Pierre Schneider

Ex oriente luxuria : a « cultural revolution » : in Imperial Rome

anthropologie globale du luxe", Marc Abélès

Introduction

In A. D. 22, Roman magistrates complained to the senate that sumptuary regulations were being ignored and that prices of luxury goods were rising daily. The senate referred the complaint to Tiberius. In his letter of reply, the Emperor explained that legislation on such private matters would be useless and counterproductive: "… for on what should I make my first effort at prohibition and return to the ancient standard? On the infinite expanse of our villas? The numbers -the nations -of our slaves? The weight of our silver and gold? The miracles of bronze and canvas? The promiscuous dress of male and female -and the specially female extravagance by which, for the sake of jewels, our wealth is transported to alien or hostile countries?"

The expression "alien or hostile countries" was by no means ambiguous. Anyone would understand that Tiberius hinted at Arabia and India, two countries supplying the Roman world with highly demanded commodities, the consumption of which nobody could stop, or reduce. It is the connections between the consumption of eastern goods and the problem of luxury that I would like to explore in this presentation. Around the mid-first century B.C., the Roman Republic had become the pre-eminent power of the Mediterranean world, having subdued the empire of Carthage, mainland Greece, Spain and Asia Minor. Pompey the Great had established Roman supremacy in the whole Near East, while Gauls were conquered by Julius Caesar. However, the growth of the Roman power created problems that the old political system of the Republic could not solve. Thus successive civil wars occurred over the first century B.C., culminating with the assassination of Julius Caesar in 44 B.C. and the battle at Actium in 31 B.C. There Octavian (later known as Augustus) defeated Mark Anthony and became the sole ruler of Rome and its empire. He also annexed Egypt with its capital city, Alexandria. This date marks the end of the Republic and the beginning of the early Roman Empire.

I -Roman luxury in

The first two centuries of the empire's existence were a period of unprecedented political stability and prosperity known as the Pax Romana (Roman Peace). The size of the empire was dramatically increased, with Britain becoming a province. After Nero's death in A.D. 68, the empire suffered a series of brief civil wars. Vespasian emerged triumphant in A.D. 69, establishing the Flavian dynasty. The succeeding Antonine dynasty gave Rome the Five Good Emperors. The empire reached its greatest extent under Trajan, the second in this line. The Roman Empire was then among the most powerful forces in the world of its time. Most importantly as regard my subject, it was not centred on the Mediterranean: by the late second century A.D., strong commercial ties had been linking the Roman Empire with the Indian Ocean area for more than two centuries.

b) From the Indian Ocean to the Mediterranean: routes and goods

Prior to the annexation of Egypt in 30 B.C., the Romans would buy eastern commodities through the Greeks who had already begun to exploit trading opportunities and acted as middle-men. This situation was to change when the city of Alexandria, which was the hub of Indian Ocean commerce, fell under Roman authority, together with the Egyptian ports on the Red Sea coast from which Mediterranean vessels headed to the Horn of Africa, Arabia and India. This resulted in a shift in trade activities. To put it in other words, the quantities of luxuries imported from the Indian Ocean increased dramatically. There is no point in detailing this historical disruption now. Suffice it to say that the geographer Strabo, who toured Egypt around 26 B.C., stresses the involvement of Roman administration in developing trade relations with Indian Ocean countries. For instance he reports the land routes from Berenike and Myos Hormos to Koptos on the Nile being equipped with water cisterns, storage depots, and being secured by armed guards. The same Strabo states that under Augustus the number of vessels leaving Egyptian ports quickly increased: one hundred and twenty ships a year leaved Myos Hormos for India, as compared with the twenties which left several years before, when the Greeks ruled Egypt.

At that time several routes joined the Mediterranean world and Indian Ocean places. However, during the period under consideration, the flow of East African and Asian commodities imported to the Mediterranean was channelled through the corridor of the Red Sea. Several sources document this pattern of trade, most notable among is the Periplus ( = navigation) of the Erythraean Sea (= the ancient name of the Western Indian Ocean). This treatise, written in the mid-first century A.D. by a Greek merchant, provides information about the ports of trade along the coast of East Africa, southern Arabia, and western India. Its unknown author also lists the valuables commodities that each place supplied, and details the lines of trade. Some ships would skirt the Arabian and East African coast to sell and buy merchandise; others would use the monsoon winds to reach north-west India or Kerala via a direct route across the ocean. Another most important source is the Roman writer Pliny the Elder, who flourished around the mid-first century A.D. and witnessed the boom in Roman consumption of eastern luxuries that I have just mentioned. In fact, his Natural History omits none of the Indian Ocean luxuries demanded at his time.

What were the so-called Indian Ocean luxuries? To begin with, animal products formed a very important part of them, being materials much in demand among the wealthy for purposes of decoration and personal adornment. Besides ivory, one must mention tortoise shell, pearls and silk. Then comes a wide range of highly prized plant-products that were bought from various peoples, such as pepper, ginger, cinnamon, nard and all kinds of "gum-resin". The category of mineral products comprises the precious gemstones mostly supplied by the Indians such as diamonds, agates, beryl, emeralds, amethyst, "murrhine" and so on. Finally one must not forget the large array of fine cotton garments that were produced in India.

c) The volume of Indian Ocean trade

So far I have hinted twice to a dramatic increase in importation and consumption which occurred after the annexation of Egypt. It is necessary to expand on this statement. Although all statistics that existed (e. g. customs accounts) have been lost, there is enough written and archaeological evidence dating back to the first and second centuries A.D. to support the existence of this phenomenon. One of the most often quoted text comes from Pliny's Natural History: the complains that not least than one hundred million sesterces a year is drained from the Roman Empire for the purchase of exotic luxuries. Another example is the topography of the city of Rome. Due to the growing demand for eastern luxuries, the number of retailers increased. These tended to cluster in particular areas of the city, hence some significant toponyms. We are aware of a vicus Turarius where the dealers in aromatics, spices and perfumes flocked to. The trade in pearls and gemstones was concentrated in a Porticus margaritaria (in Latin margarita means pearl), close to the monumental centre of Rome. Not far from there Emperor Domitian (A.D. 81-96) had the Horrea Piperataria (piper = pepper) built, a huge house intended for the sale and storage of spices from Arabia and India. The last piece of evidence I would like to mention is a partly damaged papyrus recovered in Egypt, which proves that enormous vessels had been designed to carry large amounts of Indian merchandise: this legal document contains cargo details from a ship called the Hermapollon that were checked customs officials and by agents of a money lender. The list of commodities established for payment of import tax includes pepper, malabathrum (a kind of cinnamon), nard, ivory and an unknown item -tortoiseshell is the best, if not the only candidate. Scholars have concluded from the surviving section that the Hermapollon was carrying over 600 tons of Indian commodities which had been loaded at Muziris, a mjor port of trade on the Kerala coast. This equals the capacity of the 16 th century Portuguese vessels. Another impressive fact is that the cargo was worth 7 million sesterces.

This flourishing trade was fuelled by the high purchasing power of Roman upper and "uppermiddle" classes. This shift in the distribution of wealth results from the growing prosperity of the city and the greater amount of liquid assets following the imperial expansion of the Roman State. In particular the annexation of Egypt caused a great amount of cash to flood the city. At any rate, we must keep in mind that non-elite classes were involved in the consumption of exotic luxuries.

To sum up: the consumption of Indian Ocean luxuries was so a novel phenomenon in the history of Rome that A. Wallace-Hadrill is right in coining the expression "consumer revolution". I order to better understand this crucial phenomenon, I suggest that we have a closer look on an emblematic luxury item, namely pearls imported from the Persian Gulf and South India.

II. The circulation of luxuries. A case study: Rome's craving for Indian Ocean pearls a) The stages of Roman consumption

Unlike most of eastern luxuries imported to the Mediterranean, the stages of Roman consumption of pearls can be reconstructed with reasonable certainty, thanks to Pliny the Elder. As a witness of the craving for pearls which characterises the early imperial period, he thought it important to have a chronological perspective. Three dates appear in his brief history of consumption: the time of the Jugurthinan war (111-105 B.C.); the time of Sylla (say the 80's BC.); the annexation of Egypt following the battle of Actium (30 B.C.). Clearly quality pearls begun to be demanded in the late second century B.C. This implies that the Roman taste was well formed at that time. The next stage was an increasing consumption in the early first century B.C., with the consequence that pearls of various qualities -from small to splendid -were available on the market in the Late Republican Roman society. Finally Pliny gives a third date which is pivotal: he mentions the annexation of Egypt and the Roman control over of the infrastructures of the eastern trade. As a consequence, local markets at the mouth of the Gulf and in South India had become much more accessible. Thus the Roman demand could be satisfied, in terms of quantity and quality. From then on the Roman demand, or Roman craving for pearls, rose so rapidly that the word "madness" (insania) shows up in several documents, such as: "Why is it that you have cast out your wife? Is she burdensome in her expenditure, does she weigh hard on your income? Certainly that is what the age is like-luxury spreads from bad to worse, and the ambitions of women, competing with each other, bring madness to private households-and harm to the state." (Seneca the Elder) "I see pearls-not single ones designed for each ear, but clusters of them, for the ears have now been trained to carry their load; they are joined together in pairs, and above each pair still others are fastened; feminine folly could not sufficiently have overwhelmed men unless two or three fortunes had hung in each ear." (Seneca the Younger)

b) Fashion

The word insania points to a tremendous demand, or fashion, which was to last for at least a century. Interestingly, the archaeological documentation echoes the literary evidence. Many items of jewellery made of pearls have been recovered: earrings, necklaces, rings, hair ornaments and so on. Actually it is earrings that largely predominate. They come in an incredible variety of forms: in particular the so-called crotalia were a very popular type. It is not easy to explain why earrings with pearls became so popular. Certainly fashion was a driving force. But another fact seems important to me: no doubt, a wealthy woman could exhibit earrings with two or three gorgeous pearls; however, a bunch of small pearls of lower quality could also give an acceptable pair of earrings. The dissemination of earrings thanks to the variety of pearls reflects the intense Roman demand. It also proves that pearls had become a "popular luxury" -we will go back to this issue later.

It is important that we should keep in mind pressure generated by fashion (the female "madness" referred to in literary sources). Fashion played a prominent role. This novel phenomenon in the social and cultural history of Rome is closely related to the disintegration of the old model of parsimonia, of which Cato the Elder was the emblematic supporter. This model was to be wiped away in a society characterised by rising prosperity. Roman consumers gradually switched to highly desirable objects, with a particular interest in those imported from the Indian Ocean area, that one can liken to a kind of brand. The development fashion rests on the exquisite taste of Roman consumers, which is attested by literary documents. Let us look, e.g., at the following excerpt from Pliny: "Their (=pearls) whole value lies in their brilliance, size, roundness, smoothness and weight, qualities of such rarity that no two pearls are found that are exactly alike. The longer ones also have a charm of their own. Those that end in a wider circle, tapering lengthwise in the shape of perfume-caskets, are termed elenchi. There is also a great variety in their actual brilliance; it is brighter with those found in the Persian Gulf Sea, whereas those found in the Indian Ocean resemble flakes of mica, though they excel others in size. The highest praise given to their colour is for them to be called alum-coloured."

III. Consuming eastern luxuries: new issues and new debates

I hope that the case of Indian Ocean pearls has helped understand the so-called "consumer revolution", which phenomenon is closely related to Rome' trade exchanges with the Indian Ocean world. Given this novel context, it is unsurprising that new issues and new debates about luxury came to light. This is the subject of the last section.

a) New social issues: trickle down and democratization of luxuries

Prior to the "consumer revolution", the consumption of luxuries concerned only the ruling class. The sumptuary laws that were passed in the flourishing Roman Republic (third-second centuries B.C.) aimed to limit the influences of noble families by restraining the display of the power, influence and prestige of the group, in particular in the banquets that they offered. The key factor about such laws was that they applied to food rather than clothes, because of the central role that banquets played in electoral politics. For instance, we hear of a law which forbade magistrates to accept invitations to dinner in private houses, in order to control corruption in office. To sum up, it is agreed that Rome was struggling with conspicuous consumption primarily to (try to) protect itself from destructive political competition rather than to fight luxury per se.

Given that the spreading consumption of eastern luxuries represent a novel form of luxury, and given that the Roman Imperial society had not much in common with that of the Late Roman Republic, it does not come as a surprise that the social issues were of different nature. Once again the decisive piece of evidence is provided by Pliny the Elder. At some point in his account of Indian Ocean pearls he says: "Fenestella records that they came into common use at Rome after the reduction of Alexandria under our sway."

Pliny uses two significant words : frequens means "common, usual" ; promiscuus also means "common or usual" but it also conveys social connotations : it implies that pearls had become a commodity distributed among various social strata. Today we would speak of democratization. Later on Pliny adds: " and nowadays even poor (pauperes) women covet them". Now what does pauperes mean? For sure, the have-nots, the destitute are not meant. Pliny certainly points to the plebs media, in other words the "middle class" women. These "poor" women claimed this sign of social status to display it in the streets. However, besides the middle class and the Roman elite referred to by Pliny, affluent freedmen must also be counted in the group of pearl consumers. The availability of eastern luxuries enabled them to convert his riches into social prestige. The best example is provided by Habinnas, a character from the Satyricon. Habinnas had made a success of his business (he was a lapidarius) so that he belonged to the local elite. That why not only himself but also his wife exhibited the social signs of their status, as described in the following passage: "Scintilla (= Habinnas' wife) was just as bad. She took off a little gold box from her neck, which she called her lucky box. Then she brought out two rattling earrings, and gave them to Fortunata (Trimalchios' wife) to look at in her turn, and said, "Thanks to my husband's kindness, nobody has finer ones." "What?" said Habinnas, "you cleaned me out to buy you a glass bean. I declare if I had a daughter I would cut off her ears. If there were no women, we should look on all gems as so much dirt; as it is, we piddle warm and drink cold."

This phenomenon is a form of trickle down: pearls were originally sought by the senatorial elite but gradually women from other social strata (knights, rich freedmen, plebs media) joined the group of pearl consumers. How can we explain this phenomenon? Maybe prices went down, following an increasing supply. But we have also seen that a wide range of pearls came on the market. Substandard items -I mean small pearls, uneven pearls, dull pearls and so on -were imported. Even if they were not perfect items, it was possible for a middle class woman to exhibit earrings made of true pearls. While perfect pearls were expensive, small pearls cost much less. Thanks to the large range of prices and kinds, this sign of social status was in the reach of the pauperes. The trickle down had been made possible by differentiation in the quality of commodities.

We must not draw the conclusion, however, that this wider consumption tended to bridge the gap between the upper orders and the plebs, for there ought to be a divide between these social layers. So how did upper classes react to democratization of pearls? In fact the same reason that enabled the trickle down -the variety in kinds of pearls -allowed upper classes to distinguish themselves by adjusting their consumption to their higher status: in particular they bought upmarket pearls, or adorned their shoes with them.

Other eastern commodities were similarly adopted by a wider range of Roman consumers, especially pepper, but due to the time limit I will not embark on this topic.

b) The political debate about eastern luxuries: thinking about Rome's universal domination

Octavian's victory over Mark Antony and Cleopatra, as I said earlier, marked the end of the Civil Wars and the beginning of the Roman Imperial period. A few years before he died Augustus published the Res Gestae Divi Augusti ( the Deeds of the Divine Augustus), where he gives a first-person record of his life and accomplishments. By its very nature the Res Gestae is a form of propaganda for the government that Augustus instituted. The third section describes his military deeds and the established alliances he established with other nations during his reign. Two passages are noteworthy. In the first one Augustus refers to military expeditions to Saba, a country situated in Southern Arabia; Saba was famous for producing aromatics such as frankincense and myrrh: "At my command and under my auspices two armies were led almost at the same time into Ethiopia and Arabia Felix (...) In Arabia the army advanced into the territory of the Sabaeans to the town of Mariba."

Later on Augustus states that Indian envoys were sent to Rome several times: "Embassies from kings in India were frequently sent to me. never before had this been seen with any commander." Actually Augustus failed in subduing Arabia and the Indian embassies were not followed by any alliance treaty. So why did Augustus recall these event? In doing so, he wanted to affirm the completion of the conquest of whole inhabited world (known the Romans). In other words, August express the notion of Rome's universal domination. Even if Arabia and India had not been integrated in the Roman Empire, they had to admit that Rome was the most powerful state in the world. Most importantly, the concept of universal domination was adopted by the Augustus' successors: all of them conceived of Rome as an empire holding sway on the whole oikoumenê (inhabited world), although this supremacy was in some cases more symbolical that actual. Now how does this political agenda relate to the consumption of eastern luxuries? Two texts by Pliny the Elder will help us understand the case. " What is called the ' royal ' unguent (...) made of behen-nut juice, costus, amomum, Syrian cinnamon, cardamom, spikenard (and others ingredients, most of them imported from India). And none of the components of this scent is grown in Italy, the conqueror of the world, and indeed none in the whole of Europe excepting the iris in Illyria." "By the lowest reckoning India, China and the Arabian peninsula take from our empire 100 million sesterces every year -that is the sum which our luxuries and our women cost us."

These passages point to the same idea: the consumption of eastern luxuries appears as a major thorn in the side of Roman imperial discourse and ideology, for it clearly expresses Rome's dependency. Not being capable of producing the precious commodities which are strongly demanded, as we have seen, the Roman empire is completely dependant on eastern peoples who in addition drain its riches. In other words the intense consumption of eastern luxuries made explicit the very limits of Rome's claim for universal domination and was a grain of sand in the machine of the imperial propaganda. The debate on luxury had never before been pushed on such a political and ideological level.

As a matter of fact, this ideological problem was later solved by claiming that eastern countries had no choice but sending their precious commodities to Rome, where both merchants and consumers of luxuries gathered. The universal supremacy of the Empire was redefined in presenting Rome as the world centre for consumption of luxuries, while the exporting countries (Arabia, India, China) were likened to peoples paying a tribute to Rome.

c) Nature and luxury: a shift in the philosophical debate

Nature had always played a central role in the philosophical debate on luxury. This can be traced back in classical tradition as far as Plato in the fourth century B.C. Since the couple Nature-Luxury penetrated the Stoic thought, this debate still existed in the Roman society. Seneca for instance explains what 'living according to Nature' (secundum Naturam) means: there is a proper limit to meeting the body's needs, because the fulfilment of these needs is itself naturally limited. Humans have bodily needs, but nature herself fixes a boundary to their satisfaction. Since nature's needs are few, small and inexpensive, it follows that "natural life" is also simple life, unlike a luxurious life.

Pliny the Elder, who personaly witnessed the Roman dependency to eastern luxuries, also writes of the immorality of clothing the body in silk or increasing the appetite by eating pepper. Pepper, in particular, servers as an excellent example to prove that luxury goes against Nature. First, Pliny argues, it is useless regarding the fulfilment basic needs, not to mention that it is by no means tasty. Second it naturally grows wild in forests; thus it was not intended to satisfy human needs and be a source of pleasure. Elsewhere Pliny insists that the quest for luxuries is a form of violence against nature, for humans take objects and productions that belong to Nature : "This inspires us with ever greater and greater wonder that starting beginnings man has come to quarry the mountains for marbles, to go as far as China for silk cloth, and to explore the depths of the Indian Ocean for the pearl and the bowels of the earth for the emerald (n.b.: imported from India). For this purpose has been devised the fashion making wounds in the ears (Pliny refers to earrings with pearls and emeralds): it was not enough for jewels to be worn on the hands and neck and hair without making them even pierce through the body."

However there is enough evidence to prove that such ideas were being challenged following the consumer revolution. To put it in other words, the consumption of eastern luxuries weakened the arguments inspired by Stoicism. First, how could a society boasting of the universal domination content itself with a frugal life. No doubt the natural life belonged to the past. There is a significant text by Ovid, a poet who lived during the reign of Augustus: "Let ancient times delight other folk: I congratulate myself that I was not born till now; this age fits my nature well", he says. Then he adds "culture [the latin word cultus means elegance, polish, civilization, refinement], is with us, and rusticity, which survived until our grandsires, has not lasted to our days." It is not by chance that in the same passage Ovid refers to some emblematic luxuries such as pearls, silk cloth and marble (the first two items being imported from the distant East). In other words the course of history had almost annihilated the ideal of natural life. That is why, in A.D. 23, in spite of his anti-luxury position, Emperor Tiberius refused to issue a sumptuary law touching items such as gold ware, male and female clothing and jewellery. He was aware that such a law would create greater problems than luxury itself.

As for the idea that luxuries had not been made primarily for humans, it was also to be challenged and we even Pliny contradicting himself. On the one hand he claims that humans do harm to Nature when they unearth gemstones. On the other hand, as we can read in several passages, he is forced to admit that the wonders produced by Nature exist as such only because they are admired by humans. Unlike animals, humans have the sense of, and suffer from luxury. But it is luxury that gives them this unique sense of beauty. So without explicitly saying that the marvels of nature have been created to humans, Pliny implicitly admits that her grandeur is intrinsically linked with human love for beauty, which implies usage and consumption of such marvels.

d) Challenging the paradigm of corrupting luxury

It was a major and common theme of the Late Republican Latin literature, especially in work of historical narrative, that Rome had suffered a significant moral decline: the older and simpler days of virtuous men had ended up in a indulgent attitude towards material possessions. Cicero and Sallust, among others, express the view of moral decay associated with luxury, saying that the vast majority of people cannot endure abundance without turning to depravity. More important, the Roman moral decay was widely considered as a contingent result of imperial success: conquest brought with it riches, whence came luxury and moral corruption. Let us quote this excerpt from Sallust's life of Catilina : "But when our country had grown strong through toil and the practice of justice, when great kings had been vanquished in war, savage tribes and mighty peoples subdued by force of arms, when Carthage, the rival of Rome's dominion, had perished root and branch, and all seas and lands lay open, then Fortune began to be savage and to throw all into confusion. Those who had easily endured toil dangers, uncertain and difficult undertakings, found leisure and wealth, desirable under other circumstances, a burden and a curse. Hence a craving first for money, then for power, increased; ... After riches began to be held in honour and led to the acquisition of glory, positions of authority, and political influence, then virtue began to lose its edge, poverty to be considered a disgrace, blamelessness to be regarded as spitefulness malevolence. Therefore, as a result of riches, the youth were suddenly consumed with luxury and greed, together with insolence. "

However this classical paradigm of the origin of luxury was no longer relevant at the time when the Roman society bought and consumed eastern luxuries. First goods such as pearls, silk cloth, pepper and so on had not been introduced to the city by means of military conquest. They were brought by merchants. It was not war but maritime communications and market forces that were responsible of the flow of eastern luxuries into the Mediterranean world. Second, and more importantly, the consumption of eastern luxuries did not result from the negative contact of Roman legions with the corrupted peoples that they would have defeated. The true reason for consuming such luxuries was, as we have seen earlier, that the Romans desired and loved them. Their exquisite taste for luxuries proved that the Romans had turned into a refined society. Even those who expressed concern about the spread of luxury, such as Pliny, could not deny the exquisite taste of the Roman consumers. Finally the classical paradigm of moral decay, of men weakened by their demand for exotic luxuries was seriously challenged by the actual military and economic power of the Roman Empire. Thus a fully negative view on luxury was no longer tenable.

Conclusion

Most scholars contrast the pre-modern (Greek and Roman) paradigm of luxury with the new ideas that emerge as soon as the seventeenth century. It is generally agreed that throughout antiquity luxury was unanimously decried as a cause of individual dissipation and social disintegration: luxury needed to be purged because it is unhealthy; luxury and softness were censured because they make people cowardly. In contrast the modern period would initiate a new paradigm, exemplified by Mandeville's Fable of the bees, which represents the initial step of luxury's demoralization and even democratization. The decisive break from a moral approach that viewed luxury as a threat to an orderly society result from the increased circulation of commodities (especially commodities from abroad) that takes place in the modern times. I my opinion this so-called classical paradigm had been already questioned in antiquity, following the massive consumption on eastern commodities. The first step of luxury's demoralization and democratization may well date back to the early imperial period.

  a connected world: a glimpse on Rome's long distance trade with the Indian Ocean area (early 1 st century B.C.late 2 nd century A.D.) a) The historical backdrop: from the late Roman Republic to the early Roman Empire


Yves Bertot

Safe smooth paths between straight line obstacles

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

into a discrete problem, using sorting and a recursive treatment of the obtained sorted list. The result of the first two phases consists in a collection of cells, where each cell is a convex polygon connected to neighbors by safe boundaries that can be crossed without collision with the obstacles.

For our running example, the result of these first two phases is described in the following picture, where safe boundaries are shown as dashed lines and cell numbers have been added solely to help our understanding. The problem becomes one of moving between cells. The collection of cells and safe boundaries is used to construct an abstract view of the trajectory, viewed simply as a path between the cross marks and cell boundaries. For this phase, a simple graph algorithm is used. The trajectory is built first as a broken line from the cross marks to safe boundaries of the cells that contain them, and then between safe boundaries. This broken line already represents a possible path, and we hope to prove that this path is safe thanks to the fact that cells are convex and the segments are each included in one cell at a time, except for their extremities which are members of a cell boundary or equal to one of the cross marks. However, such a broken line is unsatisfactory as a robot trajectory, because each corner would impose that the robot stops to rotate and change direction. A last phase is added to find smooth curved lines in the vicinity of the broken line, and extra proofs are required to show that the smooth trajectories still avoid collisions with the obstacles. This phase can itself be decomposed in two parts, where one proposes a Bézier curve for testing, and the other checks that the given Bézier curve is safe. When he check fails, a curve that is closer to the initial broken line is proposed. In case where the repetitive process does not converge fast enough, it is possible to use the initial broken line as a fallback solution.

Producing a sequence of events

The first objective is to produce a collection of cells with the following properties:

1. The interior of each cell is free of obstacles 2. The parts of the boundaries of these cells that can be crossed safely to reach other cells are known precisely

To achieve this objective, we took inspiration from an algorithm known as vertical cell decomposition [START_REF] Latombe | Robot Motion Planning[END_REF]. The intuitive operational model of this algorithm is that a vertical line is sweeping the working space from left to right. When the extremity of an obstacle segment or the intersection between two obstacle segments is encountered, some internal view of the cells is updated, ultimately producing a collection of closed cells. To represent this sweeping operations, we chose to implement a sorting algorithm that takes as input the list of segments and produces as output a list of events. Each event corresponds to an obstacle extremity and is annotated with the sequence of segments that have this event as leftmost extremity.

Here is an example with three segments and the corresponding list of events. For this example, we write (a, [s 1 ; s 2 ]) for an event whose location is givent by point a and whose sequences contains s 1 and s 2 . At this point in our explanation, note that we chose to simplify the problem by assuming that segments only intersect at their extremities. This assumption is not hard to satisfy, as it is possible to take a collection of segments which do not satisfy this assumption and produce a new collection of segments that contain exactly the same points of the plane but satisfy the assumption.

The sorting algorithm works as follows: every time an new segment is processed, two events are considered for addition in the sequence of events. The first event is for the leftmost extremity of the segment. If an event already exists at the same location, then the current segment is simply added to the annotation of the existing event. If not event exists with that location, the event is simply added, again with the current segment in the event annotation. The second event for the rightmost extremity of the segment is added similarly, except that nothing is done for event annotations. For instance in our example, if segments s 1 , s 2 , and s 3 are processed in this order. The sequence of events after processing s 1 is

(a, [s 1 ]); (b, []).
After processing s 2 , the sequence is

(a, [s 2 ; s 1 ]); (b, []); (c, []).
We see that in this case, no new event is added for the leftmost extremity of s 2 , but s 2 is added to the sequence of segments annotating the event a location a.

Another simplification to our presentation of the problem is that vertical segments are not allowed in our input. Again, we think that it is possible to work around this limitation: If one wishes to add vertical obstacles, only there extremities should be added as events without outgoing edges in the sequence of events. We expect the cells to be created where a safe boundary appears between the two events. It is enough to remove this safe boundary from the cells to avoid collisions with the corresponding vertical obstacle. This approach is not implemented in our current version of the algorithm.

We assume we are working in some abstract ordered field to represent the coordinates of points. For the algorithm itself, we shall only use the field operations. For the proofs we need to use some properties of the order. The data types that we need to create to describe our algorithm are as follows:

• A datatype of points, which are ordered pairs of coordinates in the field,

• A datatype of edges (for the obstacles), which are ordered pairs of points, with the invariant that the first coordinate of the first point is strictly smaller than the first coordinate of the second point,

• A datatype of events, which have two components, the first is a point and the second is sequence of edges, all of which have this point as first extremity.

For this algorithm to be correct, we need to guarantee the following properties:

• All segments of the initial collection of obstacles must be present among the outgoing edges of events in the output

• All segments attached to a given event have this event as first extremity

• The sequence of events is strictly sorted lexicographically, as a consequence each edge appears as outgoing edge of a single event.

Processing the sorted list of events then implements the idea of the sweeping line from left to right. In our example, this means that the next phase of the algorithm will be processing events located at a, d, b, c, e in this order.

Producing cells

To understand how cells are produce, we can keep the mental model of a sweeping line, which is the vertical line passing through the location of current event. When producing cells, we distinguish between closed and open cells. The closed cells lie to the left of the sweeping line, they are complete and have well-defined left and right sides. The open cells intersect the sweeping line. They have a well-defined left side, but the right side is unknown, as it will be fixed when processing later events.

At a given position of the sweeping line, there is an ordered sequence of open cells. For each of these cells, the bottom and the top boundary are given by edges. Each of the open cells also has a left side, which was defined at the time the cell was created, possibly when processing several examples that all lie on the same vertical.

The processing of a new event works as follows:

1. The sequence of open cells is decomposed into three parts: a first sequence of untouched cells, a second sequence of cells in contact with the given event, and a third sequence of cells of untouched cells 2. All cells in contact are closed: their right side is described and they are added to the collection of closed cells 3. If the given event has n outgoing edges, then n + 1 new open cells are created. The outgoing edges are first sorted vertically. The first cell has for its low edge the same low edge as first cell in the first of the contact cells, and as high edge the first of the outgoing edges (after sorting). The last new cell is similar but with the last outgoing edge and the high edge of the last contact cell. All other newly created cells have a left side that is reduced to a single point, because the low and high edges meet at the event that is currently processed.

For this algorithm to work, we add an extra constraint on the data: the whole working space is enclosed between a bottom and a top edge that do not cross and are long enough so that they define a quadrangle with no contacts with all other edges. Special care must be taken to handle the case where events are vertically aligned. When this happens, a naive version of our algorithm produces cells that have the left side and the right side on the same vertical line. These cells are flat and do not satisfy two important properties that we like about nonempty cells. First in a non-flat cell, any path from a safe point on the left side to a safe point on the right side is safe. Second, in a non-flat cell, any path to a safe point on any side to the center of the cell is safe.

To recover these properties, we modified our algorithm to memorize the position of the last processed event, together with the last opened cell and the last closed cell. When the current event is vertically aligned with the last processed event, we can avoid creating a flat closed cell by updating the last opened cell and the last closed cell. For last closed cell, only the obstacles on the right side need to be updated. However, this adds complexity in the proofs about the algorithm.

At the moment of writing these lines, this modified vertical cell decomposition algorithm is still being proved correct. We proceed by showing that a collection of properties are invariant through the processing of events. The properties that we have proved invariant so far are as follows:

• The sequence of open cells is sorted vertically, the cells are adjacent, and they cover the whole space between the bottom and top edges,

• All low and high edge of the current open cells contain a point that is vertically aligned with the currently processed event,

• All cells are disjoint,

• The obstacles are covered by the high edges of existing cells, closed or open and the outgoing edges of events that have not been processed yet.

The last two properties put together should be enough to guarantee that the interior of all closed cells is safe. We have yet to understand how we will express and prove safety properties for points on the left and right sides of closed cells.

For now, the left and right sides of closed edges are equipped with list of points, where the first one belongs to the cell's high edge and the last one belongs to the cells low edge. The other points on these lists correspond to extremities of edges that end or start on this side. The points on the side of an edge that are distinct from the elements of these lists are also safe. The intervals between the points of these lists correspond to vertical segments that are safe boundaries of the cell. From now on, we will call these doors between the cell and neighbor cells. There is a safe horizontal passage between two cells if the left side of one cell and the right side of the other have such a door in common. We use this criterion to construct a graph whose nodes are the cells and whose edges are doors between one cell and the other.

Making a broken line trajectory between two points

With the graph of cells and doors between these cells, we can already construct safe paths between two points in the space, which we shall the source and target points for clarity. We only need the source and target to be either inside the same polygon made of obstacles, or inside the work space and outside of any polygon.

The first step is to find the two cells that contain the source and the target. If the source and target are in the same cell, it is enough to draw a straight line segment from the source to the target.

If the two cells are different, we can explore the graph to check for the existence of a path in this graph from the source cell to the target cell. This path is composed of edges of the graph and each of the graph edges is actually a door between two cells. It is enough to construct a sequence of paths between the middle points of each door that is an element of this path. If two successive doors are not on the same vertical line, these two doors are on both sides of a same cell. A straight segment from the middle of the first door to the middle of the second door is a safe path in the interior of the cell (this property would not be satisfied if the cell was flat).

When two successive doors are on the same vertical line, a straight line between the center of one door and the center of the other is bound to be unsafe, because it will touch the lowest bound of the highest of the two doors, and this lowest bound is the extremity of some obstacle. To avoid this, we build a path composed of two segments. The first segment goes from the middle of the first door to the center of the cell. The second segment goes from the center of the cell to the middle of the second door. Such a two segment path is safe because the two extremities are safe, and the interior of these paths lie inside the interior of the cell, which is safe.

An example of safe path between two doors on the same side of a cell is given in the following figure :  For the formal proofs of this part, we have not started working on this, but this should rely on the properties of convex sets, in particular that any segment between two points of a convex set is included in this convex set.

Making a smooth trajectory

The trajectory described in the previous section is given by a sequence of segments, where each segment shares an extremity with its predecessor and the other extremity with its successor. We shall call these extremities the corners of the trajectory. We wish to replace each corner by a smooth curve and to connect each curve fragment in a smooth fashion, in other words, so that a point can travel the curve with continuous speed (in fact, we also wish to have continuous acceleration).

We expect that the smooth connection constraint can also be rephrased with the following sufficient criteria: the curve fragments are connected, their tangents at the connection are parallel, and these curves are locally on opposite sides of a straight line that crosses these tangents. This is illustrated in the following figures, each where two curves connect at the point where several straight lines cross. These straight lines are the two tangents of the curves at the point of intersection (two distinct dashed lines) and the line crossing the tangents that separates locally the two curves (dotted line). In the first figure the two connected curves are on both sides of the tangent. In the second figure, the two connected curves are on the same side of the tangent. Such a property is easy to achieve with Bézier curves. In our case, it is enough to use quadratic Bézier curves. A quadratic Bézier curve is given by 3 points (in the same way that a polynomial of degree 2 is given by three coefficients). Given three points A, B, C a point on the corresponding Bézier curve is given by choosing a parameter t between 0 and 1 and computing three more points: D is the barycenter of A with weight 1 -t and B with weight t, E is the barycenter of B with weight 1 -t and C with weight t, and F is the barycenter of D with weight 1 -t and E with weight t. The point F is the point on the Bézier curve for the parameter t. When t varies between 0 and 1, the point moves on the Bézier curve from A to C, generally without passing through B. The curve is tangent to the segment (A, B) in A and tangent to the segment (B, C) in C. and(C, D) by the two Bézier curves given by points A B M on the one hand and M C D on the other hand. These two curves connect in M so that their tangents are respectively parallel to the line BM and the line M C, which are the same line. Moreover, any line that crosses BC in M has B on one side and C on the other side. Locally, the points of the first curve close to M are on the same side as B and the points of the second curve close to M are on the same side as C. We hope to be able to prove this thanks to the properties Bernstein polynomials, which are useful in describing the properties of Bézier curves. Our Bézier curve construction starts with a broken line that is safe, but the curves stray away from the corners, and in doing so they may collide with the obstacles. We designed two more algorithms, one to check that a Bézier curve stays in safe territory, and the other to repair trajectories whose Bézier curve do collide with the obstacles.

These algorithms are based on a strong property of Bézier curves, which we shall call the dichotomy property. In the description above where we showed how to construct the point F which belongs to the Bézier curve, it turns out that the triplets A D F and F E C define two new Bézier curves that decompose the original Bézier curve in two complementary subsets. The triplet A D F defines a curves that covers all the points of the curve defined by A B C for a parameter t ≤ t, while the triplet F E C defines a curves that covers all the points of the curve defined by A B C for a parameter t ≥ t. The convex hull for A D F is much smaller than the convex hull for A B C, and thus the dichotomy operation makes it possible to replace a gross approximation of the Bézier curve by a more precise one.

Our checking algorithm repeats this dichotomy operation until it gets enough information to conclude that there is a guarantee that the Bézier curve is included in the union of the interior of two cells and the door between these cells, or that there is a guarantee that the Bézier curve steps out of this union, in which case we have proved that the Bézier curve is unsafe.

When we have an unsafe Bézier curve, we can repair the situation by replacing the Bézier curve A B C by the composite path obtained by adding two new points M and N which are the centers of (A, B) and (B, C) respectively, and taking the segment (A, M ), the Bézier curve M B N , and finally the segment (N, C). This composite path is smooth, because the junctions at M and N have the same tangent and direction, and it is also closer to the initial broken line trajectory. This new path needs to be checked again, and maybe the replacement process can be repeated. Ultimately, if the replacement process were repeated indefinitely the limit of the smooth path would the broken line path.

The following figure illustrates the case where the faulty Bézier curve drawn as a dotted line is repaired into the path composed of two straight line segments and a smaller Bézier curve. The vertical dashed line represents a door and the solid straight line below represents an obstacle.

In our program, we first produce a candidate Bézier curve for each corner, and we then check whether this curve is safe. If not, we produce Bézier curves, that are closer to the broken line and repeatedly check their safety. After a limited number of iterations, the program falls back to the initial broken line corner if need be. We only wish to prove that the checking procedure is correct. For this program, we will not be able to prove that the resulting trajectory is smooth.

Exploiting the program and visual feedback

This program can be run inside Coq, using rational numbers for the number system. To compute vertical cells, most computations are ring operations, except when we want to know the vertical projection of a point on an edge, where we use a division operation, so the exact division operation provided for rational numbers suffices.

When defining Bézier curves, we only need to construct midpoints of existing points to define the new control points for the successive Bézier curves that we consider. Working with rational numbers suffices to have exact computation.

The output can be displayed by simply traversing the output data structure and generating simple postscript commands for each of the straight line or Bézier curve segment. Our program uses quadratic Bézier curves, while postscript supports cubic Bézier curves, but converting from one to the other is easy. In the end, Postscript uses limited precision numbers so approximations are needed, but these approximations are only useful for display purposes.

The Coq system also provides an extraction facility, so the algorithmic part of our program can be translated to a conventional programming language. We used this to produce OCaml code, which was later translated to Javascript, and we developed a hosting webpage for this Javascript code, visible at the following link: https://stamp.gitlabpages.inria.fr/trajectories.html.

An example of trajectory computed in this setup is given by the following illustration:

Related work

In this section, we mostly concentrate on related work that entails formal verification.

The computation of sub-areas of the plane is already studied in work on convex hulls [START_REF] Pichardie | Formalizing convex hull algorithms[END_REF] and triangulations [START_REF] Dufourd | Formal study of plane Delaunay triangulation[END_REF][START_REF] Bertot | Formal verification of a geometry algorithm: A quest for abstract views and symmetry in coq proofs[END_REF]. The algorithm we use to decompose the workspace into cells relies extensively on the orientation predicate taken from the work of Knuth [START_REF] Knuth | Axioms and Hulls[END_REF].

We argued in this article that the program we designed only uses rational computations. Up to the problem of reasoning about collisions between the broken line trajectories and the obstacles, rational numbers are sufficient. However, smooth curves based on Bézier curve pose a new problem. The coordinates of potential collision points are real algebraic numbers. So while the program description can rely on rational numbers, the proofs concerning collision avoidance require working in a real closed field. Work on the decidability of equality between algebraic numbers is related to the work done in CoRN on the fundamental theorem of Algebra [START_REF] Geuvers | A constructive proof of the fundamental theorem of algebra without using the rationals[END_REF]. The Mathematical Components library also contains descriptions of procedures to descide problems with polynomials [START_REF] Cohen | Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination[END_REF].

Bézier curves can actually be viewed as parametric curves, where the polynomials giving each coordinate are easily described from the control points using Benstein polynomials as a basis of the vector space of polynomials, so this work is also related to previous work on Bernstein polynomials [START_REF] Bertot | A formal study of Bernstein coefficients and polynomials[END_REF].

A piece of work whose motivation has many similarities with ours is an experiment combining a motion planner programmed in Matlab with a formal verification performed by Isabelle [START_REF] Rizaldi | A formally verified motion planner for autonomous vehicles[END_REF]. This experiments describes a system that is more powerful than ours, as it relies on analysis, especially tools to reasons about solutions to differential equations, to check solutions that have been elaborated by a Matlab program. However, our objective is to describe a program that is eventually independent from the interactive theorem prover being used (in our case Coq, in their case Isabelle). To our understanding, their approach is to have Matlab and Isabelle being used at runtime to produce trajectories that are formally verified. On the other hand, our work contains a formal verification of the computations that happen at the time of elaborating the trajectories, while their work receives the results of these computations as if they were provided by an oracle.

Conclusion

It is unfortunate that this presentation is more a description of work in progress than a statement of achievement. At the time of writing these lines, the formal verification is focusing on proving the correctness of the vertical cell decomposition algorithm. The naive solution seems rather easy to verify, but the constraint of obtaining cells adds a lot of complexity in the program, as we need to pay attention to the case where two successive events are vertically aligned.

The full program can be seen on the public repository https://github.com/math-comp/ trajectories. The incomplete proofs of the first two phases can be seen on the public repository https://github.com/ybertot/VerticalCells.

Bézier curves provide a nice trade-off between the desire to provide smooth trajectories, the expected difficulty of proofs, and the techniques involved in trajectory repair. However, they may turn out to be unsatisfactory with respect to the smoothness of acceleration. For instance, road and railway designers rely on a different kind of curves, known as clothoids or Euler spirals. These are more specific to trajectories for wheeled vehicles. We plan to integrate their study in our work.

  [s 2 ; s 1 ]); (d, [s 3 ]); (b, []); (c, []); (e, [])

  line A B C D can be transformed into a sequence of smoothly connected Bézier curves by adding a point M on the segment between B and C and replacing the three segments (A, B), (B, C),

Acknowledgements

The initial work on the vertical cell decomposition algorithms was done by Thomas Portet. Studies of potential collisions between Bézier curves and straight line segments were done by Quentin Vermande. Laurent Théry added the possibility to visualize the results on a web-page. x


The use of the finite-state machine design can improve the performances during the grip and release phases for the two groups Participants were performing better during the release phase compared to the grip phase

The Riemannian feature extraction pipeline, the alignment method and the selection of a subset of electrodes all increased this di erence for able-bodied while only the Riemannian pipeline did for the amputee.

While the best performances were obtained with finite-state machine for the BB (Box&Blocks) & CS (Clothespin) objects, results are more nuanced with the CUP.

A finite-state machine design leads to a simplification of the decision process

The simplification does not prevent the system to raise errors during classification, as reflected by the low performance obtained when using the CUP object in the grip phase.

Classification performances for able-bodied (n = 19, left) and amputee (n = 1, right) using di erent feature extraction and decision-making methods where CSP = Common Spatial Pettern; WD = Wavelet Decomposition; BDM = Binary Decision-Making; SVM = Support Vector Machine; RBF = Radial Basis Fucntion); TS = Tangent Space; FSM = Finite State Machine. Error bars represent average and standard deviation calculated on all of the subjects.

Able-bodied are performing better than the amputee

More sessions are needed to see an effect of sensory feedback

The di erence could be explained by the di erent EEG system used for each group.

Online performances are getting closer to offline ones

With an exception for the CUP object, especially for the grip phase. • The goal of developing a Brain-Machine Interface (BMI) control system for medical prosthetics is to provide robust, comfortable and easy-to-use devices for the concerned user. They need to be fast and have high performance in estimating the user's intention in real-time.

• Yet, the transition from o ine to online settings often proves to be challenging (Deo et al., 2023), with a drop in performance when testing the BMI system in the latter.

• In the literature, there are numerous examples of methods and techniques aimed at targeting this issue, here applied to the control of a prosthetic hand.

• Online classification performances were obtained with 19 able bodied and 1 amputee subject who controlled the modified Myobock prosthesis by using the EEG data recorded in real-time and compared to the performances obtained o ine.

• The e ect of two di erent feature extraction methods and of a new decision-making process on the classification performances were studied for the same classification algorithm (SVM RBF and TS-SVM).

• The decoding of object release is always better than the object grip with both able-bodied and amputee subjects for all di erent combinations.

• Using a Feature Extraction method based on Riemannian Geometry and alignment of covariance matrices, and a Finite-State Machine design as decision-making process in online allows to reach classification performances obtained in o ine except for the CUP object.

Highlights

• The implementation of a finite-state machine design on the control of the prosthesis led to an increase in the classification performance on both the able-bodied and the amputee while simplifying the decision-making process after the classification.

• In real-time control, able-bodied subjects obtained higher performance compared to the one amputee, it could be due to the di erence in EEG systems used with the groups (64 ActiCHamp vs. 16 electrodes OpenBCI).

• The sensory feedback system did not improve the classification performance in able-bodied and for the amputee.

Acknowledgments

This work has been partially supported by the French National ResearchTechnology Association and Otto Bock France in the framework of the CIFRE project n°2018/1620.

To access my PhD thesis on the control of prosthetic hands using non-invasive BMIs Scan the QR code : • Real-time control has been performed with amputees and able-bodied subjects during two sessions without and with the use of a sensory feedback device provided by a vibrating bracelet.

• The experiments consisted in doing a sequence of Grasping-Moving-Releasing three objects, using a modified Myobock® prosthesis controlled by EEG data.

• The e ect of three di erent methods of feature extraction and decision-making process was tested on the online control performance and compared to the o ine.