Enemy Spirits, Allied Spirits: the Political Cosmology of Arunachal Pradesh Societies. * The NEHU Journal, Janv. 2005, vol.3, n°1, p.1-28

Philippe Ramirez

In memory of Verrier Elwin

Since almost a century ago, few theorists have basically called into question the image which Max Weber proposed of a State monopolizing the use of legitimate violence. [START_REF] Weber | [END_REF] Could it be inferred that the State-free societies would allow each individual a free access to violence? If the postulate may be acknowledged as true in principle, the common and dangerous Hobbsian image of stateless societies in prey with the free course of individual passions and interests must be seriously questioned. Several recent works have shown the eminently positive role of a social violence, which weaves and vitalizes the social fabric, notably by the sublimation of a sovereign value, the prestige. [START_REF] Myers | For instance[END_REF] In certain Himalayan societies, violence definitely plays a central role, defining a privileged context where interpersonal bonds are build and destroyed, individual status assessed, and highest social ideals revealed. So we would like here to try understand what certain civilizations of the Eastern Himalayas say themselves about violence: how is violence conceived, represented, where does it starts, where does it lead to? What is its meaning and specially what is its role in the human world? To answer such questions, i.e. to understand an ideology from within, one of the most accessible as well as reliable methods lies in its religious expressions. We will agree on this point with Durkheim, for whom religion is first of all a discourse of the society on itself, but also -closer to the Himalayan cultural domain -with Edmund Leach, who suggested to understand the nat, these supernatural beings of the Kachin world, as "ways of describing the formal relationships that exist between real persons and real groups in ordinary Kachin society". [START_REF] Leach | [END_REF] We will consider the case of several Tibeto-Burmese languages speaking groups of Arunachal Pradesh who, having for long remained away from both Tibetan and mainland Indian cultural influences, are of a particular interest for the anthropology of the whole Himalayan area. [START_REF]Paul Mus considered the mountain dwellers of continental Southeast Asia as the last stronghold resisting the formidable Sino-Indian cultural "pincers[END_REF] Through them we have access to ideologies which, elsewhere in Nepal, Bhutan, and Himalayan India, had been reshaped in the moulds of Hinduism, of Buddhism, and above all, of the State.

I. The Populations of Arunachal Pradesh

In the heart of Arunachal Pradesh (Subansiri, Siang and Dibang river basins), on a relatively limited (25000 sq. km) and geographically homogeneous area, demographic, economic and settlement features are so diverse that one would be tempted, at first sight, to make final distinctions. Cattle breeders of shifting cultivators inhabiting "long-houses" scattered in small hamlets, sedentary farmers practising rotation of crops and living in centralized villages, massive concentrations of wet rice growers organized in clan clusters, one cannot of course treat without reserves this whole as a monolithic block. Other criteria however go in the direction of homogeneity. The languages for instance are all part of the North-Assam group of the Tibeto-Burmese family, [START_REF]On the Himalayan languages[END_REF] and are closely related. As for the identity, each ethnic group claims a common origin with most neighbouring groups, so that the label "Tani group" is now being widely used to designate them. Similarly, descent and kinship systems have many features in common, although not specific to this particular region. But the unity is above all apparent in the ritual and mythological domains, as the following pages will show. Similarities can be put into light not only in the nature of elements taken separately, but as well in the structural relations among them.

Despite the anthropological importance of the region, the literature concerning this part of the Himalayas, until 1971 the "North East Frontier Agency" (NEFA), is very recent and not very profuse, especially if one considers full-fledged social or cultural anthropology. Beside G Dunbar's 1915 invaluable but relatively short report on the Siang area, the only pre-independence significant material was provided by C. von Fürer-Haimendorf on the Subansiri area in 1947. 6 In the 1960-70's, the Arunachal Pradesh "Directorate of Research" series offered a precious set of monographs on almost every community of the State, although focusing mainly on material culture and generally disregarding belief systems. One of the noticeable exceptions was the contribution of S. Roy on the Padam and Minyong. 7 The last twenty-five years have experienced a blossom of substantial anthropological works, particularly under the impetus of Itanagar's based researchers, around Parul Dutta, Tamo Mibang and B.B. Pandey. [START_REF] For Instance Pandey | The Buguns, a Tribe in Transition[END_REF] Beside anthropology per se, Arunachal cultures have been approached from the angle of folklore, following the lead of Verrier Elwin. [START_REF] Bhattacharya | See for example Elwin[END_REF] This made available a sum of oral traditions, which, by the use of mythological analysis, may give valuable insights into the local ideologies.

In central Arunachal, four cultural units may be distinguished from West to East: Nishi, Apa-Tani, Adi and Mishmi. Nishi are both cattle-breeders and shifting cultivators, organized in scattered "long-houses" each forming an autonomous social and political entity. [START_REF] On | Fürer-Haimendorf[END_REF] Insulated within the vast Nishi country, the small Apa-Tani valley offers a somehow antithetic picture: a mosaic of irrigated rice fields and dense villages, divided into strong clan quarters.

Each village is governed by a council, the buliang, which authority on the quarters is only relative. [START_REF] On The Apa-Tani | Fürer-Haimendorf[END_REF] Such is not the case of the Adi villages of the Siang valley, centred around their dormitory (moshup), in front of which seats a full sovereign council, the kebang. Adi economy is based on cyclic shifting cultivation, giving way to irrigated rice culture in the densely populated foothills. [START_REF] Southern | On Northern Adi, see for example Dhasmana[END_REF] At the eastern end of the Himalayas, the Dibang and Lohit valleys shelter the Mishmi, who may be roughly qualified of standing economically and politically in between the Nishi and the Adi.

Cultivation is shifting on long cycles; elders' councils do not seriously entail the large autonomy of the households. [START_REF] On The Mishmi | The Mishmis of the Lohit Valley, Assam[END_REF] Thus, several criteria make it relevant to distinguish a number of large cultural units in the concerned area. However this diversity hides a fairly high homogeneity in other features, particularly in the worldviews. As will be shown in the following lines, it may be assumed, without taking too much risk, that all the inhabitants of central Arunachal Pradesh share the same broad ideological background. It looks as if the economic or social particularities had been unable to affect a set of deep anthropological structures common to all these seemingly different communities. And we will see how many conceptions in force on this side of the Himalayas are as much central in the hierarchical and State-subjected societies of the Northern side. The materials used in this article were all collected before the 1980's. Thus, the "world views" which will be addressed here may slightly differ from the present ones, specially following the emergence of a new religious revivalism around the cult of Donyi-Polo. 14

II.

The men hunt the animals, the wiyu hunt the men When a Nishi dies at home, all men in the house seize their spears and swords. One has to remain alert, the ghosts are there, close by, and try to approach the cadaver. Without respite, the swords are shake in each recess of the room, spears pierce the walls. Outside, arrows are drawn in all directions, particularly towards the roof. Later, the corpse will be buried at the outskirts of the hamlet. The threat however will not be removed. For a whole night, the relatives will guard the grave, climbing on the funerary structure, throwing embers in the darkness, shouting, humming, stripping arrows in the night... 15 The atmosphere of fear and violence which surrounds the Nishi funerals suggests two clues: First, certain invisible forces, in this case ghosts, seem strongly attracted by an inanimate human body. Then, it is possible to counter these attacks by using material weapons, and thus to frighten invisible beings with physical threat. This brings us to a first set of interrogations:

Who are these threatening beings? Do they represent a particular category? And does such violence correspond to a common attitude towards the invisible world?

The available data suggest a relatively general dichotomy: On one side, a very limited number of omnipresent and overwhelming entities, characterized by their stability and their neutrality. The notion of "deity" may faithfully qualify them. Sometimes insulated, such as Ringya, the sun of the Mishmi, but very often presented in the shape of a couple, Donyi-Polo, the "sun-moon" of the Adi, Nishi and Apa-Tani, their principal role is to guarantee an oath or to reveal the veracity of an allegation. Until very recently, [START_REF]Donyi-Polo has become in the very last years the central deity of a revivalist cult promoted by ethnic leaders; see Regunathan[END_REF] they were specifically called to witness a treaty or an ordeal. Out of these occasions, they were propitiated only during the great annual festivals, together with other deities to which the protection of cattle and harvests was demanded.

The second category consists of a number of "spirits", designated by the same generic term: wiyu (pronounced ui by the Apa-Tani and uyu by certain Adi). Within this broad class, the Nishi distinguish the orum, "souls" of those defuncts who did not have access to the land of the dead; their relationship with the living men however, is not peculiar compared with other wiyu. Most spirits are located in space, i.e. each one of them evolves in a specific natural environment: forest, rivers, hilltops, air, underground... Some have settled in the villages of men, sometimes even in the houses themselves or in the granaries. By essence, no spirit is strictly malevolent or benevolent. Of course, the spirits of the wild world are more feared that those residing near or within settlements. However, not only is it common to call "wild" spirits for assistance and protection, but the "domestic" spirits may also be prone to revolt, sometimes causing calamities. Although not exclusive to the societies of Eastern Himalayas, the ambivalence of spirits has to be emphasised.

The Arunachali is primarily a farmer. But he is also a hunter, and very often a tradesman, who travels either up to the Tibetan border or down to the plains of Assam. He thus frequently traverses long distances through deserted mountains, dense forests and infuriated rivers. Everywhere, the spirits are nearby and threaten him. They are not "devilish" beings, seeking evil for the sake of evil; simply, as expressed by a Nishi: "the men hunt wild animals, the wiyu hunt the men". 17 The number and the diversity of invisible beings are too large to enable an exhaustive account here. Instead, let us take a particular example by considering what the world of the wiyu consists of for a Shimong Adi of North-Eastern Siang: In the underground live Kine-Dene and Sitking-Kedeng who harass men by shooting arrows. This makes the legs shake, gives indigestions and dysentery. The epom reside in the forest in great numbers. They often take human form, wearing helmet, sword and spear; one can hear them sharpen their dao (machete) on the rocks. They are great fisher(men), and where they have placed their traps, not a single fish is left for men. They use to abduct humans without killing them. For example, they hide at the top of a tree and imitate a crying child. The imprudent traveller approaches and finds at the foot of the tree a decapitated wildcat. Taking advantage of his amazement, the wiyu jumps on him and carries him far deep in the forest.

For the sleeping man, the greatest danger is the wiyu of high summits. They seize his soul, which they enslave or sell to other wiyu. When this occurs, one does not awake any more. No sacrifice can prevent them from harming. In addition there are the nippong, spirits of the women who died in delivery; they disturb the childbirth and cause abundant menses to the girls; the wiyu of the rivers who make fall people from the bridges; those of the sky, among which Banji-Banmang, the vampire, which always carries its gourd of human blood, and Doying-Botte, which gives strong headaches and epilepsy. 18 This description of the Shimong wiyu world, although partial, corresponds, often in its least details, to what the other groups depict. 19 It already suggests some aspects of the relationships between men and spirits. The wiyu attack men in various ways.

According to their nature, they strike with disease, weapons, create accidents, prevent the birth of children, suck blood, seize "souls", or capture people in flesh and bone. At the least they represent economic competitors, depriving men of their means of subsistence, in particular by monopolizing game. Another characteristic of the spirits, quite revealing, is that they use various tricks to entrap men, thanks to their capacity to take all kind of animal and human forms.

Of these first images, let us retain also what does not appear.

First, attacks by the wiyu do not apparently respond to the violation of any social prohibitions, and thus do not take the shape of "supernatural sanctions". Then, if the spirits sometimes use extraordinary powers, in most circumstances they resort to the same means as men: weapons, abduction, and enslavement. Up to this point we have considered contexts where the man kept a passive role with regard to the spirits, that of the victim. Let us now introduce other instances, where man is depicted as an acting being, which defends himself against attacks, who intercedes with the aggressors, and even call them as allies to help him fulfil his own designs.

Among the Adi, in case of difficult delivery, and if the presence of a disturbing wiyu is suspected, armed men surround the house, hold up their weapons while shouting and strike the walls with ferociousness. In the same way, when fire strikes an Padam Adi village, armed men climb to the roofs and brandish their swords to stop the blaze. 20 The villages themselves are defended by a magic enclosure; each entering path is barred by an imposing gate, equipped with bows and arrows, and on which carcasses of pigs and dogs hang. [START_REF] Dhasmana | Fürer-Haimendorf[END_REF] And the shaman, first intercessor with the invisible forces, never officiates without his sword and dagger hanging on his belt.

It seems that the societies of central Arunachal-Pradesh conceive their relationships with the supernatural in the terms of a power balance, and by no means in those of a submission to a sovereign or higher authority. If the spirit attacks, one does not bow down, one defends oneself; if it remains deaf to a request, one gets rid of it. In a village struck by epidemic, a Shimong shouts in revolt: "the wiyu are devouring us. They take no notice of our offerings. Do not believe them! They eat our gifts and they eat us as well. Let us give them no more". 22 The attacking spirit is not the enforcer of a punishment, it is indeed an enemy, and thus in a way a similar being, with which one has to fight on an equal basis. For the same reason it is a possible ally, which power will be used against a third party. In 1944, Fürer-Haimendorf experienced -and was involved in -a quarrel between the Apa-Tani and the Nishi village of Licha. At the time when Apa-Tani warriors prepared for a raid against Licha, their priest called upon spirits: "May no evil befall the Apa-Tani participating in the expedition, may their swords and spears be sharp, but may the swords of Licha be blunted and their arrows miss their mark". 23 Haimendorf interpreted this invocation as a call on the spirits "to bear witness that the fault of the quarrel lay with Licha". It is difficult to agree. The context as well as the data produced do not suggest that in this raid the spirits were considered as a kind of judges. If this was the case it may have been somehow anomalous. In these feuds, "divine justice" has no role to play, because the fight does not aim at pointing a "culprit", a transgressor of the norm. One fights possibly to affirm his prestige, but not to prove the righteousness of one's own claims. The supernatural being does not "sanction" the vanquished by imposing defeat on him. It simply acts under the terms of a request by one of the opponents. The other party may as well call upon it. It is common, in troubled times, to ask the shaman to bounce back against the enemies the forces that they themselves had previously solicited: Ramo Adi may call upon Singro to obtain the destruction of an individual or even of his entire clan. With the help of a veteran priest (nyibo), a tree and a big-sized dog are cut down each evening during three successive nights. To counter this type of assault, one will perform the same rite, but during the day. 24 Significantly, the same competition for the propitiation of an helping force may oppose a man, or a group of men, and a spirit: a Pailebo Adi who falls sick should as soon as possible propitiate Sibo-Rabro, before the attacking wiyu does the same. If the wiyu is the first to put Sibo-Rabro on its side, the patient will perish. [START_REF] Kumar | [END_REF] Some men permanently benefit from the power of an allied spirit. The shaman entertains a particular relation with a guardian wiyu, which revealed his vocation by appearing in his dreams. During healing cures, it remains near him to assist in the search of abducted souls and in fights against other spirits. The shaman can request his guardian to wound or kill his personal enemies, or to capture them so that he devours them himself, although such behaviour is generally disapproved. [START_REF]On the religious specialists in Arunachal, see Fürer-Haimendorf[END_REF] Beyond the particular case of shamans, do men and spirits entertain alliance relationships?

III. The spirit, brother of man

The close affinities of patterns in Central Arunachal mythologies conduce to investigate further on about their ideological coherence, or "structure". Oral traditions of the different groups will be contemplated here as informing each other. [START_REF] Roy | On the contexts where these myths are narrated, little information is available. Elwin[END_REF] First of all let us try to uncover the most frequent models pertaining to our main concerns, on the one hand the relationship between men and spirits, on the other hand the circumstances presiding over conflict and alliance. From this point of view, two aspects emerge at first sight: the very rare intervention of higher or creating figures, and the original indistinctness of nature between men, spirits and animals.

Stemming from the original mother, from the sea, from the encounter of sky and earth, or from unspecific creatures (named without being described), the three categories of beings are invariably introduced as kin, children of the same father, the same mother, or offsprings of three brothers. Originally, men, animals and spirits were not distinct, neither in their appearance nor in their respective aptitudes. At a certain chronological stagewithin a true "evolution" -each one progressively developed specificities: the spirits became invisible to the men, the animals lost their houses, their weapons and the speech. But all remained on an equal footing in their competition or co-operation for the exploitation of natural resources. A Shimong Adi myth describes the links that Tani, ancestor of men, maintained with Epom, wiyu of the forest, and Jimu-Tayang, wiyu of the high mountains, all three sons of Peddong-Nane, the original water: Tani reaches the lair of Jimo-Tayang on the snow peaks, where he finds the bones of its two devoured sons. Tani gets enraged and asks Jimu-Tayang a compensation (midum) for the loss of his sons. Initially reticent, Jimu-Tayang then concedes to Tani the ibex, the musk deer, the tanang wild goat, the mountain reed (emo), the aconite, and the wild garlic. 29 Tani accepts and they sit together to share apong beer. Tani then goes down, bringing back his new belongings... However, none of these mountain creatures manages to adapt in the warm and wet country of Tani. Soon he decides to bring them back to their original environment, where he regularly comes to collect them. Jimu-Tayang however keeps a grievance towards this brother who enjoys his own former properties: each time Tani, and later his offsprings, visit the summits, searching for the deer or the aconite, Jimu's sentinel, Poli-Adi, jumps on them, tears off their eyes, makes their nose and ears bleed, twists their testicles and throws their body in chasm. 30 With only a few words the framework is set: man and spirit are true economic competitors, exploiting the same resources, in particular wild animals. [START_REF] Adi | [END_REF] In this instance competition follows the division of domains. Elsewhere the process is opposite: it is the scarcity of land or game, on a previously undivided space, which induces spirits, animals and men to gather in a meeting (kebang) and partake the space. [START_REF] Guha | [END_REF] Competition may be of various natures, according to the respective needs of each category of beings:

-Some spirits are cannibals or vampires, other just hunters or farmers, and all seek wives and slaves, possibly human.

-Predatory animals may have the same preys as men, they may also be found of cattle, or simply of human meat.

-Men are primarily hunters and must target animals. As farmers they may also encroach on spirits' domains. And for sake of breeding, they must get wives, possibly among the wiyu.

In this struggle for survival and perpetuation, no category definitely takes over. Each has its own assets: the spirits have the capacity to be incarnated in various forms, the animals to fly or to smell at long distance, the men have the weapons. As we will see it, it is the possession of weapons which makes the specificity of the man. Indeed, neither intelligence, which is far from being poorer among spirits and animals, nor sociability constitute human prerogatives. The myths depict an omnipresent society, independent of superficial differences among the three categories. All marry, all owe matrimonial payments to their father-in-law, all meet in councils to deliberate… [START_REF][END_REF] Let us pursue our classification of the raw material of the myths, to now consider which place is held by alliance. Alliance, in particular matrimonial, is always conceived under an ambivalent aspect. Basically, it takes the form of a union between two beings or entities previously separated, a union which brings fruitfulness and prosperity. However, this union is considered by both sides with suspicion: the partner is likely to be a disguised predator and remains in any event an avid competitor, which benefits from the confidence granted under the terms of the new relationship. The fertilizing union is depicted by the myths as a universal process;

we would say in our language that it is not only biological and social, but also physical. Thus, far from limiting itself to the three types of individualized beings, it may as well involve cosmic entities, spaces. The most concise illustration is found in a myth of the neighbouring Miji, but the themes are common in Central Arunachal traditions:

"At first the Earth and Sky were far away from each other and the Sun and Moon lived separately. In those days the Sun and Moon did not move. It was the Earth and Sky that moved. For the Earth has many children whom she held in her lap. She was afraid that her husband, the Sky would approach her and gives her more babies and was always moving round to escape him.

But from time to time the Sky succeeds in going to the Earth and when this happens there is an earthquake. At such a time many new trees grow up and there are fine harvests" 34 The picture cannot be more explicit, and we will see that its range exceeds simple poetry. Sexual relations may as well involve dwellers of the earth and planets. Thus, Taming-Tamo, an enormous wiyu overflowed with virility, unceasingly chases his fellow females as well as human women, defying Donyi the sun by regularly capturing its wife, Polo the moon, to rape it, causing eclipses. 35 Elsewhere it is a cock which is caught having love with Polo. 36 Fecundation is not reduced to contact-coupling. Remote fertilization is also a possibility. For instance, bees take a tiny fragment of dirtiness on the skin of a man and will deposit it on the body of a woman who lives on the other side of the mountain: soon she becomes pregnant. 37 Elsewhere, a woman immobilized by ropes is fertilized by the cries of hunters passing at distance, just like the hen by the songs of the cock. 38 But the function of union is not limited to biological breeding. When it implies a set of exchanges and mutual obligations, acquiring a dimension of continuity and turning into a true alliance, the union between two beings possess other virtues. In Arunachal, numbers of accounts pertain to the unceasing quest for the bride, which compels the hero, an animal, a man or a spirit, to face all kinds of dangers and to deploy many artifices to obtain a partner. Tapeng, the bat, leaves to search a wife for Mirung-Guppu, disfigured by goitre. Reaching the place of the wiyu Sitking-Kedeng, in the underground world, it notices a girl husking rice. Pretending to be sick, Tapeng asks the girl's parents to organize a sacrifice for its healing. He then requires that all share the meat with him.

Tapeng finally addresses the girl: "Your parents ate my food, it was your bride-price, now you must follow me and marry my brother". 39 The trick -but this is also a pure political act -thus constitutes one of the means to obtain a wife. If a gift was accepted, voluntarily or not, a counter-gift must be returned. And a bride is a common compensation.

Abo-Teni, the first man, recklessly marries the fire. As could be expected, hardly has he embraced his new wife, that his genitals start to burn; soon, nothing remains of his virility out of a worm born from the wound. Searching a more lenient wife, he successively crosses several villages inhabited by animals, where each time a similar scenario replicated. A cock swallows his worm, his only wealth. Threatening to kill it, Abo-Teni compels the cock to follow him. Then the dog comes, which devours the cock and has to give Teni a whelp-slave. The history continues, through successive repetitions and interlockings. Lastly, Teni, provided with his new -but again single -belonging, the tree which crushed his mithun, reaches the village of Daini the sun. The sister of Daini, Mumsi, climbs on the tree, which breaks. Teni, once more, get angry and shouts at the culprit: "Marry me or I will kill you!". 40 Marriage, although the apex of reciprocity, does not conclude it. Quite to the contrary, it may be considered as its (new) starting point. The bride's parents, and later on the maternal uncle, become allies which will provide material help in time of shortage, and a political support in time of conflicts. 41 The maternal uncle constitutes, in violence, the ally par excellence.

Tikdeng, the Shimong frog, calls its "maternal-uncle-the-Siangriver" to engulf the village which refuses a wife to it. Then two other of its "maternal uncles" come to help, the Stone, which causes a landslide, and the Laughter, which gives spasms. 42 This leads us directly to one of the favourite mythological themes in Arunachal.

Caused by an insult, a personal animosity, a theft, jealousy, adultery, homicide, or defiance between leaders, appears the conflict. Among the Adi, another motive may be added: the control of land, trade routes and villages. The content of the accounts may reflect specificities in the social structures, as the importance of the Adi village solidarity. However, it is the fundamental values of prestige and reciprocity which strikingly emerge.

The myths explain how, following an original murder, the succession of revenges can last for eternity. It is, among others, the case of the constant fight between snakes and men.

Nibo, one of the first ancestors of the Ramo Adi, catches his sister flirting with a human-headed snake. Apparently revolted by this union, Nibo kills the two lovers. Unfortunately the idyll had been already consummated, and the girl had, a few hours earlier, given birth to a litter of small snakes. Since then, the snakes bite the men to avenge the murder of their original parents. 43 Conversely, the same Ramo claim that, when hunting the tiger, they avenge their ancestor, devoured by his own brother the first tiger. 44 But often, feud is limited in time, causes few wounds, and -essential aspectconcludes itself by a treaty and an alliance. In contrast with the Indian classical epics for example, which describe in length the bloody and morbid circumstances of war, details on physical violence occupy little place here. This is in tune with the morphology of traditional warfare in this area, based on ambush and raid and avoiding direct physical confrontation. Nevertheless, what matters before all is to stage the universal principles of feud and alliance, two principles which, in the logic of the myths, preexisted to the appearance of animals and men. The immediate motive behind feud is revenge in response to a murder. But the enemy -"opponent" would be a more exact termnever vanishes; in a form or another it always reappears out of its ashes so that finally alliance is possible between the two sides, generally through marriage. Murder is thus not restricted to destruction. From the corpses emerge new beings, from inflicted death -and not from any death -life is renewed. Revenge in itself is insufficient, and it is by the alliance, which comes to conclude the cycle of violence, that long-term prosperity is ensured, here thanks to the feeder rain.

Our preliminary aim was to analyze the concepts of conflict and alliance as depicted in the relationships between men and spirits.

In the pieces which we presented up to now, the spirits appeared on several occasions. It could be reasonably objected that these instances concerned a "mythical" space and time where the wiyu stand as a symbolic substitute of "the other", in order to describe the "real" interactions between men. Thus it is necessary to go ahead and to show how certain patterns obviously put in scene the relationship with the invisible beings, either implicitly, or explicitly like the following example, drawn from an Pasi Adi myth.

At that time, the world had already been divided between the men and the wiyu. The first lived in the fertile lowlands, the seconds in the high mountains and the forests. One day the wiyus captured a man and retained him captive. To release him, his kin offered to the kidnappers poultries, pigs and a mithun which they brought to the mountains. The wiyus accepted the ransom and promised to free their prisoner. But in the evening, instead of their brother, the men saw returning the cattle. Immediately, the animals were brought back to the country of the wiyus. But soon after they reappeared in the plains [where the Pasi Adi live]. For the third time, the men went to the mountains, and tried to settle the affair with the wiyus:

"We have given you, twice, those that you asked, but each time you have driven them back to us, and still you will not release our brother". The wiyus explained that as long as the cattle always fled away, they had to keep the prisoner. The men had an idea: "We will kill the animals, their soul will go to you, and return to us no more". The first sacrifices appeared and the prisoner was released. [START_REF]the terms "gods" and "spirit" in Elwin's translation have been replaced by respectively "spirits" and "soul[END_REF] The brilliant invention of sacrifice... From the time of its creation, this offering to the wiyus has been included in reciprocity. Sacrifice is more than mere propitiation and in any case not a mark of devotion. Like feud, it is concerned with a debt, with a cycle of exchange. Through reciprocity, the enemy, whether human or wiyu, becomes a partner, and later on, by the perpetuation of exchanges, an ally; from violence, exchange is born, from exchange, solidarity.

First of all, the ally provides material assistance: according to the Minyong Adi, when the men started to clear land for cultivation, they asked Kine-Dene, an underground wiyu, to provide them seeds. The dog, their messenger, was sent for that purpose.

Since then, the annual sacrifice of Mopum is offered to Kine-Dene in exchange for good harvests. 47 But the ally is also a mediator, who will help you to win over in litigation. When Kebung the rat and Petang the bird violently clashed over the possession of a salt mine, a kebang council was called to decide between them. Each one was represented by a spokesman, Dobak and Pukso for Petang, Nyoning for Kebung. The council admitted only spokesmen, not lawyers: Dobak defended the bird so much that all the members were exceeded and struck it on the beak. Nyoning also was beaten for the same reason.

Only Pukso did not take party and advised with the two adversaries to cease their quarrel. This was appreciated by Pedong-Koniyong, the delegate of men, who lined up in this opinion. [START_REF] Shimong | [END_REF] On behalf of other particular allies however, one awaits a total solidarity, up till violence: Robo, one of the original men, envied Doni, a deaf-mute being, whose exceptional qualities as a hunter, made him prosperous. All being jealous, lined up with Robo against Doni. To defend their insulated son, his parents called upon Tori-Mone, a wiyu full of wisdom, which sat on his shoulder to advise him, then upon Ute and Pore, to persuade the animals, especially the tigers and the snakes, not to assault him. Her mother made a shield of ekkam leaves, "like the darkness of night", so that he would be invisible to all, even wiyu. Doni finally concluded a treaty (omum) with Dadi, Master of the mithun, who would provide him with the required cattle. The fight then began: Ute and Pore are said to have covered Doni "as a shield"... [START_REF] Dhasmana | [END_REF] The multiple functions of the ally are thus specified: we saw him as a mediator; here he is a military advisor, a supplier of cattle -the nerve of war in Arunachal -, a spokesman, and finally a true mercenary. The allies of Doni are wiyu, but hardly symbolic ones: here the myth describes very concrete situations. The day before a raid, on behalf of the warriors, the priest call upon the allied spirits, which will protect them directly, and also call upon the enemy's allied spirits so that they will betray him. 50 Taru-Tase is a wiyu which can take the shape of a hawk or of a firefly. When two enemies are in war, it is incarnated in falcon and fight for those which offered him a sacrifice. But the other part may also offer a similar sacrifice. It then takes the shape of another hawk which will fight the first one. Thus, the victory on the ground of one or the other of the opponents depends on the fight occurring in the sky. Sometimes still, Taru-Tase also fights on the ground: it takes the shape of a firefly and blinds the enemy by landing on his eyes. [START_REF] Bori | [END_REF] Things can't be more explicit: certain spirits behave as pure mercenaries, serving those who spend for it. However, there are more clues in this piece. The hawk, indeed, does not appear haphazardly.

Of course, it is a tough fighter, with exceptional vision and speed.

But this predatory directly evokes the fighting function of the shaman. The wings of hawks, fixed on the back and held by hand, is an essential part of Nishi priest's panoply. [START_REF] Pandey | [END_REF] Before Apa-Tani raids, wings of a hawk are moved before the snoot of the dog which will be sacrificed. 53 One may wonder whether the hawk of the myth represents a wiyu or the officiant himself. As a matter of fact, the shaman may be identified with its guardian wiyu, which is obviously a warlike wiyu. We have seen that he officiates with his sword. This weapon is directly useful in his functions, for example while protecting the newborn child from the attacks of the spirits.

After his elder brothers had been, one by one, devoured by the Atu spirits at the time of their birth, Atungi Aru was saved by the dao that her mother used to keep the predatories away. The son of Atungi Aru, Aru Pe, became the first igu priest of the Idu Mishmi. Everywhere where a child had just been born, Atu Pe came. He stayed near the baby and cut the Atu into pieces. 54 The shaman is undoubtedly an intercessor, but some wiyu remain insensitive to his diplomacy: so he should be able to handle weapons, even beyond his own death.

Under the ground lives an enormous creature, Modi-Mobe-Ane, which carries the world on its back. When a famous shaman dies, he meets this monster which tries to bar the road. The shaman must then strike it with his dao. Angrily, Modi moves its spine violently, causing earthquakes. 55

IV. A political cosmology

How to organize the different patterns that have come to light till now? Is it possible to outline the broader ideological structure into which cosmological conceptions and the mechanisms of violence and alliance are connected? Let us proceed from general to particular, i.e. in the time of the myth, from the origins towards the present time.

The universe consists of beings with similar behaviours and concerns. Their shape, their external appearance may be different, but their essence is identical, and this also pertains to "inanimate" objects. Descending from the same original father or mother, the animals, men and spirits remain brothers, therefore of similar statute. The death is not itself characteristic of the sons of Tani. Spirits, like men, live several successive lives in distinct worlds, where each time they marry, cultivate new lands and face new enemies. 56 Just as visible and invisible worlds are hardly different, life after death does not differ from life before death.

In this non-divided universe, all beings are on an equal footing, whatever their respective form, therefore in competition one with the others, but also in complementarity: they can devour anyone, be devoured by anyone, trap, be trapped, help, marry, reject, and especially fight anyone without hesitation.

As a hunter, man cooperates with spirit to set traps and to beat up the game. But each partner may deceive or be deceived by the other, falsely claim that it is its own arrow who killed the deer, or empty the nets of his rival... As farmers, man and spirit help each other in the fields; but soon quarrels emerge, and each one must mark its territory. Lastly, both seek the wife who will give them children; if coveting the same girl, they will have to use bravery and tricks to win the decision of the father-in-law.

In short, in this particular context of generalized nondivision, [START_REF]On the "non-division" as a fundamental concept in some stateless societies[END_REF] it is necessary to defend one's own ambitions, and unceasingly assert one's own identity. Therefore, the son of Tani send back the spirits in the mountains, cut the tongues of animals, burn their houses and deprive them of their weapons. It is not meaningless that monkeys are the principal targets of this original segregation: their particular form makes the man doubt of his own specificity. [START_REF] Adi | [END_REF] However, although a constant sublimation of the distinction towards the equal-other is necessary, as a mean for each individual and each group to work out his own identity, this Other cannot be radically rejected. As a father of the wife who will give you children, as a talented mediator who will defend your claims before the assemblies, and as an ally which will provide you with help during scarcities and war, he is essential. However threatening he may be, he must exist, first as an opponent, then as a partner, allowing union, complementarily, thus fertilization, from which will emerge prosperity. We saw how the earth was opposed to the sky, feared it, and unceasingly tried to escape him: however, it was out of their encounter that the trees, the crops and all kinds of wealth sprung up.

Formerly were Buru-Gudu in the water and Doje-Karangu in the sky. [START_REF]the late 1940's to find out these animals, without success… see R. Izzard, The Hunt for the Buru[END_REF] Each one had a girl, but wished a wife. Doje went to see Buru: "Let me marry your daughter and you can marry mine". Buru answered: "How can such a thing be? You are in the sky, I am in the water". Doje gave an idea: "I will make a path from the sky to the earth and in one day you will be able to fetch my daughter and I will fetch yours". Following the path, Doje went down towards the water and Buru went up towards the sky. On their way back, each one with his new wife, they met "in the space between earth and heaven": "We must do no work, nor approach our wives, nor go hunting. If we fail in this, there will be disaster for us both"... 60 The acquisition of prosperity implies a meeting between two separated entities. Consequently, the construction of a path which will connect them and allow exchanges proves vital. In Northeast India -as in neighbouring areas -, every element evoking a communication between separated domains, like a rope, a ladder or a bridge, is directly associated to the concepts of prosperity, abundance, vitality. Shimong Adi mention "the long bridge connecting the country of the Nimat (Northern spirits) to that of Tani" as one of the elements which, resulting from the dismemberment of a gigantic original creature, formed the land relief. 61 According to the Singpho (of Eastern Arunachal), the first men landed on the Himalayas along silver and golden ladders. [START_REF] Macdonald | This obviously reminds the silver ladders along which Thai kings used to come down on earth ; for the Ahom see[END_REF] For the Shimong, those who dream of building a bridge or fighting an enemy -please note this association -will soon catch a big game. 63 It should be made clear that connecting two entities does not aim at confusing them. Alliance should mean neither obliteration of the limits nor submission of a protagonist to the other. Before "meeting the other" and exchanging with him, it is essential to show who you are, to be recognized by him and others as a specific individual, member of a specific group. As a matter of fact, who is a best ally than the one you fought without mercy and who far from submitting to you, took up arms to defend his prestige through violence? And thanks to him, you could publicly and dramatically express your own otherness and your own claims to independence. This is the logic which, it seems to us, explains why the majority of feuds ended in an alliance treaty. 64 It should be stressed that, in the symbolic system of these myths, violence in itself, as well as the death which it brings sometimes, are in themselves sources of wealth. Out of each cadaver a tree inevitably grows or an animal comes out. The process leading from violence to prosperity is often depicted by a short cut, in order to better point its essence. But sometimes, the precision of the details does not leave any doubt about the cause-effect link between violence and prosperity, as when murder is directly identified with the acquisition of a wife. 65 Midang-Kanga avenges his two sons, devoured by a large wiyu, his own friend in addition. After killing it, he throws the body in the jungle... Later, coming back to the spot, he discovers the desiccated skeleton, which he brings to the great blacksmith Ningnu-Botte. [START_REF]It is likely that "Botte" refers here to a Tibetan or at least to a Tibetan-like character. "Botte" closely resembles "Bhotya" or "Bhote" by which North Indians call the people of Tibetan culture. Moreover, most of the valuable metallic objects owned by Arunachal people were formely bought from the Northern fringes. On the trade of valuable objects in Arunachal, see[END_REF] The blacksmith transforms the skull into a cauldron (danki), the arms into rifles, the fingers and the teeth into beads, the palm into a beyop belt and gives these objects to Midang-Kanga... [START_REF]The beyop is a belt of metal disks worn by pubescent Adi girls, and strongly connected with fertility[END_REF] The Tibetan cauldron, danki, the rifle, the beads and the beyop belt are all part of the dowry of the Adi bride. [START_REF] On | [END_REF] Revenge is thus explicitely equated to a matrimonial transaction between the wiyu and Midang.

The realization of alliance, possibly marking the temporary end of a feud, thus allows the initiation of a cycle of exchanges of goods and services. It does also set up a situation in which the attitudes between new partners remain fundamentally ambivalent. This is particularly true of matrimonial alliance. Through his daughter, who entered your home, who cooks your meal, raises your children and approaches your treasure, the adversary, under its mask of fatherin-law, constitutes a persistent threat. Because he may use his new statute, in particular the commensality now granted to him, to better abuse your confidence, impede your freedom, if not slaughter you. The wife herself, due to the powers linked to her feminine nature, is prone -according to the very Adi formulations -to "block your path", to "make you loose your way", i.e. to prevent you asserting your identity in war, or simply making your arrows miss the game.

The types of taboos imposed to the women in period of hunting or war makes it possible to draw up an image of the dangerous aspects of wives [START_REF] Particularly | India's NE Frontier in the 19th Century[END_REF] : The woman is endowed with the uterus which, although carrying the baby may as well prevents its birth.

Throughout the day, while cooking, she rubs elbows with these containers of which she "closes" the lid. When weaving, she handles the shuttle, which passes through the weft to "tighten" the fabric.

She prepares the beer, which makes man loose their lucidity. Lastly, she produces this menstrual blood, which brings sterility... 70 It seems that the woman is symbolicaly associated with inhibiting acts, such as closing a space, barring the road, disturbing the sight, sterilizing... The man on the contrary, would be endowed with the opposite powers: to open, move out, release... The confrontation between these two opposite dynamics appears clearly at the time of a difficult delivery, here among the Adi:

An egg is posed on the ground. The mother must break it with her knee, then drink cold water. The husband makes a grass wristlet, places it in front of the woman and proclaims: "If I forgot anything, if something remained closed, if an earthenware jar or a basket were not opened, it is my own fault". He then slices the wristlet with his dao. 71 The powers associated with women may not be regarded as only negative: when it is necessary, in the case of an invisible threat, to "bar the path", to close up the socialized space of the village or the house, women play an essential role. It is they, in particular, who will welcome the warriors coming back from the raid, and who are followed by the malicious spells cast by the enemy, or against the enemy. When Padam-Minyong warriors are out for war, the use of the weaving shuttle (sumpa) is strictly forbidden in their village, so that the advance of the attackers will not be blocked; on the contrary, as the warriors reach home, women hold their sumpa, enabling a magic enclosure of the village against the return of the warlike spirits. 72 Thus the nature and functions of man (male) is defined dialectically by opposition to the domestic and enclosing functions of the woman: he is facing outside, his function consists in "opening", for the renewal of his group's vitality. [START_REF]This illustrates quite faithfully the general principles enunciated in Leach[END_REF] Rabbo, ancestor of the spirits and eternal brother-enemy of Nibu, ancestor of the men, succeeds in locking him between two dugouts, which it pushes in the river. Nibu, terrified, shouts: "I want to go on living, I want to create the race of men...". 74 This idea may be considered as a central one: wealth, prestige, fertility, power, are acquired by leaving the civilized/socialized world, or in other words, by opening it to the external and dangerous world. To "leave" or "open" may be in fact summarized under a single act: to communicate with the Other World or with the Other, to draw the substance of survival, of perenniality. In this respect, all major acts, whether economic like hunting or trade, social like marriage, political like war or alliance, or religious like sacrifying or fighting the invisible, are conceived as transactions with the outside. Logically, dangers are conceived as external, and defence consists in closing the socialized space, in cutting it from the external world. At the end of the rite marking the control of war trophy, Apa-Tani burn it and bury the ashes under a stone: "Go to your place, we send to you under the ground, we close the door of the earth... Till ten generations you will not be able to hurt our descendants…". 75 The ambivalence of the ideal act reflects the ambivalence of the Other, living in the external world. It is on this precise point that the political values are connected which those that social anthropology use to call "religious". Indeed, the invisible spirit is not essentially different from the alien human. Like him, it is both an incontrovertible means of prosperity and a permanent source of danger. The attitude towards it will thus follow the same lines:

seeking transaction but protecting oneself, and especially keeping one's integrity, one's own freedom. According to this ideology, one does not submit to a deity, one fights it, one pursues it with the help of the priest, one treats it on an equal footing, one pays ransoms to it, not devout offerings. The spirit is called upon like the maternal uncle, to help fight the enemy, not as a redresser of wrongs.

Arunachali myths outline a true political cosmology, at the same time deeply egocentric -everything is centred on me and my fellows -and eminently egalitarian -no higher authority is able to infringe on individual designs. From this point of view, the priest is much more than a mere sacrificator or a therapist. One of the dreams which predicts the vocation of shaman among Gungu Nishi is crossing a large river in a raft [START_REF] Pandey | [END_REF] : the shaman is the one who "knows how to cross", like the hunter, like the warrior whose outfit he wears. Therefore the Nishi priest may be proud to exhibit in his house, on the wall devoted to hunting trophies, the scapulas of animals sacrificed under his authority. [START_REF] Shukla | [END_REF] Obviously, war and sacrifice are not similar in their practical aspects. However it would not be completely unwise to suggest that in both instances the same representations and values are at stake.

Are the ideological structures put in light here restricted to Central Arunachal Pradesh? And would they provide useful clues for the study of political and religious ideologies of neighbouring societies? One knows the importance of reciprocity in Ancient China's wars. If "eating the enemy" was a common expression -to say the least -, the same enemy frequently became a son-in-law, or was a son-in-law already, and in all cases finally turned into an ally, after the treaty closing the feud. 78 But it is primarily in Tibet that the ideas highlighted here find an echo, not only in the great number of symbolic elements, but strikingly, in their structural arrangement.

Tibetans, including members of the nobility, trace their origins from union between human and invisibles spirits. Original beings, which populated the world before the appearance of men, had both human and divine features. The bon tradition describes the conception of its saints as an association between the "sperm-arrow" of their father and the "blood-spindle" of their mother. [START_REF] Stein | [END_REF] Are we far from the Arunachali opposition between the hunting male and the weaving female? On the same point, it is said that husband and wife are opposed "like sky and earth, right and left, maternal uncle and uterine nephew". 80 Similarities are confirmed in the human attitudes towards invisible forces. In the event of an attack, one tried to "entrap" the spirits, to "crush" them, not -at least in the ancient periods -to submit oneself by offerings. In troubled times, two favourite techniques were used: "closing the gate of the sky" and "cutting the rope to the sky". When, on the contrary, it became necessary to communicate with the invisible, in order to enquire about the god's aims or "bring the luck", the rope linking sky and earth (mu) was "fasten". 81 Thus, in Tibet as in Arunachal, prosperity is conceptualized as resulting from a communion with the outside of the socialized space, although "outside" is there identified with the sky: appearing in fundamentally egalitarian societies, the State would impose to the political and religious perceptions a new dimension, vertical and exclusive.

In Tibet similarly, the invisible being is dangerous, therefore powerful, it is possibly an enemy, to which the bon priest pays "ransoms". In the event of conflict, the warlike gods stand "on the shoulder of man", like Tori-Mone on the shoulder of Doni, to protect it from the enemy. And the individual is "structurally" defined as the outcome of the relationships between "the god of man, the god of woman, the god of maternal uncle, the god of war and the god of land". 82 This statement of the pillars of society could have perfectly applied to Arunachal, if it had not included the "god of land". But in a sense, its absence in the South of the Himalayas does nothing but reinforce the similarities between the two ideologies: ritual fixing within a given territory, which expresses the concrete attachment to a particular area, is undoubtedly the principal characteristic which distinguishes the populations of Eastern Tibet from those of Arunachal. In Eastern Himalayas, the indigenous histories tell of a constant mobility. 83 Local migrations were still going on recently; shifting of whole villages in the 1960's were reported to me in the Galo-Minyong area (Lower Siang).

The Apa-Tani's sedentarity would slightly diverge from this model: their way of life is indissociable from a very specific space, that of a flat valley convenient for irrigated rice cultivation. And the myth of foundation of the site thus tells how the first two heropioneers drained the marshes covering the country, edifying the cradle of this civilization in a particular point of space. 84 But, to our knowledge, no localised and sovereign divinity sublimates this territorial rooting. The absence of a "god of the land" and the vivid memory of the migrations which led them to their present valley, 85 show that the Apa-Tani sedentarisation has not yet been translated mythologically and ritually.

Our preliminary interrogation on the perception of violence in Arunachal may be answered to by reinstating violence in its broader ideological framework. This series of discourses on the world -can one still speak about "myths"? -make us discern an undivided world, deeply egalitarian, in which each one is able, but also compelled, to assert his own identity. If the universe is ordered, it is not by a superimposed hierarchy, but indeed by the sovereign principle of reciprocity. Building a relationship with the Other means communicating with the Outside, trying to make coincide disjoined spaces, but taking care not to blend them. To work one's way in the world, to assert one's vitality, confrontation with fellows is unavoidable. Because it is by the fight, and then by the gift, in short by the reciprocity, that prestige and prosperity are acquired.

From such a point of view, how can religion be demarcated from politics? Possible answers obviously stumble on the concept of "sacred". If the sacred is a domain "unreachable" to man, governed by transcendent rules, then there is no "religious act" in these societies. If the sacred on the contrary means an external but penetrable world, the relationships with the invisible may be qualified as "religious", but it is as well the case of feud and matrimonial exchange, which, as we saw, consist also in an opening, in a communication with the Outside. And thus the proposal by which Durkheim defined the nature of the religious act, would perfectly account for the originality of the political ideas in this part of the Himalayas: "This placing in relationship [with the sacred] in itself is always a delicate operation that requires precautions and a more or less complex initiation. Yet such an operation is impossible if the profane does not lose its specific traits, and if it does not become sacred itself in some measure and to some degree".

  Alliance presumes firstly reciprocity, i.e. a mutual exchange of services, and possibly of goods, between the partners. And furthermore the bond should be a lasting one, and not only a shortterm contract. When the spirits defend the men, what do they receive in exchange? It would be tempting to claim that the sacrifices offered at the time of the invocations are intended to bind the invisible forces into a debt bond. Available descriptions of rituals are too scarce and superficial to go that far. Continuity of the relationship must also be questioned. During the annual festival of prosperity, certain Nishi go down on the banks of large rivers to ask the aquatic wiyu to prevent the use of the bridges by hostile warriors.27 Does such a ritual point to a relationship of lasting alliance or is it just a recurrent request? Additional details on the content of the ritual would have helped to answer. On this point, as in others, investigation on the indigenous concepts is hampered by the lack of data on religious practices, which have for long characterized cultural studies on Arunachal. However, thanks to the "tribal romanticism" inspired by J. Nehru and V. Elwin a rich mythological corpus has been accumulated, and usefully balances our ignorance on rituals.

  Tani and Epom live as neighbours, settled respectively on the banks of the Siang river and in the large trees of the forest. Tani and Epom hardly commiserate and carefully avoid glancing at each other. So Tani covers his eyes behind a leave, which will prevent him, and his offsprings to come, to see Epom. But Epom does not find anything else that a basket, through which he is still able to see Tani. The two brothers decide to share game: each one thus sets his own traps. Tani however, jealous of the preys of Epom, uses to steal them. The two elder sons of Tani disappear while hunting in the forest. Helped in his quest by the bird Sigo-Pereng,

  So he decides to throw the corpse from the top of a hill over the village of Nite-Nippong. Enraged by the offence, the villagers pursue the boy, seize him and kill him. Thus Langkon must himself assume the revenge. So he submits Nite-Nippong. Back home, he finds his

	brother Langtang assassinated by a wiyu. After many investigations, he
	finds out the murderer and kills it. Immediately, an impressive tree
	emerges from the corpse of the victim. Langkon, irritated, sets on the tree
	and cuts it down. From the trunk comes out a beetle, but hardly had the
	insatiable avenger taken his sword that the insect metamorphoses into a
	pretty girl. Langkon, moved by such a sight, decides to marry her. The
	gives birth to Pedong-Nane, mother of waters, and rains start fertilizing
	the ground. Then two sons come, Tani, ancestor of the men, and Taro,
	ancestor of the spirits. 45
	At that time, the earth, dry and barren, carried neither tree,
	neither grass, nor the smallest insect. Langkon-Tumbo had a son, and his
	brother, Langtang-Kenne, had nine. The nine sons of Langtang are one by one
	poisoned by an old wiyu which cuts them into pieces that it makes dry. The
	single son of Langkon sharpens his sword and set out to search for his

cousins... He arrives at the cave of the wiyu, which like usual invites him to a meal. But Langkon surreptitiously swaps the two shares, and it is the wiyu who this time is poisoned... "If I bury this evil body, who knows what will grow from it?"


J A De Doná

‡ F Suryawan
email: fajar.suryawan@newcastle.edu.au

M M Seron

J Lévine

A flatness-based iterative method for reference trajectory generation in constrained NMPC

Keywords: flatness, trajectory generation, B-splines, Nonlinear MPC

This paper proposes a novel methodology that combines the differential flatness formalism for trajectory generation of nonlinear systems, and the use of a model predictive control (MPC) strategy for constraint handling. The methodology consists of a trajectory generator that generates a reference trajectory parameterised by splines, and with the property that it satisfies performance objectives. The reference trajectory is generated iteratively in accordance with information received from the MPC formulation. This interplay with MPC guarantees that the trajectory generator receives feedback from present and future constraints for real-time trajectory generation.

Introduction

Differential flatness [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF] is a property of some controlled (linear or nonlinear) dynamical systems, often encountered in applications, which allows for a complete parameterisation of all system variables (inputs and states) in terms of a finite number of independent variables, called flat outputs, and a finite number of their time derivatives. We consider a general system ẋ(t) = f x(t), u(t) ,

where x(t) ∈ R n is the state vector and u(t) ∈ R m is the input vector. If the system is flat [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF], we can write all trajectories (x(t), u(t)) satisfying the differential equation in terms of a finite set of variables, known as the flat output, y(t) ∈ R m and a finite number of their derivatives:

x(t) = Υ y(t), ẏ(t), ÿ(t), . . . , y (r) (t) , u(t) = Ψ y(t), ẏ(t), ÿ(t), . . . , y (r+1) (t) .

(

) 2
The parameterisation [START_REF] Piegl | The NURBS Book[END_REF], afforded by the flatness property, allows to simplify (especially in the case of nonlinear flat systems) the generation of reference trajectories (trajectory planning). Typically, some 'desired' reference trajectory is prescribed for the flat output, y ref , and the corresponding input and state trajectories for the system are obtained from (2); namely, u ref (t) = Ψ y ref (t), ẏref (t), . . . , (y ref (t)) (r+1) , x ref (t) = Υ y ref (t), . . . , (y ref (t)) (r) . However, a very common requirement in engineering applications is for some of the variables of the dynamical system to satisfy a number of constraints, usually expressed as inequality constraints. For example the input and state of the system can be required to satisfy u ∈ U and x ∈ X, where U ⊂ R m and X ⊂ R n are specified constraint sets. The presence of such constraints makes trajectory generation for nonlinear systems (in general) a highly nontrivial task, due to the ensuing nonlinearity of the mappings Υ(•) and Φ(•) in ( 2). (In particular, it is typically very difficult, to specify constraint sets for the flat output variables y in terms of the constraint sets for u and x, respectively, U and X.)

In this paper, we propose a novel methodology that exploits the flatness parameterisation (2) for trajectory generation and the use of the Model Predictive Control (MPC) strategy for constraints handling. The methodology consists of a trajectory generator module, that generates a reference trajectory y ref (t) with the property that it satisfies performance objectives (e.g., satisfies given initial and final conditions, passes through a given set of way-points, etc.). There are points of contact between some aspects of the approach advocated in this paper and, for example, the work in [START_REF] Flores | Trajectory generation for differentially flat systems via NURBS basis functions with obstacle avoidance[END_REF] where the problem of generation of a reference trajectory for a nonlinear flat system subject to constraints is formulated as a NonLinear Programming (NLP) problem. One of the main drawbacks of posing the problem as a NLP optimisation problem is that, in general, it is very difficult to prove convergence, or convergence to a global optimum. Hence, in this paper we explore an alternative algorithm for trajectory generation for nonlinear flat systems, in the presence of constraints, that is based on the information provided by a model predictive control (MPC) formulation. The approach is, to the best of the authors knowledge, the first attempt to combine the differential flatness formalism with model predictive control techniques in an iterative algorithm for constrained nonlinear trajectory generation. No proofs of convergence are available at present, due to the challenging nature of these problems, and this will be of concern in future work. However, simulation results, as the ones presented in this paper, are promissory and indicate that the effort of developing such algorithms and investigating formal proofs of convergence is worthwhile.

Thus, in the methodology investigated in this paper, the reference trajectory y ref (t) is generated iteratively in accordance with information (predicted in real time) received from an MPC formulation. That way, the trajectory generator receives "feedback from the (present and future) constraints" of the system while generating the desired trajectory. Thus, the proposed method unites two important properties. Firstly, since the trajectories are generated via the flatness parameterisation (2), with "feedback from the constraints," they constitute natural trajectories for the nominal model to follow. And, secondly, the information generated by an MPC formulation (via the solution of a Quadratic Programming optimisation, based on the linearised dynamics around the given reference trajectory) ensures that the system constraints are taken into account.

Flatness and trajectory parameterisation

We consider the problem of steering system (1) from an initial state at time t 0 to a final state at time t f . Note that, in a Model Predictive Control context, this fixed interval problem is one window of a bigger scheme, implemented repeatedly in a receding horizon fashion. In order to generate a suitable reference trajectory, we will use a spline parameterisation, as explained in the following sections.

Parameterisation of flat outputs and their derivatives

We parameterise each of the flat outputs y j (t), j = 1, . . . , m, as

y j (t) = N i=1 λ i (t)P ij ; t ∈ [t 0 , t f ], (3)
where λ i , i = 1, . . . , N , is a set of basis functions, which is the same for each flat output y j . The basis functions are assumed to be

λ i ∈ C r+1 [t 0 , t f ], i = 1, . . . , N .
This reduces the problem of characterising a function in an infinite dimensional space to finding a finite set of parameters P ij . In a discrete set of M +1 sampling times, t 0 , t 1 , . . . , t M = t f , this parameterisation becomes

Y j = G 0 P j , (4)
where Y j [y j (t 0 ), y j (t 1 ), . . . , y j (t f )] T , P j [P 1j , . . . , P N j ] T is a vector containing the parameters P ij , i = 1, . . . , N , defined in (3), and

G 0    λ 1 (t 0 ) . . . λ N (t 0 ) . . . . . . . . . λ 1 (t f ) . . . λ N (t f )    (5)
is the basis function matrix (also known as blending matrix ). Collecting all the m flat outputs, we have

Y Y 1 Y 2 . . . Y m =    y 1 (t 0 ) y 2 (t 0 ) . . . y m (t 0 ) . . . . . . . . . . . . y 1 (t f ) y 2 (t f ) . . . y m (t f )    = G 0 • P 1 P 2 . . . P m = G 0 P = Y (P ), ( 6
)
where Y is an (M +1)×m output matrix, G 0 is the (M +1)×N blending matrix, and P P 1 P 2 . . . P m is an N × m matrix containing the coefficients P ij of the parameterisation (3). The rows of P are m-dimensional vectors called control points.

Furthermore, we can also build the time-derivatives of y i at discrete points in time, by successively differentiating (3) followed by time-discretisation. Doing this and using the notation as in (6), we obtain

Y (1) = G 1 P ; Y (2) = G 2 P ; Y (3) = G 3 P ; . . . Y (r+1) = G r+1 P ; ( 7
)
where

Y (q) [Y (q) 1 Y (q) 2 . . . Y (q)
m ], and

Y (q) j     d q
dt q y j (t) t=t0 . . .

d q dt q y j (t) t=t f     ; G q     d q dt q λ 1 (t) t=t0 . . . d q dt q λ N (t) t=t0 . . . . . . . . . d q dt q λ 1 (t) t=t f . . . d q dt q λ N (t) t=t f     , (8)
with j = 1, . . . , m and q = 1, . . . , r + 1.

Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008

Trajectory parameterisation using splines

Given a reference trajectory parameterised as in (6), Y ref = G 0 P ref , with specified reference control points P ref , in this section we will show how to parameterise variations around that reference trajectory using splines. In this paper, clamped B-splines [START_REF] Piegl | The NURBS Book[END_REF] are chosen as basis functions which results in the blending matrix G 0 having a particular structure. Namely, G 0 has only one non-zero element in the first row (which lies in the first column) and only one non-zero element in the last row (which lies in the last column). The matrix G 1 has two non-zero elements in the first row (which lie in the first and second column) and two non-zero elements in the last row (which lie in the last and second-last column). The matrix G 2 has a similar property with three non-zero elements, etc. More properties of B-splines can be found in, e.g., [START_REF] Piegl | The NURBS Book[END_REF]. Notice from (3) that,

d q dt q y j (t) t=t0 = N i=1 d q dt q λ i (t) t=t0 P ij , (9)
for q = 0, 1, . . . , r + 1; j = 1, . . . , m. We can see from (9) and the structure of G 0 discussed above that y j (t 0 ) = λ 1 (t 0 )P 1j , j = 1, . . . , m, and hence, by fixing the first row of P , P 1j = P ref 1j , j = 1, . . . , m, the flat outputs at time t 0 are fixed and equal to the corresponding values of the reference trajectory. Fixing more rows of P (up to the order of the B-spline) fixes the flat output derivatives at time t 0 (e.g. fixing two rows fixes the first derivatives, three rows fixes the second derivatives, etc.). This property (made possible by the structure of G q , q = 0, 1, . . . ) can be used to maintain fixed end-points. For example, prescribed position and first and second order derivatives of the flat output at times t 0 and t f , as in the rest-to-rest case, can be maintained by holding the 'external' control points (the three topmost and the three bottommost rows of P in Eq. ( 6)) fixed. This can be achieved by reparameterising P as:

P = P ref + ρ P ; ρ = [0 I 0] T , (10)
where matrix P is an [N -(l 1 + l 2 )] × m matrix that parameterises the deviation from the 'internal' control points of P ref and ρ is an N × [N -(l 1 + l 2 )] matrix with the l 1 top rows set equal to zero, the l 2 bottom rows set equal to zero and the identity matrix of dimension [N -

(l 1 + l 2 )] × [N -(l 1 + l 2 )
] in the middle.

3 Using MPC to shape the reference trajectory

In this section we will develop an iterative algorithm for trajectory generation for nonlinear systems, subject to constraints, that is based on information provided by model predictive control (MPC). The main motivation for resorting to MPC is to exploit the well-known capabilities for handling constraints of this control technique. The basic idea is to propose an initial reference trajectory based purely on performance considerations, parameterised as in (6), i.e. Y ref,0 = G 0 P ref,0 (it is assumed here that an initial set of reference control points P ref,0 is specified), and to then use an MPC formulation to give information as to how well that trajectory can be followed in the presence of constraints and, moreover, which parts of the original trajectory are problematic and should be Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008 modified. Then a new reference trajectory is generated based on a trade-off between the information obtained from MPC (this information can be regarded as the feedback from the constraints) and the original performance specifications. This interplay between performance objectives and MPC (feedback from constraints) is then iterated, and the challenge is to devise an algorithm such that the iteration converges to a suitable reference trajectory.

MPC formulation

We will assume, for simplicity, that the flat output is given by a (possibly nonlinear) combination of the states:

y(t) = h x(t) . (11)
Note that, although this is not the most general case for flat systems, many examples of practical interest satisfy this assumption (e.g., models of cranes, nonholonomic cars, etc.). Given a specified reference trajectory for the flat output, parameterised by control points P ref as explained in the preceding section:

y ref j (t) = N i=1 λ i (t)P ref ij ; t ∈ [t 0 , t f ], (12)
for j = 1, . . . , m, we compute the corresponding state and input reference trajectories, x ref (t) and u ref (t), respectively, from (2). Note, in particular, that the flatness formulation implies that these trajectories satisfy the system's equation ẋref . The resulting linear time varying system is then discretised in time, so that the following time varying discrete time system is obtained:

(t) = f x ref (t), u ref (t) .
xk+1 = A k xk + B k ũk , ỹk = C k xk . ( 14
)
In the discretisation (14) we consider a sampling interval T s (t f -t 0 )/M , so that exactly M sampling intervals fit in the interval of definition of the splines, [t 0 , t f ]. Moreover, we define a grid of equally spaced sampling times, t k = t 0 + kT s , k = 0, . . . , M . Note that the variables in (14) (cf. ( 13)) are measured with respect to the reference trajectory. Thus we will consider an MPC formulation for the time varying system (14) where the performance objective is regulation to the origin (this will ensure tracking of the respective reference trajectories).

Given the current state of the plant at time t, x(t), we compute x0 x(t)x ref (t 0 ) (where x ref (t 0 ) is obtained from (12) using ( 2)). The aim is to find the M -move control sequence {ũ k } {ũ 0 , . . . , ũM-1 } that minimises the finite horizon objective function:

V M ({x k }, {ũ k }, {ỹ k }) 1 2 xT M P xM + 1 2 M-1 k=0 ỹT k Qỹ k + 1 2 M-1 k=0 ũT k Rũ k , (15)
Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008 subject to the system equations ( 14) and x0 x(t) -x ref (t 0 ), and where P ≥ 0, Q ≥ 0, R > 0, and M is the prediction horizon. Using the standard vectorised notation

x [x T 1 . . . xT M ] T , ũ [ũ T 0 . . . ũT M-1 ]
T , the cost function (15) can be written in compact form as:

V M = 1 2 xT 0 C T 0 QC 0 x0 + 1 2 xT Qx + 1 2 ũT Rũ, (16)
where

Q diag{C T 1 QC 1 , . . . , C T M-1 QC M-1 , P } and R diag{R, .
. . , R}. The system's state evolution from k = 0 to M can be expressed as x = Γũ + Ωx 0 , where Γ and Ω are formed from the system's A k and B k matrices (see, e.g., [START_REF] Goodwin | Constrained Control and Estimation: An Optimisation Approach[END_REF]). Substituting this expression for x into (16) yields:

V M = V + 1 2 ũT H ũ + ũT F x0 , where V is a constant term, H Γ T QΓ + R and F Γ T QΩ.
If the problem is constrained, for example with input constraints |u(t)| ≤ u max , then the solution is obtained from the following quadratic program:

ũopt = [(ũ opt 0 ) T . . . (ũ opt M-1 ) T ] T arg min ũ 1 2 ũT H ũ + ũT F x0 subject to |u ref + ũ| ≤ U max , (17)
where

u ref [(u ref (t 0 )) T (u ref (t 1 )) T . . . (u ref (t M-1 )) T ] T , U max [u T max . . . u T max ]
T , and the absolute value and the inequality are interpreted element-wise. (Other types of constraints, e.g., state and output constraints, can be incorporated in (17) in a straightforward manner.)

The corresponding j-th flat output trajectory, j = 1, . . . , m, obtained by MPC is computed from the result of (17), using (13) and ( 14). Using the expression x = Γũ + Ωx 0 , the MPC flat output trajectory can be expressed as:

Y mpc j C j x0 Γũ opt + Ωx 0 + Y ref j , (18)
where Y mpc j and Y ref j are the j-th flat output sequences stacked over time [defined similarly to Y j in (4)], and C j diag{C 0,j , . . . , C M,j }, where C k,j is the j-th row of the time-varying matrix C k , defined in (14) for k = 0, . . . , M . In an MPC implementation, one then applies the first control move obtained in (17), ũopt 0 , and the process is repeated in a receding horizon fashion. However, in our proposed implementation (see next subsection) this process is iterated before the actual control input is applied.

Iterative method for reference trajectory generation

In this section we present the iterative algorithm that is proposed in this paper. The algorithm starts from a set of specified initial control points P ref,0 that parameterise an initial reference trajectory Y ref,0 = G 0 P ref,0 which is generated based on performance considerations, and then it utilises the information about the effect of the constraints, provided by the MPC formulation, to update the reference trajectory through successive sets of control points, P ref,0 , P ref,1 , . . . , P ref,k , . . ., etc.

Step 1 Given a set of control points P ref,k ;

Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008

Step 2 Compute, from (6), Y ref,k = G 0 P ref,k ;

Step 3 Compute Y mpc,k j from (12)-( 18). Note that Y mpc,k so obtained is a (in general nonlinear) function of

P ref,k , that is, Y mpc,k = G P ref,k .
Step 4 Given Y mpc,k , find the variation of the 'internal' control points in the parameterisation (10), denoted P mpc,k , that gives a reference trajectory that is closest in a least-squares sense to Y mpc,k . Namely,

P mpc,k j = (G 0 ρ) T G 0 ρ -1 G 0 ρ T Y mpc,k j -G 0 P ref,k j . ( 19
)
Step 5 Update the control points according to: P ref,k+1 = P ref,k + ρ P mpc,k .

Step 6 While (a weighted 2-norm of) the difference P

Simulation Example

In this section we will test the previous algorithm on a classical example of a flat system, a nonholonomic car system. The system is modeled by the equations: ḋx (t) = u(t) cos θ(t), ḋy (t) = u(t) sin θ(t) and θ(t) = (1/l)u(t) tan ϕ(t), where the state d x (t) is the displacement in the "x-direction", the state d y (t) is the displacement in the "y-direction", the state θ(t) is the angle of the car with respect to the x-axis, the input u(t) is the velocity of the car, the input ϕ(t) is the angle of the steering wheels, and l is the distance between the front and the rear wheels. It is straightforward to determine that the flat output for this system is given by y(t) = d x (t), d y (t) . A matrix of initial control points, P ref,0 , is chosen so that, together with the parameterisation (12) using cubic B-splines λ i (t), gives the initial reference trajectory y ref,0 shown with a dotted line in Figure 1(a). The control inputs are assumed to be subject to the constraints u(t) ≤ 0.8 and |ϕ| ≤ 0.45. The inputs corresponding to the initial reference trajectory y ref,0 are shown with dotted lines in Figures 1(c) and (d), far exceeding the constraint limits. The result after 2, respectively 50, iterations of the algorithm is shown in Figures 1(a),1(c), and 1(d) with dashed, respectively solid, lines. Notice that the algorithm produces a final reference trajectory which is close to the initial reference trajectory and with associated inputs only mildly exceeding the constraints. In addition, the initial and final end-point conditions are maintained. A measure of convergence of the algorithm, strategy for constraint handling has been proposed. The methodology consists of a trajectory generator that generates a reference trajectory parameterised by splines, and with the property that it satisfies performance objectives. The reference trajectory is generated iteratively in accordance with information received from the MPC formulation. The performance of the iterative scheme has been illustrated with a simulation example. Future work will focus on investigating the conditions required to establish the convergence of the iterative algorithm, and on evaluating its computational performance for real-time applications.

A

  novel methodology combining the differential flatness formalism for trajectory generation of nonlinear systems, and the use of a model predictive control Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008

Figure 1 :

 1 Figure 1: Initial reference trajectory, 2 nd and 50 th iteration. (a) Flat output y = d x , d y ; (b) Measure of convergence η k ; (c) Input u(t); and, (d) Input ϕ(t).

  ref,k+1 -P ref,k is larger than a prescribed tolerance level and the maximum number of iterations is not reached: assign P ref,k ← P ref,k+1 and go to Step 1. Note from Steps 1-5 that the proposed algorithm implements a recursion P ref,k+1 = F P ref,k , whose complexity depends predominantly on the (in general, nonlinear) mapping Y mpc,k = G P ref,k . The convergence properties of the recursive mapping, P ref,k+1 = F P ref,k , will be investigated in future work.

Int. Workshop on Assessment and Future Directions of NMPC Pavia, Italy, September 5-9, 2008