age, due to degenerative phenomena of the spine: bone remodeling, degenerating disks and facet joints along with sarcopenia. It appears that thoracic vertebrae tend to rotate forward while the ribs rotate upwards. This phenomenon may be explained, through a contribution of the costovertebral joints, as a prevention of TC subsidence into the abdomen, in order to maintain sufficient TC volume for respiratory function. The results of the present study corroborate Weaver et al. conclusions [START_REF] Weaver | Morphometric analysis of variation in the ribs with age and sex[END_REF]. Indeed, it was found that the thoracic cage ages mainly through increasing its anteroposterior diameter (thickness) which is logically correlated with increasing thoracic kyphosis; hence augmenting TC osseous volume. Thickness grows constantly, whereas width increased only during skeletal maturation. Conversely, SPI seemed to be more specific of aging as it starts decreasing after adolescence. This phenomenon increases as well TC volume as thoracic spine occupies less space within the rib cage. These variations may be explained by aging phenomena of lungs with diminution of reserve volumes which impact rib cage anatomy as well as weakened inspiratory muscles, leading to decreased vital capacity [START_REF] Turner | Elasticity of human lungs in relation to age[END_REF]. Indeed, in healthy subjects, aging is associated with changes in thoraco-pulmonary mechanical properties [START_REF] Verbeken | The senile lung. Comparison with normal and emphysematous lungs. 2[END_REF][START_REF] Galetke | Reference values for dynamic and static pulmonary compliance in men[END_REF]. Whether this may explain the noticeable differences observed on thoracic parameters in the senior group remains to be studied.

Differences between gender specifics were also observed. Females had more penetrating spines into the rib cage (higher SPI), ribs sloping more downward (higher rib relative sagittal angles), associated with smaller TK than male subjects. Males presented larger thoracic cages in terms of width, thickness and volume. These differences between genders mainly lie in the size difference between males and females as demonstrated by multivariate analysis, with a body size effect seven times higher than gender effect.

Limitations

This study analyzed transversally a cohort of healthy subjects. A longitudinal study would overrule the risk of selection bias and may allow more accurate description of aging phenomena. Another limitation of this study is the absence of anthropometric data and pulmonary function tests, such as functional residual volume, to be correlated with radiological anatomy. New inter-disciplinary longitudinal studies would be interesting to further analyze relationships between rib cage and spine, and correlate anatomical features with functional data.

Conclusion

The results of this study demonstrate the correlation between thoracic cage anatomy and spinopelvic parameters, confirming that rib cage can be considered as a part of the spinopelvic chain of balance. Indeed, higher spinal curvatures were associated with lower SPI and higher TC thickness, TC volume and rib absolute sagittal angles. Furthermore, rib cage anatomy is also related to age and gender. Males have larger thoracic cages in terms of width, thickness, and volume while females have higher SPI and ribs relative sagittal angles. TC mainly ages through an increase in anteroposterior diameter and decrease in SPI. These results suggest that thoracic cage could be taken into account when addressing sagittal alignment issue as trunk anteroposterior diameter evolves. Further studies including pulmonary function tests in adult spinal deformity patients could help spine surgeons improve surgical management.

Fig. 1

 1 Fig. 1 Anteroposterior and lateral views after pelvis, spine and thoracic cage reconstruction

Fig. 2

 2 Fig. 2 Spinal and pelvic parameters representation on lateral view TK: T1-T12 kyphosis, CSL: Curvilinear T1-L5 spinal length, LL: L1-S1 lordosis, SS: sacral slope, PT: pelvic tilt. Pelvic incidence = PT + SS.

Fig. 3 a

 3 Fig. 3 a Thoracic cage width and thickness measure. Spinal Penetration Index (on the right) is obtained as follows: Interior surface × 100 / Total surface. b Rib parameters. 1.: Absolute sagittal angle / 2.: Relative sagittal angle / 3.: Umbrella angle. H: Horizontal line

Fig. 6

 6 Fig.[START_REF] Ignasiak | Thoracolumbar spine model with articulated ribcage for the prediction of dynamic spinal loading[END_REF] Boxplot representations of absolute and relative sagittal angles and umbrella angles according to age. Age brackets: Children (8-12 years), Adolescents[START_REF] Assi | A Novel Classification of 3D Rib Cage Deformity in Subjects With Adolescent Idiopathic Scoliosis[END_REF][START_REF] Holcombe | The effect of age and demographics on rib shape[END_REF][START_REF] Weaver | Morphometric analysis of variation in the ribs with age and sex[END_REF][START_REF] Yeung | Accuracy on the preoperative assessment of patients with adolescent idiopathic scoliosis using biplanar low-dose stereoradiography: a comparison with computed tomography[END_REF][START_REF] Delin | Ionizing radiation doses during lower limb torsion and anteversion measurements by EOS stereoradiography and computed tomography[END_REF][START_REF] Courvoisier | EOS 3D imaging: assessing the impact of brace treatment in adolescent idiopathic scoliosis[END_REF][START_REF] Bouloussa | Biplanar stereoradiography predicts pulmonary function tests in adolescent idiopathic scoliosis: a cross-sectional study[END_REF], Young[START_REF] Janssen | Differences in male and female spino-pelvic alignment in asymptomatic young adults: a three-dimensional analysis using upright low-dose digital biplanar X-rays[END_REF][START_REF] Humbert | 3D reconstruction of the spine from biplanar X-rays using parametric models based on transversal and longitudinal inferences[END_REF][START_REF] Vergari | A novel method of anatomical landmark selection for rib cage 3D reconstruction from biplanar radiography[END_REF][START_REF] Laouissat | Classification of normal sagittal spine alignment: refounding the Roussouly classification[END_REF][START_REF] Ilharreborde | Use of EOS imaging for the assessment of scoliosis deformities: application to postoperative 3D quantitative analysis of the trunk[END_REF][START_REF] Melhem | EOS(®) biplanar X-ray imaging: concept, developments, benefits, and limitations[END_REF][START_REF] Katz | The effect of body position on pulmonary function: a systematic review[END_REF][START_REF] Diebo | Recruitment of compensatory mechanisms in sagittal spinal malalignment is age and regional deformity dependent: a full-standing axis analysis of key radiographical parameters[END_REF][START_REF] Kent | Fatality risk and the presence of rib fractures[END_REF][START_REF] Verbeken | The senile lung. Comparison with normal and emphysematous lungs. 2[END_REF][START_REF] Galetke | Reference values for dynamic and static pulmonary compliance in men[END_REF](31)(32)(33)(34)(35)(36)(37)(38)(39), Middle-aged (M.A.: 40-59) and Seniors (60 +)

Table 1

 1 Cohort demographic dataBold values correspond to totals (by gender or age categories)

		Females	Males	Total	%
	Children	28	10	38	15%
	Adolescents	18	17	35	14%
	Young	41	41	82	32%
	Middle-aged	34	30	64	25%
	Senior	19	18	37	14%
	Growing	46	27	73	29%
	Mature	94	89	183	71%
	Total cohort	140	116	256	100%

Table 2

 2 Correlation coefficients between spinopelvic and rib cage parameters. Only significant p values and variables presenting correlations are reported (*: p < 0.05. †: p < 0.001). TC-Thoracic cage. SPI-Spinal penetration index Bold values correspond to significant correlations T1-12 kyphosis T4-T12 kyphosis L1-S1 Lordosis Pelvic Incidence Sacral slope Pelvic Tilt Spinal length

	TC volume	0.3 †	0.2 †	0.2*	0.1	0.1	0.3 †	0.8 †
	TC thickness	0.3 †	0.3 †	0.1	0.1	0.1*	0.4 †	0.5 †
	TC width	0.2 †	0	0.1*	0.1	0	0.2 †	0.8 †
	SPI	-0.3 †	-0.2 †	-0.2*	-0.1	-0.2 †	-0.3 †	-0.2 †
	Upper ribs absolute sagittal angle	0.1	0.2 †	0.1	-0.1	0	-0.1	0.1
	Middle ribs absolute sagittal angle	0.1	0.1*	0.1	-0.1	0	-0.1	0.1
	Lower ribs absolute sagittal angle	-0.1	-0.1	0.3 †	-0.2 †	0.1*	-0.1	0.04
	Upper ribs relative sagittal angle	-0.4 †	-0.5 †	0	-0.1	-0.1	-0.3 †	-0.2 †
	Middle ribs relative sagittal angle	-0.2 †	-0.4 †	0	-0.1	-0.1	-0.2 †	-0.2 †
	Lower ribs relative sagittal angle	0	-0.1	-0.2 †	0	-0.2 †	-0.3 †	-0.2 †
	Upper ribs umbrella angle	0	0.1	0	-0.1	0.1	-0.1	-0.1*
	Middle ribs umbrella angle	-0.1	0.1	-0.1	-0.1	0	-0.1*	-0.2 †
	Lower ribs umbrella angle	-0.1*	0	-0.1	-0.1	-0.1	-0.1*

Table 3

 3 Quantitative

	Upper ribs	37 ± 6	35 ± 7	37 ± 6	37 ± 8	36 ± 7	0.70
	Abs. sagittal angle
	Middle ribs	38 ± 6	37 ± 8	38 ± 6	38 ± 7	37 ± 6	0.65
	Abs. sagittal angle
	Lower ribs	46 ± 5	45 ± 7	45 ± 5	46 ± 6	44 ± 5	0.30
	Abs. sagittal angle
	Upper ribs	17 ± 8	22 ± 9	19 ± 8	21 ± 7	20 ± 9	0.08
	Rel. sagittal angle
	Middle ribs	35 ± 8	37 ± 10	36 ± 8	38 ± 9	38 ± 8	0.20
	Rel. sagittal angle
	Lower ribs	56 ± 7	56 ± 8	58 ± 7	59 ± 7	60 ± 7	0.02*
	Rel. sagittal angle
	SPI	2.2 ± 1.8	3.2 ± 2.2	3.0 ± 1.8	3.7 ± 1.8	3.3 ± 1.5	< 0.001*
	Volume	7569 ± 2144 5960 ± 2316 7141 ± 1718 5808 ± 2216 6717 ± 1565	< 0.001*
	Thickness	169 ± 27	152 ± 26	165 ± 21	148 ± 26	158 ± 24	0.001*

results of rib and thoracic cage parameters according to the five types of Roussouly's classification. P values resulting from ANOVA are given, marked with a "*" if significant. Abs.-Absolute. Rel.-Relative. 3A-Type 3 anteverted. SPI-Spinal penetration index Italics was used to describe p-values Roussouly type 1 (n = 73) 2 (n = 13) 3 (n = 91) 3A (n = 26) 4 (n = 53) p

Table 4

 4 Correlation table between spinopelvic parameters and age. Pearson coefficients for the growing (age 8-19 years) and mature skeleton (age > 19) groups are reported, with corresponding p values (marked with a "*" if significant)

	Correlation with Age	Growing p-value	Mature p value
	Pelvic incidence	0.25	0.03*	0.06	0.40
	Pelvic tilt	0.26	0.02*	0.41	< 0.001*
	Sacral slope	0.06	0.61	-0.24	< 0.001*
	L1-S1 Lordosis	-0.04	0.73	-0.27	< 0.001*
	T1-T12 Kyphosis	0.12	0.31	0.29	< 0.001*
	Curvilinear spinal length	0.78	< 0.001* -0.18	0.02*
	Italics was used to describe p-values

Table 5

 5 Quantitative results of thoracic cage morphology and spinopelvic parameters according to gender. Results are expressed by mean ± standard deviation Italics was used to describe p-values

	Females	Males	p value

Funding No funding.

Availability of data and material Data will not be deposited.

Declarations

Conflicts of interest No conflicts to declare by any of the authors.


Claude Jauffret
email: jauffret@univ-tln.fr

Denis Pillon
email: denis.pillon@fr.thalesgroup.com

Annie-Claude Pignol
email: pignol@univ-tln.fr

Bearings-Only Maneuvering Target Motion Analysis from a Nonmaneuvering Platform

Keywords: Tracking, Bearings-only TMA, Maneuvering target, Observability, Cramèr-Rao Lower Bound

come

I -Introduction

For the last four decades, the so-called bearings-only target motion analysis has been the subject of constant and intensive interest. If, at the very beginning, this problem was posed in an acoustic context (submarine with the use of passive sonar systems, see [START_REF] Kolb | Bearings-Only Target Motion Estimation[END_REF] for example), it is encountered in the electromagnetic (aircraft surveillance using ESM sensor) and optic domains (satellite using an infrared camera). The "standard" version of BOTMA assumes that the two mobiles are on the same plane and that the target has a constant course and speed (rectilinear movement) during the observation duration. As a consequence, the aim of the classical BOTMA algorithms is to estimate four parameters (two coordinates for geographical position, speed and course of a target) from a set of available bearing measurements collected by the observer.

Under those classical assumptions, it is well known ( [START_REF] Nardone | Observability Criteria for Bearings-Only Target Motion Analysis[END_REF], [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF]) that if it maintains a constant velocity vector, the own ship cannot determine the trajectory of the source. In the relative coordinate system linked to the current position of the observer, the trajectories producing the same bearings are homothetic. As a consequence, in an absolute coordinates system, the family of these trajectories is defined by three parameters ( [START_REF] Lindgren | Properties of Bearings-only Motion Analysis: An interesting case study in system observability[END_REF], [START_REF] Le Cadre | Discrete-time Observability and Estimability Analysis for Bearings-Only Target Motion Analysis[END_REF]): the missing dimension causes the lack of observability. This observability status is the fundamental difference between BOTMA and other TMA problems, such as the TMA based upon measurements of time difference of arrival (TDOA) or measurements of frequency difference of arrival (FDOA). In these latter cases, the source's trajectory is not observable if the source travels on a symmetry axis (for example, in TDOATMA, the source moves along the line defined by the two sensors or along their mid-perpendicular [START_REF] Arnold | Target Parameter Estimation Using Measurements Acquired with a Small Number of Sensors[END_REF]). As soon as the target leaves this axis, the problem becomes observable. In short, the observability is guaranteed most of the time in TDOATMA and FDOATMA while this property is hard to obtain in BOTMA since the observer must maneuver in an efficient fashion: in BOTMA, the quality of the estimation depends highly on the observer's maneuver. Some papers propose interesting solutions to answer the difficult question: how to maneuver to get the best from the collected bearings (see [START_REF] Passerieux | Optimal Observer Manoeuver for Bearings-Only Tracking[END_REF] and [START_REF] Le Cadre | Optimizing the Receiver Maneuvers for Bearings-Only Tracking[END_REF])? On the other hand, there exist a variety of maneuvers which maintain the trajectory of the source unobservable ( [START_REF] Nardone | Observability Criteria for Bearings-Only Target Motion Analysis[END_REF], [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF]).

Any maneuver is costly for the observer which prefers to reach a point in the shortest time, i.e. by traveling along a straight line. Indeed, any zigzag will slow down the own ship. Moreover, maneuvering causes a loss of discretion with regard to another platform (which can be the target itself). For example, when maneuvering, a submarine can emit transient signals while an aircraft changes its infrared signature.

As a consequence, in BOTMA, a non-maneuvering observer which would still be able to estimate without ambiguity the trajectory of a source, would benefit from a precious advantage over many users. The way to achieve that goal is to take prior information into account. Prior information can be extra measurements, for example a given speed drawn in an admissible range (the ship's speed never exceeds 20 m/s, while the aircraft's is greater than 30 m/s) or a given course.

Another way is to take other measurements into account such as the frequency line; this supposes that the source emits pure unknown and constant tones and makes a dedicated processing necessary (see [START_REF] Passerieux | Target Motion Analysis with Bearings and Frequencies Measurements[END_REF], [START_REF] Maranda | Localization of a Manoeuvring Target Using Simulated Annealing[END_REF] and [START_REF] Becker | A general approach to TMA observability from angle and frequency measurements[END_REF]). Frequency measurements have also been exploited for localization problems (i.e. stationary target) without the requirement of constant frequencies [START_REF] Becker | Passive localization of frequency-agile radars from angle and frequency measurements[END_REF].

Bearing measurements collected by another non-maneuvering sensor make the trajectory of the source observable [START_REF] Guelle | Inter-array Multitracks Association[END_REF], but the price to pay is the existence of an efficient communication between the two platforms and a reliable fusion center. This link can be wireless (two submarines which communicate at a low rate) or not (for example, a towed array). In that case, except for some "pathological cases", the observability is guaranteed [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF].

In this paper, we propose to consider a new piece of information when the trajectory of the source is composed of two legs ("segmented" or "piecewise rectilinear" trajectory is also the vocabulary used in the literature): the time of maneuver can be interpreted and used as an extra measurement. An assumption is required: the speed of the target must be equal to a constant.

The segmented trajectory model for the target has been widely employed in papers dealing with the problems of detecting maneuvers (see [START_REF] Holst | On Target Manoeuvres in Bearings-Only Tracking[END_REF], [START_REF] Koteswara Rao | Modified Gain Extended Kalman Filter with Application to Bearings-Only Passive Manoeuvring Traget Tracking[END_REF] and [START_REF] Koteswara Rao | Algorithm for Detection of Manoeuvring Targets in Bearings-Only Passive Target Tracking[END_REF]). Concerning the own ship maneuver, this model has been used in the past in order to propose an approximation of the CRLB [START_REF] Nardone | Fundamental Properties and Performance of Conventional Bearings-Only Target Motion Analysis[END_REF], or to optimize the observer's maneuver [START_REF] Le Cadre | Optimizing the Receiver Maneuvers for Bearings-Only Tracking[END_REF], or to improve the optimization algorithms carried out in BOTMA (see for example [START_REF] Pham | Some Quick and Efficient Methods for Bearings-Only Target Motion Analysis[END_REF], [START_REF] Nardone | A Closed-Form Solution to Bearings-Only Target Motion Analysis[END_REF] and [START_REF] Blanc-Benon | Fusion of Reduced-Rank TMA Estimates[END_REF]). This model is justified since in practice, a ship or a submarine prefers to change its heading at a constant speed.

Of course, a more realistic model must incorporate a part with constant turn rate between two legs: this model is widely adopted for the own ship (see [START_REF] Koteswara Rao | Modified Gain Extended Kalman Filter with Application to Bearings-Only Passive Manoeuvring Traget Tracking[END_REF], [START_REF] Sanjeev Arulampalam | Bearings-Only Tracking of Manoeuvring Targets Using Particle Filters[END_REF]) and for the source as well (see [START_REF] Kirubarajan | D Bearings-Only Tracking of Maneuving Targets Using a Batch-Recursive Estimator[END_REF], [START_REF] Sanjeev Arulampalam | Bearings-Only Tracking of Manoeuvring Targets Using Particle Filters[END_REF] and [START_REF] Ristic | Tracking a Manoeuvring Target Using Angle-Only Measurements: Algorithms and Performance[END_REF]). Our proposed algorithm will be applied to some scenarios based on this model. Note that some authors have considered scenarios with a circular motion for the observer (see [START_REF] Farina | Target Tracking with Bearings-Only Measurements[END_REF] and [START_REF] Ristic | Tracking a Manoeuvring Target Using Angle-Only Measurements: Algorithms and Performance[END_REF]).

In all papers dealing with bearings-only maneuvering target motion analysis (BOMTMA), the observer is supposed to maneuver before the source ( [START_REF] Holst | On Target Manoeuvres in Bearings-Only Tracking[END_REF], [START_REF] Koteswara Rao | Modified Gain Extended Kalman Filter with Application to Bearings-Only Passive Manoeuvring Traget Tracking[END_REF], [START_REF] Koteswara Rao | Algorithm for Detection of Manoeuvring Targets in Bearings-Only Passive Target Tracking[END_REF]). Most of them consider recursive solutions. Here, we claim and prove that BOMTMA is feasible for a non-maneuvering own ship under non restrictive conditions. And a batch estimator reached by a numerical routine is also proposed.

The paper consists of four main sections and a conclusion:

Section II is devoted to the presentation of BOMTMA when the source's trajectory is composed of two legs. A criterion of observability is given with its practical consequences.

The maximum likelihood estimator is proposed in section III, the observability condition being satisfied. Some numerical details are given.

Monte-Carlo simulations are also presented together with the ultimate performance given by the CRLB. Robustness to a non abrupt change is illustrated by an example.

Section IV presents an extension of the BOMTMA to the cases of a smooth change of heading. Robustness to a small change in speed is also illustrated.

Section V presents some tactical considerations: the respective performances of BOTMA and BOMTMA made by each mobile against the other are compared.

The conclusion follows.

This paper is the extended version of [START_REF] Jauffret | Bearings-Only TMA Without Observer Maneuver Proceedings of the International Conference on Information Fusion[END_REF].

II -BOMTMA Observability analysis and associated criterion

A. Notations and problem formulation of the

BOMTMA

We consider an observer (or own ship) and a source (or target) moving on the same plane. In this section, the Cartesian coordinates will be used.

The source moves with a constant velocity vector and changes suddenly its course at time to have a new heading up to the end of the scenario. More precisely, its complete motion is composed of two legs: during

M t [ ] M t , t 1
, the source is traveling with the velocity vector and during

t ( ) ( ) ( ) ( ) ⎪ ⎩ ⎪ ⎨ ⎧ - + = - + = i , S M M S S O M M O O V t
( ) ( ) ( ) ( ) ( ) [ ] [ ] ⎪ ⎩ ⎪ ⎨ ⎧ = - = = - = T i , R i , R O i , S i , R T R R O S R y x V V V t y t x t P t P t P & & (2)
Hence, the relative motion equation is given by

( ) ( ) i , R M M R R V t t t P ) t ( P - + =
with the same convention as previously :

1 = i if M t t ≤ and 2 = i if . M t t >
As usual in BOTMA, the noise-free bearing is then given at time t by

( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = - t y t x t θ R R 1 tan (3)
A typical scenario and the associated notations are given in Fig. 1.

B. Bearings-equivalent trajectories on each leg

For the sake of clarity, we will need the two following definitions:

( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ] ⎪ ⎩ ⎪ ⎨ ⎧ - = - - = - . 1 2 1 2 t y t y λ t y t y t x t x λ t x t x O S O S O S O S
The parameter is called the homothetic ratio.

λ

For one leg only, it is well known (see [START_REF] Lindgren | Properties of Bearings-only Motion Analysis: An interesting case study in system observability[END_REF], for example) that any homothetic trajectory is bearings-equivalent to the actual one: Indeed, from (3), we get

( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ - - = - - t y λ t x λ t y t y λ t x t x λ t θ R R O S O S 1 1 tan tan . Conversely, if
is not a constant (which will be assumed subsequently), any bearings-equivalent trajectory is a homothetic trajectory. This (intuitive) statement is not trivial and we propose a proof in the appendix.

( ) t θ

As a consequence, on each leg, the sole trajectories at a constant velocity vector and producing the same data given by (3) are defined, in the relative coordinate system whose origin is the current location of the observer, by their relative positions and velocity vectors denoted and

) λ , t ( i R P ( ) i i , R λ V as follows ( ) ( ) ⎪ ⎩ ⎪ ⎨ ⎧ = = i , R i i i , R R i i R V λ λ t P λ ) λ , t ( V P ( 4
)
The homothetic ratios are strictly positive.

i λ

The corresponding trajectory is then given by (in an absolute coordinate system)

( ) ( ) ( ) ⎪ ⎩ ⎪ ⎨ ⎧ + = + = t P ) λ , t ( ) λ , t ( V λ λ O i R i S O i i , R i i , S P P V V (5)
Using ( 4) for ( 5), we get

( ) ( ) ⎪ ⎩ ⎪ ⎨ ⎧ + = + = t P ) t ( P λ ) λ , t ( V V λ λ O R i i S O i , R i i i , S P V (6)
Then, using ( 2) for ( 6), we obtain

( ) [ ] ( ) ( ) [ ] ( ) ⎪ ⎩ ⎪ ⎨ ⎧ + - = + - = t P t P t P λ ) λ , t ( V V V λ λ O O S i i S O O i , S i i i , S P V (7)
Since the source's position at the maneuver time is unique, i.e , we have the equality

) λ , t ( ) λ , t ( M S M S 2 1 P P = λ λ λ = = 2 1 . So, (7) is simplified, as follows ( ) [ ] ( ) ( ) [ ] ( ) ⎪ ⎩ ⎪ ⎨ ⎧ + - = + - = t P t P t P λ ) λ , t ( V V V λ λ O O S S O O i , S i , S P V (8)
Fig. 2 illustrates the source's trajectory and a homothetic source's trajectory in the relative coordinate system linked to the observer.

C. Observability analysis when is known

M t

The question of observability can be reduced to the question concerning the uniqueness of the parameter λ : if another parameter λ such as

( ) ( ) λ λ , S , S 2 1 V V =
(fundamental assumption of constant speed) exists, then the trajectory is not observable, otherwise it will be.

So, let's consider the following difference:

( ) ( ) 2 2 2 1 λ λ ) λ ( v Δ , S , S V V - = (9)
Using ( 8) for ( 9), we get

( ) ( ) 2 2 2 1 O O , S O O , S V V V λ V V V λ ) λ ( v Δ + - - + - =
which can be written as

( ) ( ) 2 2 2 1 1 1 O , S O , S V λ V λ V λ V λ ) λ ( v Δ - + - - + = . Using the classic formula Y X Y X Y X T 2 2 2 2 + + = + , we get ( ) ( ) 2 1 1 2 1 2 , S T O , S T O V V λ λ V V λ λ ) λ ( v Δ - - - = , or after factorizing ( ) ( ) 2 1 1 2 , S , S T O V V V λ λ ) λ ( v Δ - - = .
We are interested in cases where

0 = ) λ ( v Δ . If ( ) 0 2 , 1 , ≠ -S S T O V V V , then 0 ) ( = Δ λ v if and only if or 0 = λ 1 = λ . The solution 0 = λ must be discarded (degenerated case). The other solution ( 1 = λ
) is acceptable and corresponds to the actual trajectory. So the condition of observability follows:

( ) 0 2 1 ≠ -, S , S T O V V V (10)

D. Practical consequences

The previous analysis implies some practical consequences:

1) The BOMTMA system is not observable if a)

, i.e. the observer is motionless,

[ ] T O V 0 0 = b) or
, the source does not change its course, which is not our assumption,

2 , 1 , S S V V = c) or ( ) 2 , 1 , S S O V V V - ⊥
. Note that this condition holds whatever the initial positions of the two mobiles.

2) The "special symmetric geometry" of the scenario proposed in [START_REF] Nardone | Fundamental Properties and Performance of Conventional Bearings-Only Target Motion Analysis[END_REF] (see Fig. 3)

satisfies the last condition ( ) 2 , 1 , S S O V V V - ⊥
: when the role of target and ownship are inverted, such a scenario allows one to perform an efficient BOTMA of our observer, while our observer cannot estimate the source's trajectory! Fig. 3 : The "special symmetric geometry" (copy from [START_REF] Nardone | Fundamental Properties and Performance of Conventional Bearings-Only Target Motion Analysis[END_REF] page 780)

3) In practice, a maneuver detection informs us about the existence of the maneuver between and . Nevertheless, the two-leg trajectory of the source

1 t F t
can still be unobservable, see for example, the "special symmetric geometry" of the Fig. 3.

4)

If the trajectory of the source is composed of several legs, the BOMTMA observability is guaranteed if two consecutive velocity vectors satisfy the criterion [START_REF] Arnold | Target Parameter Estimation Using Measurements Acquired with a Small Number of Sensors[END_REF].

5) This conclusion is not in contradiction with previous analyses presented in [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF] and [START_REF] Payne | Observability conditions for angle-only tracking[END_REF], for example. Indeed, in those papers, the trajectory of the source is defined by a polynomial model, which is not the case here.

E. Observability analysis when is unknown

M t

We are going to prove ab absurdo that is observable. If is not observable, then there exists another instant, say (arbitrary

M t M t N t N M t t < ),

] ] ]

] ]

] N t , t 0 [ F N t , t F M t , t N t During [ then [
, the two trajectories are homothetic. More precisely, the two homothetic velocity vectors of the ghost source are homothetic to the same one. In mathematical words,

N M t , t F N t , t
during , the actual source has a constant relative velocity vector :

[ F M t , t 2 , R V during then [
, the ghost source has two constant velocity vectors, homothetic to the actual source's velocity vector : and , linked to by

[ N M t , t ] ] F N t , t ' , R V 2 ' ' , R V 2 2 , R V ⎪ ⎩ ⎪ ⎨ ⎧ = = 2 2 2 2 , R ' ' , R , R ' , R V ' ' λ V V ' λ V At
, the position of the ghost source is unique, so

N t λ ' ' λ ' λ = =
. We deduce that . Hence the ghost source does not maneuver at , which is in contradiction to our assumption. Hence, is observable.

'' , R ' , R V V 2 2 = N t M t
Note that in this proof, the condition of observability ( 10) is not used. As a consequence, can be estimated even though the whole trajectory is not observable.

M t

III -Performance and robustness of the

BOMTMA

We are going to construct an estimator under the observability condition (10):

( ) 0 2 , 1 , ≠ -S S T O V V V .
The observer collects the measured bearings at time :

k t ( ) k k k ε t θ β + = , for K , , k L 1 =
where k ε is the additive noise corrupting the data. As usual, the random vector is assumed to be Gaussian. It is zero-mean and its covariance matrix is equal to

[ T K ε ε L 1 ] ( )

A. known

M t

So far, the two-leg trajectory of the source is defined by three 2D vectors:

and . Because ( ) ( ) ( ) [ ] 1 , S T M S M S M S V , t y t x t P = 2 , S V S , S , S v V V = = 2 1
, any five component vector

( ) ( ) [ T , S , S S n S n S n h , h , v , t y , t x Z 2 1

=

] can be helped as state vector ( is the source's heading during leg #i ). For convenience, we have chosen the state vector at time denoted

i , S h M t ( ) ( ) [ ] T , S , S S M S M S M h , h , v , t y , t x Z 2 1 = .
Considering that the noise-free bearings ( ) k t θ are a function of and of , the noise-free measurement equation is given by

M Z M t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) M k , t , Z θ t y h v t t t y t x h v t t t x t , Z θ M M , k k O , S S M k M S k O , S S M k M S M M k ≤ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - + - - + = - if cos sin tan 1 1 1 1 (leg 1), ( ) ( ) ( ) ( ) ( ) ( ) ( ) M M , k k O , S S M k M S k O , S S M k M S t , Z θ t y h v t t t y t x h v t t t x 2 2 2 1 cos sin tan = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - + - - + = - otherwise (leg2).
Note that with the choice of ( n Z M n ≠ ) as state vector, the mathematical expression of is more complicated.

( M n k t , Z θ ) 1)

Cramèr-Rao lower bound

The CRLB being equal to the inverse of the Fisher Information Matrix (FIM, see [START_REF]Van Trees Detection, Estimation, and Modulation Theory[END_REF] [29]), we compute the latter as follows

( ) ( ) ( ) M M k T Z K k M M k Z k M t , Z θ t , Z θ σ Z F M M ∇ ∇ = ∑ =1 2 1
where is the gradient operator. Then

M Z ∇ ( ) M Z F 1 - is numerically evaluated.
We can easily compute from

( n Z F ) ( ) M Z F with the classic transformation ( ) ( ) n , M M T n , M n Φ Z F Φ Z F =
where is the Jacobian of the transformation , defined by

n , M Φ M n Z Z a ( ) ( ) ( ) ( ) ( ) ( ) ⎪ ⎩ ⎪ ⎨ ⎧ - + = - + = , h v t t
1 = i M n < 2 = i ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - - - - - = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 sin cos 1 0 0 cos sin 0 1 1 , 1 , 1 , 1 , , S S n M S n M S S n M S n M n M h v t t h t t h v t t h t t Φ , if M n t t < and ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - - - - - = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 sin 0 cos 1 0 cos 0 sin 0 1 2 , 2 , 2 , 2 , , S S n M S n M S S n M S n M n M h v t t h t t h v t t h t t Φ , if . M n t t >

2) Maximum likelihood function and algorithm aspects

The maximum likelihood estimator (MLE) being equivalent to the least squared estimator if the additive noise is Gaussian and temporally uncorrelated, we have to minimize the following quadratic criterion

( ) [ ] ∑ = - = K k M M k k k M M t , Z θ β σ ) t , Z ( C 1 2 2 1 .
We have employed the classic Newton-Raphson scheme, switching during iterations with the Levenberg-Marquardt algorithm (LMA) when necessary (for example, when the Hessian is not numerically positive definite, see [START_REF] Dennis | A User's Guide to Nonlinear Optimization Algorithms Proceedings of IEEE[END_REF]). The initialization point is chosen The returned vector is denoted

( ) ( ) [ ] T , S , S S M S M S M h , h , v , t y , t x Ẑ ˆ0 2 0 1 0 0 0 0 = ( ) ( ) ( ) ( ) ⎪
( ) ( ) [ ]

3) Results

For Monte-Carlo simulations, the chosen scenario is defined as follows: The observer starts from the origin; its speed is 5 m/s and its heading is 90°.

Meanwhile, the source which has a speed of 4 m/s begins its trajectory at with the initial heading of 90°. At time The standard deviation of the measurement noise is equal to 1°. We run 500 Monte Carlo simulations.

The statistical analysis of the 500 estimates are summarized in Table 1 and illustrated in Fig. 4 where the initial positions of the source (respectively the observer) is indicated by the letter "S" (respectively the letter "O"). The given statistics concern the final position of the target. The 90% confidence ellipse (corresponding to the CRLB) is plotted together with the 500 position estimates.

We note that the biases are negligible and the standard deviation of each component is close to the corresponding element of the CRLB. Obviously, the estimator performs correctly: The relative accuracy of the estimated range is

1) CRLB

The Fisher information matrix w.r.t. the "extended state vector" ( is not defined, since the log-likelihood function depends on

) M M t , Z ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ t Δ t M (ceiling function of t Δ t M ) and is not differentiable w.r.t. : M t ( ) ( ) [ ] ( ) [ ] ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ - - - - × = ∑ ∑ + = = 2 1 2 , 2 2 1 1 , 2 1 , 1 2 1 , 1 2 1 exp Cste , , , Ln K M k M M k k k M k M M k k k K M M t Z θ β σ t Z θ β σ β β t Z L L
Consequently, the computation of the gradient of the log-likelihood function is hopeless. We have implicitly assumed that is a multiple of .

M t t Δ

The Cramèr-Rao Lower Bound previously computed is here optimistic, since M is unknown and information will be shared for its estimation. However, it keeps helping as a lower bound.

2) Algorithm aspect

The following procedure is applied: for each

{ } 2 3 - ∈ K , , M L , M Z ˆ minimizes . Then we retain ) t , Z ( C M M M Ẑ
ˆ (and the associated M ˆ) , for which

( ) M M ˆt , Z Ĉ is the least.
Remark: a test of maneuver detection (for example the one proposed in [START_REF] Holst | On Target Manoeuvres in Bearings-Only Tracking[END_REF] or [START_REF] Blanc-Benon | Trajectographie Passive en Présence d'Erreur de Modèle: Utilisation des Résidus[END_REF]) can be applied on the bearings, yielding a coarse estimate M ~ of M . Then,

a

3) Results

The Newton-Raphson (again switching with the LMA) algorithm has been initialized as previously for each M .

The statistical analysis of 500 Monte Carlo simulation runs is presented in Table 2 and illustrated in Fig. 5. In Table 2, the results about the state vector estimator and those about the time estimator are separated by three lines.

M t

The bias and the standard deviations are a little bit greater than in the previous case (see table 1). We note that the maneuver time is correctly estimated. The relative accuracy of the estimated range is

% ρ σ ˆρ 4 =
at the final time while its relative mean-square error is equal to 4.07%.

C. Robustness of the BOMTMA to a non abrupt change

In practice, the source does not change its heading instantaneously. This is why, it is highly important to check how robust the BOMTMA algorithm is when this assumption is violated. A new scenario has been considered: from now, the source has a constant turn rate between its two legs at constant speed.

More precisely, its trajectory is composed of A first leg between [ )

1 1 M t , t
Then, a arc of a circle during [ )

2 1 M M t ,

IV -Extension of the BOMTMA algorithm to a non abrupt change of heading

In the previous section (III C), we have used the leg-by-leg trajectory model. In this section, we are going to consider the correct trajectory model, taking into account the actual geometry of the source's trajectory: the source keeps a constant speed, but its trajectory is composed sequentially of a first leg at constant velocity vector, followed by a turn at constant turn rate and a second leg. The case where the beginning and the end of that turn are known will be considered; then the case where they are unknown will be studied.

A. and known.

1 2 - ≤ ≤ K k M ( ) ( ) ( ) ( ) ( ) ( ) ⎪ ⎩ ⎪ ⎨ ⎧ - + = - + = 2 2 cos sin h v t Δ K k t y t y h v t Δ K k t x t x K k K k
For (constant turn) :

1 2 1 - ≤ ≤ M k M ( ) ( ) ( ) ( ) ( ) ( ) ⎪ ⎪
⎩ ⎪ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - - + + + - + - - - - + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - - + + + - + - - - - + = ∑ ∑ - = - = k M i K k k M i K k M M M i k h M M i k M h h K M t Δ v t y t y M M M i k h M M i k M h h K M t Δ v t x t x 2 2 1 1 2 1 2 1 2 2 1 2 2 1 1 2 1 2 1 2 2 1 2 2 1 1 1 cos cos 1 1

sin sin

For (first leg) :

1 1 1 - ≤ ≤ M k ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - + + - + - - - - + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - + + - + - - - - + = ∑ ∑ - = - = 1 1 1 1 2 2 1 2 1 2 1 2 2 1 1 1 1 2 2 1 2 1 2 1 2 2 cos 1 1 1 cos cos sin 1 1 1 sin sin 1 2 1 2 h M k M M i h M M i M M h h K M t Δ v t y t y h M k M M i h M M i M M h h K M t Δ v t x t x M M i K k M M i K k
The expression of and the computation of the FIM w.r.t.

follow. We did not analyze the observability; however, the FIM is numerically regular for the simulated scenario. This guarantees local observability for this scenario [START_REF] Jauffret | Observability and Fisher Information Matrix in Nonlinear Regression[END_REF].

(

2 1 M M K k t , t , Z θ ) K Z The quadratic criterion to minimize is ( ) [ ] 2 1 2 1 2 2 1 1 ∑ = - = K k M M K k k k M M K t , t , Z θ β σ ) t , t , Z ( C .
Again, 500 Monte Carlo simulations were made. Their statistical properties are given in Table 4 and the set of estimated positions are displayed in Fig. 7. The relative accuracy of the estimated range is % 83 . 4 ˆ= ρ σ ρ and its relative meansquare error is equal to 5.02%.

) t , t , Z ( C M M K 2 1 K Z 1 M t 2 M t 1 M and 2
M in the search set { } 2 3 - K , ,L (and ),
we minimize . Again, we retain

2 1 M M < ) t , t , Z ( C M M K 2 1 1 M ˆ and 2 M ˆ
(and the associated K Z ˆ) , for which the criterion is the least. The initialization is given by the output of the BOMTMA procedure, i.e K Z ˆ as described in section III B.

The statistics of the 500 Monte Carlo simulation runs are presented in Table 5 and illustrated in Fig. 8. The biases are of the same order as those presented in Table 4, except for the biases of the second heading which are surprisingly smaller. The standard deviations are close.

The relative accuracy of the estimated range is % 48 . 5 ˆ= ρ σ ρ and its relative mean-square error is equal to 5.56%. From theses simulations, it seems that the lack of knowledge of and is not a handicap.

( ) ( ) [ ] T , S , S , S S K S K S K h , h , v , v , t y , t x Z 2 1 2 = ′ 1 M 2 M t
The algorithm presented in IV B is then applied to 500 simulated Monte Carlo runs.

The results are summarized in Table 6 and illustrated in Fig. 9. The relative accuracy of the estimated range is % . ρ σ ˆρ 81 4 =

. Its relative mean-square error is equal to 7.47%. We observe a degradation of the performance which can be evaluated by the bias and the standard deviation of each component of

K Z ˆ.
Again, the estimator can be used advantageously in a real situation.

V -Tactical aspect

Let two mobiles be denoted by A and B. Mobile A moves at a constant velocity vector while mobile B moves on a leg at a constant velocity vector, then changes suddenly its heading but keeps its speed (in Section II A, A was the observer and B the source).

Each of them shall try to localize the other: mobile A by the BOMTMA and mobile B by conventional BOTMA.

We compare the accuracy of the respective estimated range, when the initial range (of the same scenario as that given in section III A 2) varies from 6 km to 40 km. This range accuracy is evaluated with the CRLB, the standard deviation of the measurements being equal to 1° for each platform. The values are illustrated in Fig. 10. As expected, conventional BOTMA has advantages over BOMTMA; but mobile B must have maneuvered to realize a performing TMA, which implies a loss of discretion and other disadvantages from a tactical point of view.

We must emphasize the fact that, as noticed in paragraph II E, a maneuver corresponding to the "special symmetric geometry" will guarantee the supremacy of TMA of B on A, whatever the initial azimuth. An open question is then: does there exist a scenario with special geometry for which the BOMTMA overtakes the conventional BOTMA?

VI -Conclusion

Two new bearings-only tracking methods have been proposed in this paper: neither requires a maneuver of the own ship provided the source's speed does not change during the scenario and its trajectory is composed of two legs. The maneuver times need not to be known. The basic assumptions are:

-For the first method, the change of headings is instantaneous: in that case, it has been proven that the source's trajectory is observable if the bearing rate is not equal to zero and the difference of the two velocity vectors is not orthogonal to the velocity vector of the observer.

-For the second one, the two legs are separated by a turn at constant rate.

The observability has not been rigorously established; in the cases treated, we have used the non-singularity of the Fisher information matrix as local observability criterion.

In both cases, a batch estimator has been proven to perform properly, in agreement with the Cramèr-Rao lower bound. Table 7 shows that the performance is excellent since the relative mean square error on the final range (~9 km) is less than 5.6 %.

When the assumptions are violated, i.e. the speed of the target is not the same on each leg, the results are still acceptable with a relative mean square error on the final range less than 7.5 %. In the future, recursive algorithms could be applied to the same type of scenario.

For a same family of scenario, the performance analysis via the CRLB reveals the superiority of the BOTMA over the BOMTMA. This analysis must be extended to a large number of cases, before a definitive conclusion can be reached.

More generally, the observability analysis poses new tactical problems which merit a deeper analysis.

( ) 0 = t y R ( ) π t θ = ( ) 0 > t x R and ( ) π t θ - = if ( ) 0 < t x R , We start from ( ) ( ) ( ) ( ) ( ) [ F R R R R t , t t y * t t * t y x * t t * t x t θ 1 1 tan ∈ ∀ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + - + = - & & ] (A.1)
where is a reference time.

* t

The polar coordinates of the relative position vector and of the relative velocity vector are

( ) ( ) ( ) T t t θ ρ ,
and (

, respectively. The correspondence between Cartesian and polar coordinates is

) T R R h , v ( ) ( ) ( ) ( ) ( )⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ t θ t θ t ρ t y t x R R cos sin and . (A.2) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ R R R R R h h v y x cos sin & & Reporting (A.
2) in (A.1), then, the noise-free measurement is given by The set of solutions is hence We, hence, have to solve the system

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - + - + = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - + - + = - - R R R R R R R R h * t ρ v *

Fig. 1 :

 1 Fig. 1: Example of observer's and source's trajectories in an absolute

Fig. 2 :

 2 Fig. 2 : Example of source's trajectory in the relative coordinates system linked to the observer.

  and a (ghost) source whose trajectory is composed of two legs : one during [ and one during . In other words, the bearings collected during [ are generated by a non-maneuvering target (the true one) and also generated by a maneuvering target (at time).

  Here we have arbitrarily chosen the range (at) M t m ρ M 2000 =, which is far from the true one. The routine stops as soon as the last five iterations provide five quasi identical vectors (their relative distance is less than 1%) or the maximum number of iterations is reached (this number has been fixed to 20, in our program).

(

  it suddenly changes its course and its new heading is equal to 240°. The total duration of the scenario is 30 min. corresponding to a number of measurements equal to 450

)

  at the final time; its relative mean-square error is equal to the same value.Note the empirical standard deviation of the final range is defined by , where is the estimate of at the i-th run and i ρ ˆρ ρ is the empirical mean of the 's, while its relative mean-square error is defined by i

Fig. 5 :

 5 Fig. 5: Results of the Monte-Carlo runs ( unknown) M t

Fig. 6 :

 6 Fig. 6: Results of the Monte-Carlo runs with the leg-by-leg model

Fig. 8 :.

 8 Fig. 8: Results of the Monte-Carlo runs ( and unknown) with the

Fig. 9 :

 9 Fig. 9: Results of the Monte-Carlo runs ( and unknown) with a

Fig. 10 :

 10 Fig. 10: relative accuracy of the range for each platform.

  satisfies another one, for example 0 2 = z . In short, the 3-dimensional state vector Y is not observable when is constant.In this case, there does not exist an open subset of [ ] Note that because the bearing is not a constant, the contra poses of (A6) and (A7) stand:

Table 1 : performance at the final time ( known).

 1

	t	K	t	M

S O Fig. 4: Results of the Monte-Carlo runs ( known) M t M Z

  smaller search interval [

	M	Min M ,	Max	]	centered on	M ~ can be used for the
	algorithm.

Table 2 : performance at the final time ( unknown).

 2

	t	K	t	M

Table 3 : performance at the final time (non abrupt change of heading)

 3 t The other parameters are unchanged.Again, we have had recourse to 500 Monte Carlo simulations and we have used the BOMTMA algorithm of section III B. The results concerning this population are given in Table3and illustrated in Fig.6. global performance is degraded especially concerning the estimates of the headings which are now biased. The relative accuracy of the estimated range is

	And finally a second leg during [ t 2 M	,	t	K	]

Table 4 : performance at the final time ( and known).

 4

	t	K	t	M	1	t	M	2

Results of the Monte-Carlo runs with the correct model B. and unknown.

	t	M	1	t	M	2
	The minimization of	w.r.t.	,	and	follows the same
	scheme : for each

Table 5 : performance at the final time

 5

	t	M	1

Table 6 : performance at the final time K

 6

	t

Table 7 : Synthetic results about the relative range accuracy at the final time Algorithms Assumptions

 7

					Relative
			Relative range
					mean-square
			accuracy	ρ σ ρ .	error on the
					range
		M t known	3.3 %		3.3 %
	BOMTMA	M t unknown	4. %		4.07 %
		Non abrupt change
			6.45 %	7.17 %
		of heading

.

  The question is then to know if the vector Y is observable.If the answer is yes, the system is observable; otherwise it is not.Two situations can be met: the first one for which ( ) t θ is a constant and the second one when is not. We are going to analyze these two situations.

	The question of the observability can be reformulated by
	tan	-1	⎢ ⎣ ⎡	cos sin	y y 1 1	+ +	( ( t t	--	t t	* *	) )	y y	2 2	cos sin	y y 3 3	⎥ ⎦ ⎤	=	tan	-1	⎢ ⎣ ⎡	cos sin	z z 1 1	+ +	( ( t t	--	t t	* *	) ) z z	2 2	z z 3 3 cos sin	⎥ ⎦ ⎤	t ∀	⇒	{	Z	=	Y	} ?
																																							(A. 5)
																				θ	( ) t
	I -	θ	( ) t	is constant
	This case is equivalent to ( ) t θ &	=	0	∀	t	∈	[ ] F t , t 1	). We exploit this derivative :
																θ &	( ) t	=		0		∀	t		∈	[ ] F t , t 1	⇔	dt d	tan	θ	( ) t	=	0	∀	t	∈	[ ] F t , t 1
	t Using (A.4), we obtain θ tan		1				ρ ρ	* * t t	θ θ cos sin	t t	* *	t t	t t	* *	v v	h h cos sin
					y	2	sin	y	3	[ cos		y 1	+	tan ( t -	1 t		*	* y cos t θ 2 sin ) y	3	]	t -	y	* cos t 2	* [ t v ρ y sin sin y 1 3	h +	( t	-	t	*	)	y	(A.3) y sin 2	3	]	=	0	t ∀
	simplified into																				θ cos y	t 2	* sin	(	y	t 3	-	t y 1	)	=	0	cos t ∀
	In (A.3) the bearings are completely described by the 3-dimensional vector
	Y	=	θ ⎢ ⎣ ⎡	( ) * t		( ) R * t v ρ				h	R	⎥ ⎦ ⎤	T			=		[	y 1	y	2	y	3	] T
	Let		Z	=	(	z 1			z	2			z	3	)	T		be another state vector, such that
																		θ	( ) t		=		tan	-1	⎢ ⎣ ⎡	cos sin	z z 1 1	+ +	( ( t t	--	t t	* *	) ) z z	2 2	cos sin	z z 3 3	⎥ ⎦ ⎤	t ∀	(A.4)

, S

Appendix

For all the time t (except for one, at most), the bearing is given by (see [START_REF] Lindgren | Properties of Bearings-only Motion Analysis: An interesting case study in system observability[END_REF] From (A8), we deduce (the solution

is not physically acceptable, since both define the line of sight). As a consequence, (A9) becomes

From (A10), we get or

, since neither nor are allowed to be zeroed (otherwise would be a constant).

There are two possibilities only:

which is incompatible with our assumption ( ( )

As a consequence, the case

Case 2:

The solution incompatible with our assumption ( is not constant).

( )

Hence the sole solution is .

All the components of Y and Z have been proved equal. As a consequence, the answer to question (A5) is yes: The 3-dimensional state vector Y is observable when is not constant.

( ) t θ

Summary :

If the bearing rate is not equal to zero, then the set of trajectories providing the same noise-free bearings as those originating from the source, are homothetic to the source's trajectory, both w.r.t. the observer's trajectory.

More precisely, they are defined by and

where the homothetic ratio is strictly positive.