INTRODUCTION & BACKGROUND

since the 1970's, aksum stands in area under discussion since the development of culture in the ethiopian highlands appears to be, even now, the subject of much debate. What role did it play, when did it emerge to prominence, and what were its relations with pre-existing societies and cultures? and above all, to what extent was aksum a conduit for external influences, Mediterranean and south arabian, or even vice versa? the history of the kingdom of aksum, between the end of the 2 nd and the end of the 6 th century, is only known by few various sources: Ge'ez, pseudo-sabaean and Greek inscriptions found in ethiopia; sabaean and himyarite inscriptions discovered in Yemen which make mention of conflicts in the 3 rd and 6 th centuries; archaeological data, especially around aksum; Greek and roman sources which relate the conversion of ethiopia to christianity in the 4 th century, as well as the crisis of the 6 th century. several aksumite kings are though already known among others by numerous coins. In the same way, several sites of the so called "pre-aksumite" civilization have been established in tigrai region between the 8 th and the 5 th c. Bc. this civilization had significant contacts with south arabia. script, architecture and cult of several deities have been brought by sabaean immigrants in the first half of the I st millennium Bc. recent excavations at Maqabər Ga'əwa (near Wukro) revealed, for example, the remains of a temple to almaqah, whose cult existed in the kingdom of saba (see I. Gajda, Y. Gebre sellassie and h. Berhe, "pre-aksumite Inscriptions from Maqabər Ga'əwa (tigrai, ethiopia)", in Annales d 'Éthiopie, 24, 2009: 33-48). south arabian influence is as well obvious in economy, trade and social structures. though, data are still insufficient to make it clear.

this first mission (March-april 2010) consisted in a preliminary visit. the aim of it was a general survey of pre-aksumite and aksumite sites (Yeha and around, adowa and around, hawzen and Gulo Makada, the eastern part of tigrai, the region of atsbi-Dera' and the one of Medri senafe), and also a geographical study of the kingdom of aksum, to assert the identification of tribes and regions mentioned in the descriptions of military campaigns.

as several pre-aksumite and aksumite sites around aksum and within the study area proposed for a first survey have recently been excavated or surveyed by other teams, we have only paid a short visit to some of them: Yeha (Iris Gerlach, since november 2009), haulti (holger hitgen, since 2009-2010), Maqabər Ga'əwa (ricardo eichmann & pawel Wolf, since 2009). thus, in agreement with ato Kebede amare Belay, General Manager of the tigrai culture and tourism agency at Mekelle, this preliminary survey concentrated on documenting several sites first surveyed by the ethiopian team in the region of atsbi-Dera, east of Mekelle (fig. 1), and took place from 22 March to 2 april 2010. the investigations are directed by Dr Iwona Gajda of the centre national de la recherche scientifique (cnrs) in paris and team members included:

Dr Fabienne Dugast, centre national de la recherche scientifique, paris (archaeology), Yohannes Gebre sellassie, University of paris-sorbonne (history of aksumite period), hiluf Berhe, University of aksum (archaeology), habtamu Mekonnen, tigrai Bureau of culture and tourism (survey), amare, French center for ethipian studies, addis ababa (driver).

the methodological approach of this first project was to collect artefact samples, in order to make a preliminary location of pre-aksumite and aksumite sites without undertaking any excavations. the final objectives will be to enable the prioritization of areas of eastern tigrai for survey and mapping, and of sites for excavation.

I. OBJECTIVES AND METHODOLOGY

the main tasks were to initiate an archaeological survey. specific fieldwork objectives were to implement a sampling design and initiate the archaeological survey which employed geomantics techniques including Gps (Global positioning system) and GIs (Geographic Information system).

artefact samples have been systematically collected and registered: pottery shreds, iron slag, and even bones. the objectives were to try to identify each site as well as to insure their size by Gps outlining. as pre-aksumite and aksumite artefacts are still not well known, the fieldwork objectives were as well to implement a sampling of each type of ceramics in order to produce dating references. In this sight, this preliminary sampling has been completed with a visit to the national Museum at addis ababa (F. anfray's excavations at Matarā, near senafe, 1963Matarā, near senafe, -1974)), and the Museum at aksum.

II. SURVEY AND COLLECTION OF ARTEFACT SAMPLES

the survey is based on a few sites randomly found and selected by the tigrai Bureau of culture and tourism, at Mekelle (habtamu Mekonnen) and hiluf Berhe (University of aksum). local people has been interviewed to try to get some more information about any ancient objects, structures, megaliths that could have been found recently. some 15 sites have been visited (see annexes I-III): 5 aksumite, and perhaps pre-aksumite sites, 5 sites with megalithic stelae or standing stones, 5 so called "roc churches".

except adi Ba'ekel, east aksum, most of them are concentrated on the eastern side of the road from Wukro to adigrat, and on the edge of a plateau closed to the east by a mountain range, before sloping on the other side to the red sea (fig. 2).

Description of some archaeological sites of Eastern Tigrai (from south to north)

Aggula Čerkos

situated about 10 kms south of Wukro, 13°41'29"6 n / 39°35'03"2 e (altitude 2005 m).

ruins of an aksumite church, erected on a hillock about 1 m high -or even covered with alluvium (?) (phot. 1, 2). the structure was still visible at the end of the 19 th century (cf. carlo conti rossini, Storia d 'Etiopia, Milano, 1925, vol. I, pl. Xl). the author described it as a podium built in freestone, having the outer walls marked by massive corner stones, with a length of somewhat 1 or 1,50 m. the complex' indented plan resulted in a suite of rooms approached on the western side by a stairway of freestone. pillars or columns on stone bases supported the ceiling (fig. 3).

very little remains today as the ruins were further devastated since then. the actual hillock has an extension of some 20 m e/W by 10 m n/s. on top of the it, apparent walls about 0,50 m thick, delineate a complete structure, made up of a suite of two adjoining rooms about 4 x 11 m square each, leading by a kind of anteroom some 4 x 2 m square (phot. 4). a stairway of somewhat 10 steps seems to approach it on its western side, of which hardly any masonry remains (phot. 5). probably columns or pillars on stone bases supported the ceiling, one of which still remains (phot. 6). recently the inhabitants intended to construct a new church on the site. In order to prevent an eventual destruction of archaeological remains and to check the limits of the site, the tigrai Bureau of culture and tourism at Mekelle (tekle hagos and habtamu Mekonnen) made a test excavation next to the church and unearthed pottery shreds and bones. the local administration actually retains the right to inspect a somewhat large area in order to preserve the remains and hopefully allow for a better understanding of its original structure.

Phot. 1: Aggula Čerkos

samples: a few shreds have been collected. Black paste and grey slip allow to date back to the preaksumite period and orange ware to the aksumite one.

-Addi Akaweh situated about 5 kms east of Wukro and near the aksumite site of Maqabər Ga'əwa, discovered in 2007 thanks to the rescue excavation directed by hiluf Berhe (report published in Annales d 'Éthiopie, 24, 2009) and then excavated by a German team (report of r. eichmann & p. Wolf, 2009), 13°45'22"5 n / 39°34'05"3 e (altitude 1993 m).

the church of enda abba Garima in addi akaweh contains three stone incense burners with preaksumite inscriptions: see RIE 9 and RIE 10, edited by r. schneider. a third inscription, addi akaweh 1, has been published by I. Gajda and Y. Gebre sellassie. the text of the inscription is identical with the inscription RIE 9, and enables a minor emendation of the reading. three limestone architectural ornaments of different size were collected from farmlands around the village of addi akaweh and in its vicinity. two of these stones were reused in the construction of the south-west side of the church. see: I. Gajda and Y. Gebre sellassie, "pre-aksumite inscribed Incense Burner and some architectural ornaments from addi akaweh (tigrai, ethiopia), in Annales d 'Éthiopie, 24, 2009: 49-61. samples: a few shreds have been collected in front of the church area, black paste and orange slip as well as orange ware including mica (dating back to the aksumite period?).

In the area of Atsbi-Dera':

-Habes situated about 10 kms north-east of Wukro, on the way to Dera', 13°48'40"4 n / 39°41'06" e (altitude 2575 m). -May Tərəro situated about 10 kms north-east of Wukro, opposite to habes (on the other side of the road), near the village asab'ali, 13°49'13"6 n / 39°41'00"5 e (altitude 2575 m).

Megalithic stele, lying down to the northeast, and broken into two massive pieces. It is an undecorated but hewn monolith of granite, rectangular in crosssection, totalizing more than 3 m length, similar as the one of habes (phot. 8). the base is missing (slight pieces just around), and the head is buried for most of its length:

-the lower part is 1,35 m length, 0,525 m wide, and 0,30 m thick; -the top part is 2,20 m length, 0,525 m wide, and 0,15 m thick.

samples: several pottery shreds have been collected immediately around, all the most dating back to the aksumite period: black ware, of which an amphora (?) handle, black paste and orange slip, orange and grey ware. Inside, the sections are decorated with peculiar architectural ornaments (phot. 12) and paintings with colorful religious scenes that may be dating back to the 4 th century, or maybe 12 th (phot. 13).

samples: none.

-Maryam Kadih situated about 7 kms north of Wukro (1/2 an hour road), 13°52'09"9 n / 39°36'11"7 e (altitude 2266 m).

church and tombs. peculiar stones with a hole are standing round the modern church. on the opposite side of the road and the modern church, a monolith flagstone of schist is lying on the earth, partly buried, but maybe 3 m total square (phot. 14). the priest assures it a royal tombstone.

samples: none can be seen round the church. on the contrary, several pottery shreds could be collected on the opposite site of the church, in the tombstone area, all the most dating back to the aksumite period: black paste and orange slip, rough and fine orange ware.

the Italians cut the ancient site and tombs to build the actual road to adigrat. on the northern side of the road, a few bones are still visible (phot. 15): 3 samples have been collected to hopefully be analyzed in addis ababa and make sure they are human's (no possible dating 14 c). -Atsbi situated about 15 to 20 kms northeast of Wukro (1/2 an hour trail), 13°51'59"3 n / 39°44'09"9 e (altitude 2689 m).

Monolithic stele, standing on the side of the trail.

It is an undecorated but hewn monolith, rectangular in cross-section, totalizing less than 2,50 m length (phot. 16). Its head seems to have been broken, even slightly.

samples: none.

-Gulgul Na'la situated about 2 kms north of atsbi, 13°53'03" n / 39°43'57"2 e (altitude 2732 m).

three monolithic stelae, standing on both side of the trail. these are undecorated and unhewn monoliths, somewhat 2 m length (phot. 17).

samples: none.

-Zarema Giorgis situated about 10 kms northwest of atsbi (1 hour trail), 13°56'17"3 n / 39°40'30"2 e (altitude 2822 m). some 100 m north of the church, flat rocs (limestone) one on the other show a suite of holes, conic in section (phot. 18). this type of holes are also reported by researchers on different archaeological sites.

Phot. 16: Atsbi, monolithic stele.

Phot. 17: Gulgul na'la, monolithic stelae.

Phot. 18: Zarema Giorgis, flat rocs with conic holes.

ancient church: as the priest says, it shelters pre-aksumite artefacts. But no visit was possible and none would be, as, according to the inhabitants, the priests are very conservative.

samples: several pottery shreds have been collected all around, near the church as well as near the flat roc, all the most dating back to the aksumite period: black paste and orange slip, black ware, one of which with a ornament, black paste and external as well as internal orange slip.

-Maryam Madhaniti situated about 30 kms north of Wukro, 15 mn west of Der'a, 13°59'52"6 n / 39°42'42"6 e (altitude 2849 m).

Monolithic stelae, two of which are on the south side of the mountain, another one is standing near the actual church, a few meters to the south.

on the south side of the mountain, the stelae are undecorated but hewn schist monolith, rectangular in cross-section. one is still standing, leaning to the west, somewhat 4 m length, but its head may be broken (phot. 19). on the same side of the mountain and less than 5 m to the north, a second stele is lying above the ground on one side. the most part of it with rounded head is visible, leaning to the west. Its base is still in its original dressed position, leaning to the west and broken at a length of 0,50 m. Its total length is some 6 m (phot. 20).

near the church, a third stele is dressed, undecorated and unhewn, less than 3 m length (phot. 21).

samples: several pottery shreds have been collected all around, near the stelae as well as near the church, all the most dating back to the aksumite period: black paste and orange slip, black paste and external as well as internal orange slip, orange ware including mica, fine orange ware.

Phot.19-20: Maryam Madhaniti, two monolithic and hewn stelae.

Phot.21: Maryam Madhaniti, monolithic stele near the church.

The area of Senafe:

-Addi Gelemo situated about 40 kms north of Wukro, 2 hours trail from edaga hamus, and ½ an hour walk, just near Magbar Weldi negus, 14°04'27"6 n / 39°44'37"5 e (altitude 2920 m).

the site is situated in the district of tabia hawila, in the region of asseraw, at the edge of the eastern plateau of tigrai. In 1953/1954, the local people discovered by chance, under a slab of stone looking to the east, several pre-aksumite and aksumite objects of different origins (phot. 22-24). see: e. Bernand, a. J. Drewes and r. schneider, Recueil des inscriptions de l'Éthiopie des périodes préaxoumite et axoumite, paris, 1991 [RIE], who describe the site (p. 43) and the inscribed objects:

-a pre-aksumite incense burner with a royal inscription of the rbh sovereign (mukarrib) of Da'amat and of saba', reigning in the 7 th or in the 6 th century Bc: RIE 8: 76-77, pl. 5-6 (phot. 22); -a pre-aksumite statue of a woman on an inscribed base dating from the 7 th century Bc: RIE 52:

131-132, pl. 42 (pho. 23); -a bronze votive object with a royal inscription of the aksumite sovereign Gadarat reigning in the 3 rd century aD: RIE 180: 219-220, pl. 89 (phot. 24); -several bronze bowls coming from the nile valley.

It is difficult to understand why these objects which come probably from different sites and dating from different periods have been found together in this remote place in the high mountains. the researchers supposed, that it had been hidden there by robbers. the use of the site is largely hypothetic as no structures are visible (phot. 25). all the same, several pottery shreds have been collected all around, on the terraced land, and make it possible to have been inhabited in the aD 1 srt c., maybe after. the area seems to cover about 2000 m²: shreds have been collected on the slope of the hill, but result probably of erosion (phot. 26).

Addi Gelemo Phot. 22: Pre-Aksumite incense burner.

Phot. 23: Pre-Aksumite statue of a woman.

Phot. 24: Bronze votive object with royal inscription of Gadarat.

samples: several pottery shreds have been collected, all the most dating back to the aksumite period, maybe the pre-aksumite one: black paste and orange slip (1 vase's rim = pottery beaker?), black paste and external as well as internal orange slip, orange ware including mica (1 vase's rim, 1 with stripe pattern [chevron], other patterns), fine orange ware.

-Wakarida situated about 70 kms north of Wukro, 2 hours trail from edaga hamus, east and near sawna, 14°16'56"4 n / 39°43'31"9 e (altitude 2343 m) -surrounded by chain of mountains, according to habtamu Mekonnen, and from north to east and south to west: Daima, afedadae, Dagaraebe and arebata.

surveyed in 2004 by the tigrai Bureau of culture and tourism at Mekelle (tekle hagos and habtamu Mekonnen): see their report, 2004: 16-18.

several structures are still apparent on the hill, consisting on a regular suite of adjoining rooms, somewhat 3 m square, along the slope. the walls are about 0,50 m thick, and made of rubble-stone (phot. 27). the area could cover some 5 ha, from top to mid-slope of the hill, looking north and east, on the valley. trails open north and east on the red sea direction.

local people use the stones as building material for their own houses, which are erected mostly on ancient structures. a complex has been found on the north side of the hill, and consists of 4 adjoining rooms, arranged 2 by 2 (phot. 28). In the middle of one of the rooms was a pillar like wall, actually destroyed by the inhabitants. More structures arose on the both sides of the complex.

samples: local people have found several objects, most of them nearby the pillar like wall, actually destroyed (phot. 29-30):

-small stone figurine, engraved on both sides, 0,14 m length, 0,03 m wide (phot. several pottery shreds have been collected all over the site, all the most dating back to the preaksumite and aksumite periods: black paste and orange slip, orange ware including mica, fine orange ware… (phot. 36). a few fragments of a (?) footwasher have been also collected (phot. 37).

Near Feresmay:

-Adi Ba'ekel situated about 40 kms east of aksum, 1 hour trail, 14°11'42"9 n / 39°04'19"7 e (altitude 2042 m).

the site has been surveyed by hiluf Berhe and his students (University of aksum), on inquiry of local people who had excavated an inscription on a megalith stone (phot. 38).

three sites have been surveyed: 1/ near the trail, where the fragmentary pre-aksumite inscription was found, as well as several other building stones some 0,20 m underground -14°11'42"9 n / 39°04'19"7 e (altitude 2042 m): none is visible but several pottery shreds indicate an area of somewhat 2000 m² (phot. 39). the identification of the site remains hypothetic: the inscription and the building stones are maybe of a public monument. Phot 39: The "first" site where the inscription has been found.

samples: a few pottery shreds have been collected, dating back to the aksumite periods: black and orange ware (including a handle…).

2/ some 100 m northeast, at the edge of the plateau, a stelea field leaning on the slope and looking northeast -14°11'44"7 n / 39°04'32"1 e (altitude 2031 m): nearly 10 stelae are visible, broken, lying on the side. they are undecorated but hewn, and most of them have rounded head (phot. 40-42). samples: similar pottery shreds have been collected, in good condition.

3/ less than 50 m east, past the stelae field, next to a hillock which h. Berhe would like to hide a ancient structure -14°11'38"5 n / 39°04'32"8 e (altitude 2020 m): there is no structure visible, but numerous pottery shreds in front of the hillock, to the west. the area may be of some 2000 m² (phot. 43). samples: similar pottery shreds have been collected, of which different types of handles and patterns (phot. 44). the local people have also collected several artefacts (unseen).

III. STORAGE OF SAMPLES

the collection consisted, at this stage, in artefact samples with a systematic register and localization. though, as this first survey was a preliminary study, the samples have been limited (very small volume), and consisted mostly in pottery shreds, some iron slags and bones.

they are stored in plastic wrap with mention of the origin, and entrusted to ato Kebede amare, head of the tigrai culture and tourism agency at Mekelle.

Register:

Site

IV. CONCLUSION

this first visit in tigrai, at the beginning of our project, was a very short and preliminary survey, which enabled to project archaeological investigations in the future. the objectives will be to clarify the landscape-scale distribution of ancient human activity. We are convinced that beside the most important pre-aksumite and aksumite sites, numerous other sites can be investigated and can reveal more data about rural life and possibly about trade, including local exchanges and the long distance trade. the regions situated along the modern road from Wukro to adigrat, which followed probably an ancient road, deserve a detailed survey as several sites show. thus the region of senafe, including the site of Wakarida, near the village of sawna could be interesting for this study: the region of senafe is fertile, well cultivated land and it is situated on the ancient trails leading from the inland to the east, towards the red sea coast.

We have got in touch with certain teams working in tigrai on the pre-aksumite and aksumite periods. We also plan to contact very soon the other teams, as we are convinced that the fieldwork of several groups of researchers can be complementary and that their collaboration, exchange of data and of ideas can accelerate the progress of research on ethiopian history.

Figure 1 :

 1 Figure 1: The study area, Eastern Tigrai, 2010.

Figure 2 :

 2 Figure 2: The archaeological sites surveyed, Eastern Tigrai, 2010.

  , the hillock to the South. Phot. 2: Aggula Čerkos, the hillock to the East. Phot. 3: Aggula Čerkos, the site seen in 1895. Phot. 4: Aggula Čerkos, the visible structures on the hillock. Phot. 1: Aggula Čerkos, remains of a stairway. Phot. 6: Aggula Čerkos, fragment of column on a stone base.

-

  Abreha wa Asbeha situated about 7 kms northwest of Wukro (1 hour trail), 13°50'49"3 n / 39°32'11"1 e (altitude 2002 m), at the eastern side of the mountain. "roc church", rectangular in plan : its indented plan results in 3 sections, approached on its western side by a monumental stairway, and open to the west by 3 entrances, north for men, south for women (phot. 9-10). the central entrance consists in a double wooden doorway (phot. 11), and runs through the keddest. Phot. 8: May Tərəro, monolithic stele. Abreha wa Asbeha Phot. 9: Men entrance, to the North. Phot. 10: Women entrance, to the South. Phot. 11: Main entrance, double wooden doorway.

Phot. 12 :

 12 Abreha wa Asbeha, architectural ornament. Phot. 13: Abreha wa Asbeha, part of a religious scene. Phot. 14: Maryam Kadih, tombstone. Phot. 15: Maryam Kadih, remains of (human?) bones still visible on the side trench of the road.

Phot. 25 :

 25 Addi Gelemo, the place of the discovery. Phot. 26: Addi Gelemo, the place of the discovery on top of the hill.Phot. 27: Wakarida, remains of rubble stone walls.Phot. 28: Wakarida, remains of a complex north side of the hill.

  31) ; -ceramic figurine, without head, 0,08 m length, 0,06 m wide (phot. 32); -piece of a ceramic figurine (pelvis) (phot. 33); -bronze coin, diam 0,15 m (phot. 34) ; -funeral urn (phot. 35). Phot. 29: Wakarida, objects found by the local people before 2004. Phot. 30: Wakarida, objects found by the local people since 2004: Aksumite coin. Phot.35: Funeral urn.

Phot. 40 -

 40 42: Adi Ba'ekel, stelea field on the slope of the plateau. Phot. 43: Adi Ba'ekel, the "third" site, close to a hillock. Phot. 44: Adi Ba'ekel, handle and pattern of pottery shreds.

ANNEXE II. LOCALIZATION OF THE STELAE FIELDS (SURVEY 2010) ANNEXE III. LOCALIZATION OF THE "ROC CHURCHES" (SURVEY 2010)


M Lagha

M Tikhemirine

S Bergheul

T Rezoug

M Bettayeb

Performance comparison of pulse-pair and wavelets methods for the pulse Doppler weather radar spectrum

Keywords: 92.40.Ea, 92.40.eg, 92.60.gf Doppler weather radar, pulse-pair, wavelets, denoising, spectral moments

In the civilian aviation field, the radar detection of hazardous weather phenomena (winds) is very important. This detection will allow the avoidance of these phenomena and consequently will enhance the safety of flights. In this work, we have used the wavelets method to estimate the mean velocity of winds. The results showed that the application of this method is promising compared with the classical estimators (pulse-pair, Fourier).

Introduction

Mainly in civil aviation, the meteorology plays a big role in the security of flights. Aeronautics depends on reliable information related to the present and future meteorological conditions. This information is given by a meteorological radar. This last detects and locates even distant atmospheric disturbances in form of signals in microwaves.

Several methods of signal processing are worked out, others offered, to complete successfully the extraction of useful information to be transmitted to the user.

The pulse-pair method acts on the Doppler radar signal in the temporal domain by autocorrelation. As for the wavelets method, it spreads out in the spectral domain while having references to time.

The main purpose of this work is to improve the estimates of the average velocity and variances of the detected meteorological disturbances. The use of an algorithm of denoising based on the wavelet transform will lead to this improvement.

In section 2, we will introduce the estimation theory of spectral moments. This estimation will be made by two estimators, one temporal, named pulsepair and another one scalo-temporal, the wavelets. The section 3 introduces results and comments linked to calculations and simulation. Finally a general conclusion will be presented in section 4.

Estimation theory

The first three (03) moments of the power spectral density of the Doppler spectrum are directly linked up with the desired atmospheric basic parameters. The radar reflectivity (Z), the radial velocity (Vr) and the spectral width of velocities (W). [START_REF] Pazmany | Polarization Diversity Pulse-Pair Technique for Millimeter-Wave Doppler Radar Measurements of Severe Storm Features[END_REF], [START_REF] Aalfs | Signal Processing Aspects of Windshear Detection[END_REF], [START_REF] Doviak | Doppler weather radar[END_REF], [START_REF] Lagha | Doppler Spectrum Estimation By Ramanujan-Fourier Transform (RFT)[END_REF].

Since the return signal in the radar from a range cell (space dimension of an impulse) is generated by the back scattering of a big number of randomly distributed particles and/or by variations of the refraction index of the atmosphere/air, then the process of the received signal is considered (central limit theorem) or approximated by a gaussian random process.

The shape of the received signal reflected by all particles met by the transmitted beam is: [START_REF] Doviak | Doppler weather radar[END_REF], [START_REF] Bringi | Polarimetric Doppler weather radar[END_REF], [START_REF] Lagha | Doppler Spectrum Estimation By Ramanujan-Fourier Transform (RFT)[END_REF].

V (t, R) = i A i e j2πf i t-2 R i c .w(t -2 R i c ) (1)
with:

A i : Complex amplitude of the individual particle i, f i : Doppler frequency of the particle i, R i : Distance from the radar to the particle i, w(t -2 R i c ): Weighting Function of distance. At the k th To the kth radar impulse, the received wave shape may be written in the complex form [START_REF] Doviak | Doppler weather radar[END_REF], [START_REF] Bringi | Polarimetric Doppler weather radar[END_REF], [START_REF] Lagha | Performances comparison of pulse-pair and 2step prediction algorithms for the Doppler spectrum[END_REF] V k (R) = I k (R) + jQ k (R) [START_REF] Aalfs | Signal Processing Aspects of Windshear Detection[END_REF] with:

I k (R): In phase component. Q k (R): Quadrature component.
The estimation of the meteorological parameters is typically accomplished in the signal processing on the basis of range cells.

Pulse-Pair Method

The working frequencies of a meteorological radar are in the order of 10 9 Hz corresponding to wavelength of some centimeters (¡10 cm). Generally, the meteorological targets move with speeds lower than 50 m/secs. The Doppler Effect would translate these speeds into a shift of the transmitted frequency. This shift is a few hundred of Hz: much too weak to be directly measured. To reach it, we measure the phase shift between the return of two successive impulses having probed the same volume of space. Indeed, after the return of the second impulse, the target would have changed position which would be translated in a phase shift between both impulses. Hence the pulse-pair method is derived.

The pulse-pair technique

The classical pulse-pair technique gives the first two spectral moments (speed Doppler and its variance) of the weighted Doppler spectrum of the reflectivity from the function of autocorrelation of the radar complex signal [START_REF] Pazmany | Polarization Diversity Pulse-Pair Technique for Millimeter-Wave Doppler Radar Measurements of Severe Storm Features[END_REF], [START_REF] Aalfs | Signal Processing Aspects of Windshear Detection[END_REF], [START_REF] Doviak | Doppler weather radar[END_REF], [START_REF] Serafin | Meteorological radar, Chapter 23, Radar Handbook[END_REF], [START_REF] Bringi | Polarimetric Doppler weather radar[END_REF].

V [n] = I[n] + jQ[n] (3)
Components I and Q are supposed to be independent statistically [5, [6], [START_REF] Lagha | Performance Comparison of Pulse-Pair and 2step Prediction Approach to the Doppler Estimation[END_REF].

The function of autocorrelation can be expressed for the sequence V [n] of length N and whose samples are spaced by Ts, by: [3], [START_REF] Bringi | Polarimetric Doppler weather radar[END_REF]

, [6] R[1] = 1 N N -2 n=0 V [n + 1].V * [n] (4)
The mean velocity of a weather target is estimated by [START_REF] Bringi | Polarimetric Doppler weather radar[END_REF]:

v = - λ 4πT S arg( R[1]) (5)
And consequently, the spectral width of speeds is done by:

ŵ = λ 2 √ 2πT S ln R S+N [0] -R N [0] R[1] (6)
Where index S+N indicates the weather signal merging in the noise N.

Wavelet Method

The fundamental assumption in signal processing is to consider the radar echo signal consisting of two parts.

y = f a + ε (7)
Where f a produced by the Gaussian process of atmospheric diffusion and ε a noise resulting from various but mainly thermal sources [START_REF] Pazmany | Polarization Diversity Pulse-Pair Technique for Millimeter-Wave Doppler Radar Measurements of Severe Storm Features[END_REF].

In the following, we will deal with the problem by using the wavelets to filter the collected signal then apply the already seen method of processing (pulse-pair) to extract the various spectral moments of interest.

The primary reasons of the effectiveness of processing by wavelets are linked to the nature of the noise:

-it is nonstationary -it is transitory and of unknown form We will suppose that the Gaussian model fits the atmospheric signal f a and noise ε.

Discrete wavelet transform

The dyadic discrete wavelet transform (DWT) of a signal y(t) is [START_REF] Mallat | A Wavelet Tour Of Signal Processing[END_REF], [START_REF] Daubechies | Ten lectures on wavelets[END_REF]

T m,n = +∞ -∞ y(t)2 -m 2 Ψ * (2 -m t -n)dt (8)
Where: m: represents the considered scale. n: represents the translation of the wavelet Ψ(t). T m,n : represents a coefficient of correlation between y(t) and Ψ(t).

Ψ m,n = 2 -m 2 Ψ * (2 -m t -n),
represents the orthonormal basis of the mother waveletΨ.

Consequently the reconstruction of y(t) from the coefficients T m,n is then:

y(t) = +∞ m=-∞ +∞ n=-∞ T m,n Ψ m,n (t) (9)

Multiresolution analysis

If one samples a continuous signal u with a regular interval unit, one will obtain a sequence with discrete values u n =u (n) , n ∈ Z. [START_REF] Justen | wavelet-based methods for clutter removal from radar wind profiler data[END_REF] The dwt on one level breaks up the sequence u n =u (n) in two sequences a1 and d1 by a low-pass filter h and a high-pass filter g, both followed by an undersampling of order 2.

The sequence a1 obtained is called approximation and contains information of low frequencies of u n =u (n) , while the sequence d1 contains information of high frequencies of u n =u (n) called details. h and g are the decomposition filters of finite lengths (FIR).

a 1 = (u * h) ↓ 2 ( 10
)
d 1 = (u * g) ↓ 2 (11)
The operation of undersampling by 2 means to take every other sample. It is very useful during the reconstruction operation. This one is carried out by means of filters of reconstruction h and g. The filters of analysis and those of synthesis are in quadrature mirror (QMF).

Indeed, the sequence u n can be found by the expression:

u n = a 1 ↑ 2 * h + d 1 ↑ 2 * g ( 12
)
The oversampling of order 2 inserts zeros in the sequences in order to find the initial sequence u n with the same initial number of samples.

If the operation of decomposition is repeated J times, one obtains the algorithm of filter bank applied to the approximation outputs (see figure 1).

In the opposite direction, the reconstruction of u n is carried out starting from the approximation a 4 and the details d 1 , d 2 , d 3 and d 4 .

Thresholding

The wavelet coefficients correspond to the details of a signal. A detail can be been ignored without affecting significantly the data.

There exist various methods of thresholding of the wavelet coefficients: hard thresholding and soft thresholding.

Hard thresholding:

d s = 0 si |d| < λ d si |d| ≥ λ (13)
Soft thresholding or shrinking specific to the complex signals: [START_REF] Sardy | minimax threshold for denoising complex signals with waveshrink[END_REF]

d s = d 1 - λ |d| (14)
As for the various thresholds, one can use:

-The universal threshold: λ N = 2log(N log (N )) -The minimax threshold: (see table I) -The SURE threshold: (Stein's unbiased risk estimate).

Where d, λ and N are the wavelet coefficients, the threshold and the length of the signal, respectively.

Signal processing algorithm

The thresholding is efficient means to ignore the weakest details, compared to the selected threshold , λ, which one can compare to the noise and pre-serve only the most important wavelet coefficients. One rebuilds, then, the signal from the remaining coefficients which represent the denoised data (see figure 2).

Results and comments

In this work, we have generated data representing the echoes of a weather radar of wind disturbance.

For that, we have used the algorithm of Zrni à § developed in 1975 and included in works of R.D. Palmer. These data consist of signals I and Q. These calculations were carried out on a set of ten range cells taken on one radar azimuth.

In figure 3, a, b, we have represented signals I and Q and their spectra respectively for the range cells 1 and 5. As one can see it on the figure, this spectrum is a Gaussian form.

The estimates of the mean velocity and the spectral width of the weather disturbances already generated above are represented on figures 4 and 6. It is noticed that the estimate of the mean velocity of the wind by the pulse-pair algorithm is close to the real speed. On the other hand, the estimate by the method of Fourier is less close.

We also notice that the estimation of the spectral width is less important with the wavelets than with the pulse-pair algorithm.

In addition, by applying the wavelet method, we have obtained better results compared to Fourier and pulse-pair, see figure 4. This indicates the interest to have used this method. These insufficiencies of the algorithm of Fourier are inherent in his approach and they are due to:

-Resetting the frequencies which are apart from the field of work, which generates spectral losses.

-Spectral resolution (close frequencies).

Conclusion

In this work, we have re-examined the estimate of the spectral moments of the signals received by a Doppler radar, namely the moments of orders one (mean velocity of the wind) and two (spectral width).

It comes out from it, through the results obtained, that the algorithm of denoising by the means of the wavelet transform of the signal to be treated is efficient. Our results were consolidated by the comparison of the errors of the various used algorithms: pulse-pair and wavelet. The proof is the reduction of the relative error made on the parameters estimated compared to the

Figure 1 :

 1 Figure 1: DWT of u n to level J = 4: the sequence u n is decomposed into an approximation a 4 and details d 1 , d 2 , d 3 et d 4

Figure 2 :

 2 Figure 2: Synoptic diagram of the wavelet denoising method.

Figure 3 :

 3 Figure 3: signals I and Q and Doppler spectrum of the cell N 1

Figure 4 :

 4 Figure 4: Estimate of mean velocities of the wind by the Pulse-pair and wavelet estimators.

Figure 6 :

 6 Figure 6: Spectral width obtained the Pulse-pair and wavelet estimators.

Table I :

 I Extract of table 1 of[START_REF] Sardy | minimax threshold for denoising complex signals with waveshrink[END_REF]

	N	Minimax
		(λ N	* )
	16	1.763
	128	1.973
	256	2.176
	512	2.371
	1024 2.560
	2048 2.741

Acknowledgment This work was realized under the collaboration between the Aeronautical Science Laboratory of Blida University and the Search group of the College of Engineering, Sharjah, UAE.


P Petrov
email: ppetrov@tu-sofia.bg

F Nashashibi
email: fawzi.nashashibi@inria.fr

In this paper, we present a nonlinear adaptive controller for a two-vehicle automated overtaking maneuver. We consider the problem of a three-phase overtaking without the use of information obtained from road infrastructure. The only information available for feedback control is the current relative inter-vehicle position and orientation. Reference trajectories for every phase are generated in real time. An update control law for the automated overtaking vehicle is designed that allows tracking desired trajectories in the presence of unknown velocity of the overtaken vehicle. Simulation results are included to illustrate the performance of the proposed controller.

I. INTRODUCTION

HIS paper presents a nonlinear adaptive controller for a two-vehicle automated overtaking maneuver. Various control laws for different maneuvers of automated vehicles has been designed and analyzed during the last two decades. The ultimate goal in automating the driving process is to reduce accidents caused by human error and increase safety. While considerable research work has been reported on lane keeping [START_REF] Cerone | Combined automatic lanekeeping and driver's steering trough a 2-DOF control strategy[END_REF] and lane change maneuver [START_REF] Godbole | Distributed hybrid controls for automated vehicle lane changes[END_REF][START_REF] Jula | Collision avoidance analysis for lane changing and merging[END_REF], the problem of automated overtaking has attracted less attention. In contrast to lane keeping and lane change maneuvers, the overtaking is a composition of three consecutive maneuvers: lane change, followed by trajectory tracking and again a lane change, which have to be planned and coordinated. A twolayer controller architecture for overtaking maneuver is presented in [START_REF] Naranjo | Lane-change fuzzy control in autonomous vehicles for the overtaking maneuver[END_REF]. The lower level consists of two fuzzy steering controllers for path tracking and lane change, and the high-level is to evaluate the necessity and possibility of overtaking, and to switch between the low-level controllers. An on-road demonstration of cooperative driving solutions and, in particular, overtaking maneuver by autonomous road vehicles designed for the cities is reported in [START_REF] Baber | Cooperative autonomous driving: Intelligent vehicles sharing city roads[END_REF]. In [START_REF] Resende | Real-time dynamic trajectory planning for highly automated driving in highways[END_REF], two methods for real-time trajectory planning applied to automated driving, which are based on partial motion planning approach and 5 th order polynomial interpolation method, respectively are presented.

They have been implemented for the HAVEit European project. A smooth optimal lane-change trajectory for the overtaking maneuver is presented in [START_REF] Shamir | Overtaking a slower-moving vehicle by an autonomous vehicle[END_REF].

In this paper, we consider the problem of automated overtaking without the use of road infrastructure. An important issue in designing feedback controllers for overtaking maneuvers is to decide the desired trajectory of the overtaking vehicle. The approach used in this paper, is inspired by our previous works [START_REF] Petrov | Nonlinear adaptive control for autonomous vehicle following[END_REF][START_REF] Petrov | A mathematical model for control of an autonomous vehicle convoy[END_REF], in the sense that it consists in consecutive tracking reference virtual points, which are positioned at desired a priory known distances from the overtaken vehicle with a virtual reference point attached to the overtaking vehicle. In addition, similar to [START_REF] Resende | Real-time dynamic trajectory planning for highly automated driving in highways[END_REF], reference trajectories for the desired motion of the overtaking vehicle during the overtaking maneuver are generated using polynomial functions. We consider an automated overtaking maneuver with only the current intervehicle position and orientation available for feedback control. An adaptive nonlinear controller for the overtaking vehicle is designed that allows tracking desired trajectories in the presence of unknown velocity of the overtaken vehicle.

The organization of the paper is as follows: In Section II, the mathematical description of a three-phase overtaking maneuver suitable for feedback control is derived. The Problem formulation is given in Section III. In Section IV, a nonlinear adaptive control law is designed. Section V contains simulation results. Conclusions are presented in Section VI.

II. MATHEMATIC MODEL

A. Three-Phase Overtaking Maneuver

In this paper, the overtaking involving two vehicles is established as a three-phase maneuver (illustrated in Fig. 1). We assume that the following conditions are met:

--The overtaken vehicle is moving along a rectilinear route with a constant velocity.

--The left lane is free and there is enough space to complete the overtaking maneuver.

Planning and Nonlinear Adaptive Control for an Automated Overtaking Maneuver

Plamen --The only information available for feedback control is the relative inter-vehicle position and orientation.

A brief description of the three-phase overtaking scenario considered in this paper is provided below. Assume that the automated vehicle has to pass the preceding car at distance d ahead. The first phase consists of "lane-change"-type maneuver. Starting with some initial conditions, the overtaking vehicle diverts from the lane and tracks a given reference trajectory for a given time period and has to reach a preselected position behind the overtaken vehicle, (Fig. 1). At the end of the first time period, the guidance program switches automatically from Phase 1 to Phase 2. The second phase consists of driving straight in the adjacent lane along the overtaken vehicle at a prescribed lateral distance. The vehicle tracks again a given reference trajectory for a given time period until the overtaken vehicle has been passed and the overtaking vehicle reaches a preselected position on the left side of the overtaken vehicle, (Fig. 1). During the third phase, given a reference trajectory for the third phase, the overtaking vehicle returns to the lane and has to reach a preselected position in front of the overtaken vehicle. The desired position and linear velocity of the overtaking vehicle with respect to the overtaken vehicle at the end of every phase, as well the phase duration, are determined to satisfy the operational requirements imposed on such a maneuver. A reference trajectory for the overtaking vehicle is generated in real time for every phase. At the beginning of every phase, the desired initial position and velocity of the overtaking vehicle coincides with its current position and velocity. In such way, there is no restriction for the initial position, orientation, linear and angular velocities of the overtaking vehicle in the beginning of every phase of the overtaking maneuver. However, from a practical point of view, a minimal inter-vehicle distance d is introduced for the beginning of the maneuver, in order to meet some geometrical and dynamical limitations of the vehicles, (for example, limitations for the front-wheel steering angle).

B. Relative Kinematics

A schematic plan view of an overtaking maneuver involving two vehicles is shown in Fig. 1. We assume that the vehicles have non-deformable wheels which roll on a horizontal plane without slipping. In order to describe the position and orientation of the vehicles during the overtaking maneuver, we assign the following coordinate frames, (Fig. 2):

-Fxy -inertial coordinate frame in the plane of motion; -A i x Ai y Ai , (i = 1,2) -vehicle coordinate frames located at the center of the rear vehicle axels of the overtaken and the overtaking vehicle, respectively, where x i is along the longitudinal base of the vehicle; -R i x Ri y Ri , (i = 1,2,3) -virtual reference frames rigidly attached to the overtaken vehicle and located at specific places, as shown in Fig. 2. The coordinates of the origins R i with respect to A 1 x A1 y A1 are (L i 1t, L i 1n ), (i = 1,2,3) and the corresponding axes are parallel to those of A 1 x A1 y A1 . -Lx L y L -coordinate frame rigidly linked to the overtaking vehicle and located at the mid-point of the front bumper of the vehicle at a distance L 2 from point A 2 , where the x L axis is in direction of the longitudinal vehicle axle. For brevity, in the exposition which follows, we will derive in details the relative inter-vehicle kinematics with respect to the first phase of the overtaking maneuver (lane change maneuver). Similar relationships can be also derived for Phase 2 and Phase 3.

] 2 1 1 1 ℜ ∈ = T R R R y x p
, and the orientation of A 1 x 1 y 1 with respect to Fxy by θ R . If the vector

[ ] 2 1 1 1 1 1 1 ℜ ∈ - = T n t R A L L d is the vector from the origin A 1 to the origin R 1 expressed in the coordinate frame A 1 x 1 y 1 , then the coordinates of points R 1 R 3 x F y L 1 1n . . . L 3 1t n R 2 L 2 1n L 2 A 2 . A1 . L 2 1t n R 1 L 1 1t n A 1
e x e y e θ

L Overtaken vehicle

Overtaking vehicle

and A 1 are related by We note that based on the assumption of pure rolling without lateral sliding of the wheels, the nonholonomic constraints imply that in (4) the components of the velocities of points A 1 and A 2 in direction of the rear vehicle axles are equal to zero, i.e., v y = v Ry = 0.

1 1 1 1 R A r A R d R p p + = (1) where ) 2 ( SO R r ∈ is orthogonal matrix.
Remark 1: The front wheel steering angle α of the overtaking vehicle which is a control input for the system (4) is related to the angular velocity of the vehicle as follows

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = x v l a ω α 2 tan . ( 5
)
In the following, we consider an overtaking maneuver during rectilinear motion of the overtaken vehicle. In this case, the angular velocity of the overtaking vehicle ω R = 0.

Let us consider the following change of inputs in (4) where one easy verify that the transformation matrix is nonsingular when 0 2 ≠ L . Using [START_REF] Resende | Real-time dynamic trajectory planning for highly automated driving in highways[END_REF], in case of rectilinear motion of the overtaken vehicle, the inter-vehicle kinematic equations ( 4) can be written in the form

⎥

+ - = = - =    . ( 7
)
2) Phase 2 and Phase 3:

During the Phase 2, the vector 2 1 R A d which is the vector from the origin A 1 to the origin R 2 expressed in the coordinate frame

A 1 x A1 y A1 is set to be [ ] 2 2 1 2 1 2 1 ℜ ∈ = T n t R A L L d ,
where

1 1 2 1 n n L L = .
In the same way, for Phase 3, vector 3 1

R A d is set as [ ] 2 3 1 3 1 0 ℜ ∈ = T t R A L d
, (Fig. 2).

III. PROBLEM FORMULATION

In this paper, we consider autonomous overtaking maneuver without any information obtained from road infrastructure or communicated from the overtaken vehicle. The only information that the overtaking (robot) vehicle can use for feedback control is the current relative position and orientation with respect to the overtaken vehicle given by (3) which are obtained from onboard sensors.

A. Trajectory Planning

We consider in details the problem of generating smooth trajectories for point to point motion in terms of the posture errors (e x , e y ) with respect to the moving reference frame R 1 x R1 y R1 which is rigidly linked to the overtaken vehicle. Assuming rectilinear motion at constant velocity of the overtaken vehicle, we are interested in overtaking scenario for the first phase, when the overtaking vehicle starts the maneuver from an arbitrary initial position and orientation behind the overtaken vehicle with given (arbitrary) velocity. In the end of the first phase, the two vehicles have to be parallel and in the same time, the desired position of the overtaking vehicle behind the overtaken vehicle has to be shifted at distances

1 1t
L and 1 1n L in longitudinal and lateral direction, respectively, (Fig. 2). These distances are determined from considerations of safety driving and sensor requirements (in particular,

1 1t
L may be equal to zero). In addition, the relative inter-vehicle velocity in longitudinal direction must be equal to a prescribed non-zero value. For this end, we suppose that at time t 0 , the state variables satisfy

At time t f , we wish to attain the values

) ( ) ( 0 ) ( f Rx f d x f d x t v t e t e Δ = =  , 0 ) ( 0 ) ( = = f d y f d y t e t e  (9)
where

) ( f Rx t v Δ
is the prescribed relative non-zero intervehicle velocity in x R -direction at t f, , i.e., the overtaking vehicle has to start the second phase of the overtaking maneuver at prescribed velocity which is higher compared to the velocity of the overtaken vehicle. The time t f, is determined from considerations of vehicle's power and acceleration capabilities. The initial values of the program errors at t = t 0 . However, from a practical point of view, the following conditions have to be satisfied:

y y y x x x b t e a b t e a ≤ ≤ ≤ ≤ ) ( , ) ( 0 0
, where a x , b x , a y , and b y are determined from considerations of safety driving, sensor requirements, and vehicle's power and acceleration capabilities. In this paper, we consider cubic desired trajectories of the form

3 0 3 2 0 2 0 1 0 3 0 3 2 0 2 0 1 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t t
- + - + - + = - + - + - + = . ( 10
)
Differentiating [START_REF] Khalil | Nonlinear Systems[END_REF] with respect to time, we obtain quadratic polynomials with respect to the derivatives

) (t e d x  and ) (t e d y  : 2 0 3 0 2 1 2 0 3 0 2 1 ) ( 3 ) ( 2 ) ( ) ( 3 ) ( 2 ) ( t t
- + - + = - + - + =   . ( 11
)
Using ( 8), ( 9), ( 10) and (11), we obtain 8 equations for the eight unknown coefficients a ix and a iy , (i = 1,2,3,4). Similar expressions for the desired trajectories can be assigned for Phase 2 and 3.

B. Problem Statement

We assume that we are able to measure the posture coordinates Given the inter-vehicle kinematics in error coordinates (7) the control objective for Phase 1 is to asymptotically regulate to zero the coordinates of reference point L of the overtaking vehicle with respect to the coordinate frame R 1 attached to the overtaken vehicle according to desired trajectories [START_REF] Khalil | Nonlinear Systems[END_REF] for the error coordinates (e x , e y ) with initial and final conditions given by ( 8)- [START_REF] Petrov | A mathematical model for control of an autonomous vehicle convoy[END_REF]. Similar objectives can be formulated for Phase 2 and 3 where the virtual reference points associated with the overtaken vehicle are R 2 and R 3 , respectively.

IV. ADAPTIVE CONTROL DESIGN

In this paper, we consider the problem of controlling the motion of the overtaking vehicle during the overtaking maneuver, where the overtaken vehicle moves straight with constant velocity.

Consider again in details the Phase 1 of the overtaking maneuver. We make the following change of coordinates are given by [START_REF] Khalil | Nonlinear Systems[END_REF].

The system [START_REF] Shamir | Overtaking a slower-moving vehicle by an autonomous vehicle[END_REF] in the new coordinates becomes

+ - = - = - - =      . ( 13
)
The adaptive control design is based on a reduced-order system composed of the first two equations of (13) rewritten below for clarity of exposition (14)

Consider the system (14) and assume that the velocity of the overtaken vehicle 0 > = cte v Rx is unknown constant parameter. The control problem consists in finding an adaptive feedback control law for the system (14) with inputs (u 1 , u 2 ) such that

( ) 0 ) ( lim = ∞ → t x e t and ( ) 0 ) ( lim = ∞ → t x e t . (15)
Consider the control

where k x and k y are positive gains. We chose the following Lyapunov function candidate where

Rx Rx Rx v v v - = ~ (18)
is the parameter error and γ v = cte > 0 is the adaptation gain. Using ( 14), ( 16) and (18), the derivative of V is obtained in the form

) 1 ( 2 2 Rx v e Rx e y e x v x v y k x k V     γ + + - - = . ( 19
)
Choosing the update law as

e v Rx x v γ - =  ˆ (20) we get for the derivative of V 0 2 2 ≤ - - = e y e x y k x k V    . ( 21
)
The resulting closed-loop adaptive system becomes Proof. The system (22) has an equilibrium point at the origin. The function ( 17) is continuously differentiable and positive definite and its derivative (21) along the trajectories of the system is negative semi-definite. From (21), it follows that (17) is non-increasing, (

) 0 ( ) ( V t V ≤
), and this in turn implies that x e (t), y e (t) and ) ( ~t v Rx are uniformly bounded with respect to the initial conditions.

To characterize the set

{ } 0 ) ( | 3 = ℜ ∈ = z V z S  , note that 0 0 ) ( = ⇒ = e x z V  and 0 = e y . ( 23
) Hence { } 0 ; 0 | 3 = = ℜ ∈ = e e y x z S .
To prove asymptotic stability of the equilibrium point z = 0, we use the LaSalle invariance principle [START_REF] Khalil | Nonlinear Systems[END_REF]. Suppose that z(t) is a trajectory that belongs identically to S. From (22), we have in the domain of interest, (λ is a positive constant).

0 0 ) ( 0 0 ) ( 0 0 0 ) ( ≡ ⇒ ≡ ≡ ⇒ ≡ ≡ ⇒ ≡ ⇒ ≡ Rx e e e
The point e θ = 0 is an exponentially stable equilibrium point for the nominal system (26), which can be easy proved by using, for example, the Lyapunov function . If e θn (t) and e θp (t) are the solutions of the nominal and perturbed systems (26) and (25), respectively, using the Gronwall-Bellman inequality [START_REF] Khalil | Nonlinear Systems[END_REF], we have on the compact As mentioned above, at the end of the first time period, the guidance program switches automatically from Phase 1 to Phase 2. The same approach can be used to derive feedback controllers for the overtaking vehicle for Phase 2 and Phase 3.

time interval [t 0 , t f ] } 1 )] ( {exp[ )] ( exp[ ) ( ) ( ) ( ) ( 0 0 0 0 - - + - - ≤ - t t

V. SIMULATION RESULTS

To illustrate the effectiveness of the proposed controller, several simulations are carried out in order to evaluate the inter-vehicle behavior and tracking accuracy during the three-phase overtaking maneuver. In the simulation using MATLAB, a planar bicycle 2DOFs vehicle model is used. The speed of the overtaken vehicle during the three phases of the maneuver is constant and set to be v A = 4m/s. The duration of every phase of the maneuver is chosen to be t int = 5s. The coordinates of the reference frames R i x Ri y Ri , (i = 1,2,3) with respect to the coordinate frame A 1 x A1 y A1 attached to the mid-point of the rear vehicle axle of the overtaken vehicle for every phase of the three-phase overtaking maneuver is provided in Table I.

The longitudinal vehicle base of the overtaking vehicle was chosen to be L 2 = 2m. The initial position and orientation of the overtaking vehicle in the inertial frame Fxy are x A2 (0) = 0; y A2 (0) = 0; θ(0) = 0. The initial position and orientation of the overtaken vehicle in Fxy are x A1 (8) = 0; y A1 (0) = 0; θ R (0) = 0. The desired inter-vehicle distance in the beginning of the maneuver is set to be 6m.

In the first simulation, from Fig. 3, we can see the planar path drown by the vehicle guide points A 2 and A 1 of the overtaking and overtaken vehicle, respectively. As seen from Fig. 4, the relative orientation e θ (t) between the two vehicles remains bound during the three phases of the overtaking maneuver, which is consistent with the theoretical results in Section IV. The evolution in time of the inter-vehicle distance AL is given in Fig. 5. An overtaking maneuver involving two CyCab vehicles has been performed using CIVIC software, as shown in Fig. 6. The results of the simulation confirm the validity of the proposed controller.

VI. CONCLUSION

In this paper, a nonlinear adaptive controller for twovehicle three-phase automated overtaking maneuvers has been presented. Reference trajectories for every phase using 3 rd order polynomial interpolation method are generated in real-time and consecutively tracked. The only information available for feedback control is the current relative intervehicle position and orientation. The unknown velocity of the overtaken vehicle is estimated using adaptive update law. Simulation results illustrate the performance of the proposed controller. Our future work will include implementation of the proposed controller and experiments with CyCab vehicles developed at INRIA.

Fig. 1 .

 1 Fig. 1. A three-phase overtaking maneuver involving two vehicles

  linear velocity of the overtaken vehicle is unknown.

Fig. 2 .

 2 Fig. 2. Coordinate frame assignment

[

Similarly

  point A 2 with respect to the frame Fxy; the coordinates of point L with respect to the frame Fxy by orientation of A 2 x 2 y 2 with respect to Fxy by θ. If the vector from the origin A 2 to the origin L expressed in the coordinate frame A 2 x A2 y A2 , then the coordinate of points L and A 2 are related by .The coordinate frames R 1 x R1 y R1 and Lx L y L are defined to describe the relative inter-vehicle kinematics during the first phase of the overtaking maneuver. For this purpose, the coordinates and orientation of the frame Lx L y L in the coordinate frame R 1 x R1 y R1 can be expressed as follows posture.Differentiating (3) with respect to time and taking into account equations (1) and (2), after some work, the intervehicle kinematics in error coordinates is obtained as follows where v x and v Rx are the longitudinal components of the linear velocities of points A 1 and A 2 , respectively; are the angular velocities of the overtaking and overtaken vehicle, respectively.

  3), but the linear velocity v Rx of the overtaken vehicle is unknown constant parameter.

  error coordinates (e x , e y ) are defined by (3), and the desired trajectories

Proposition 1 :.

 1 Assume that the linear velocity of the overtaken vehicle is bounded unknown constant parameter If the control law given by (16) is applied to (14), where the velocity estimate Rx v ˆ is obtained from the parameter update law (20), the origin

.

  Therefore, e θ is bounded and 0 ) ( → t e θ as ∞ → t

  the overtaken vehicle (red) and the overtaking vehicle (blue)

Fig. 3 .

 3 Fig. 3. Planar path drown by the vehicle guide points (the mid-points of the rear vehicle axles): point A2 (blue line) and point point A1 (red line)

Fig. 4 .

 4 Fig. 4. Time history of the error coordinates xe(t), ye(t) and eθ(t)

Fig. 5 .

 5 Fig. 5. Time history of the inter-vehicle distance AL

Fig. 6 .

 6 Fig. 6. Overtaking maneuver involving two CyCab vehicles using CIVIC software

support from the FP7 EU project HAVEit.