The residents of 11 Rynek II in Lubartów. Lublin State Archives (APwL), 35/43/0/7/46/50-51, Lubartów population register, vol.46 © Archiwum Państwowe w Lublinie First, we can see filiation and conjugal ties. The residential status column (column 5) provides initial information regarding these ties: Chana Bajgielman was Kielman Bajgielman's "daughter" (corka), while Etla née Zandmer was Kielman's "wife" (zona). This matrimonial connection also appears in the marital status column (column 8): they were married (malz.) in 1895. The information in the occupation column (column 6) also points in this direction, since Etla is said to be "with the husband" (przy mezu), and Chana "with the parents" (przy rodzicach). The first and last names of the residents (column 2) and their parents (column 3) also confirm that Chana is Kielman's and Etla's daughter.

At the source of networks: identifying ties between individuals from a population register par Gabrielle Escaich

The population register for Lubartów lists six inhabitants residing in 1938 at 11 Rynek II, the town's market square. Kielman and Etla Bajgielman, and their daughter Chana born in 1907, were already living there when the register was established in 1932. Moszek-Hersz Kawa moved there in August 1937, and his son Chaim-Alter was born eleven months later. In the meantime, Wincenty Jezior, a 31-year-old baker, also arrived. He was the only person of the Roman Catholic faith in a dwelling where all other residents are identified as Jews. What is more, he was the only Catholic on a street where 235 of the other 240 inhabitants were Jewish (the religion of the remaining seven is not recorded). What ties did Wincenty Jezior have with the inhabitants of his home and his street? Do these ties help us understand this exceptional case, in an urban space highly segregated by religion? Beyond this case, how can we reconstitute the ties between a town's inhabitants from a historical source, a population register designed to control the movements of its inhabitants?1

Identifying the ties between inhabitants of a social space paves the way for in-depth knowledge of the town's social morphology. It is also central to linking the residential, migratory, and social trajectories of residents, as well as the social relationships in which they found themselves. A register such as that of Lubartów is a particularly rich source in this respect: many ties between the inhabitants were recorded, going well beyond kinship (such as economic or residential ties).

However, identifying them poses a number of methodological difficulties. In the Lubartów register, each double page corresponds to a dwelling; the residents were registered as they arrived, and the ties between them were recorded in relation to the "head of household." A first difficulty is therefore the scale of analysis: a source whose unit of registration is the dwelling does not encompass (or provides only rare clues about) ties between individuals who did not live together. Moreover, the categories in the source are only partially standardized: the same kind of relationship was recorded in different ways. Finally, although it is possible to glean additional information about the ties from other columns in the source, it is often partial or uncertain. What techniques can be applied to this population register to identify ties between individuals, both inside and outside of dwellings.

A diversity of ties inside dwellings

Dwellings are good starting points, especially because of the diverse ties between individuals that can be identified within them, whether they are explicitly indicated, or can be identified or assumed through various clues. Let us consider the example of 11 Rynek II (see the photograph below).

Other relationships are given with less precision: Moszek-Hersz Kawa was a "member of the family" (czl. rodziny) of Kielman Bajgielman. It can be assumed that Moszek-Hersz was Chana's partner, as they are listed as the father and mother of a child born in 1938, Chaim-Alter Kawa, whose parents' first and last names and residential status appear in the register ("son of Nr 4"syn p. Nr 4 -, i.e. of Moszek-Hersz).

Such ties were not limited to kinship, as ownership and tenancy relationships also appear: Kielman is listed as the "owner of the dwelling" (wlasciciel domu), and Wincenty Jezior as a "worker" (pracownik). The latter status also indicates a professional relationship: Wincenty's profession ("baker", piekarz), combined with his status as a worker, probably indicates that he worked for Kielman, whose profession was "bakery owner" (wlasciciel piekarni).

The configuration of ties at 11 Rynek II suggests that these six residents also had an additional co-residence tie. However, being registered on the same double page of the register does not always mean they lived simultaneously at the address, since successive residents were registered as they arrived. So how can we determine who lived together?

Identifying co-residence ties within dwellings

Co-residence is a good indicator of "shared daily life," 2 with all of the exchange and interaction that this can imply. A source whose basic unit is the dwelling can be used to identify coresidence, namely by analyzing the various dates registered. For example, when a person's date of arrival (moving to the address or arriving in town) was later than the date of departure (death, moving, leaving town, and/or removal from the register) for a resident previously registered on the same double page, it can be assumed that they did not co-reside. Let us consider the example of apartment no.1 at 8 Niecala Street (see table below). Mordko, Brucha, and Pesa-Cywja Bajgielman left Lubartów on October 6, 1932. They were replaced three weeks later by Jozej-Szczepan Kondracki, his wife Marja, and their 17-month-old daughter, Janina-Teresa. After their departure in December 1933, the apartment remained empty, according to the register, for almost four years, until the arrival of the Bogdanski family.

Stefania and her children Jerzy and Krystyna lived there for only one month, and Zdzislaw-Leon also left the house two weeks later, in mid-January 1938. Stanislawa-Tekla and her son Ryszard-Oskar had already arrived in Lubartów when the Bogdanski family moved in, and did not leave the town until September 1938; since the register does not provide information on any of the dates when the residents of this apartment moved in or out, there is no indication that the Bogdanski and Bikner families did not live there together.

However, there are limitations to identifying co-residence from the dates recorded in the register. Firstly, movements and their dates were not systematically filled in. A date of departure or death that was not recorded does not mean that the residents did not leave or die, especially in the case of the Jewish population during the flight and deportation movements of the Second World War. For example, it is likely that in some of the 118 addresses in Lubartów where there are several "owners of the dwelling" or "main tenants" whose dates do not indicate that they did not co-reside, the departures or deaths of successive residents were in fact not recorded. Moreover, even the dates of departure and arrival recorded by the administration may differ from the actual dates of these movements. Nevertheless, if we stick to this method of identification, we can distinguish, within the 1,928 residences in Lubartów, 2,225 subsets of inhabitants who had a co-residential relationship with each other.

However, the ties shown in the register go far beyond just the co-residents. The challenge is to identify, as widely as possible, what linked the town's inhabitants. So how can we systematize identification of the wide range of ties made possible by the register both inside and outside dwellings?

Identifying ties inside dwellings: from the statuses recorded in the source Two types of information enable an initial identification of ties within dwellings: residential status (left blank for only 13 individuals out of the 11,950 registered) and marital status (provided for 4,341 individuals). The challenge is to propose a coding similar to the source's categories, which is to say how individuals-the residents and local administration-qualified and categorized these ties. 3 We use identifiers to link individuals together; each individual in the database has a unique identifier. We then create a set of lines for each individual, with each line representing one of the individual's ties. The newly created database thus contains three columns: the first corresponding to the individual's identifier, the second indicating the type of tie (such as "daughter of" or "brother of"), and the third containing the identifier of the tie's recipient.

Identifying ties from residential status

The residential status entered in the register is a first type of information that makes it possible to identify ties within dwellings. This status was recorded on each double-page spread in relation to a reference person, most often the first person listed, except when the reference person was specified (e.g., "son of no. 2," syn p. nr. 2), or when new residents replaced previous ones in the dwelling. With information about residential status, linking individuals by their identifiers is in some cases relatively straightforward: for example, if a person is described as a "daughter" (corka) or "sister" (siostra), the identifier of the reference person in the dwelling will be associated. From these first ties, it is possible to identify new ones: for example, "daughters" and "sons" (syn) whose father and mother identifiers are identical can be considered siblings; each of these siblings will then be associated with the identifiers of their brothers and sisters.

However, not all residential statuses explicitly refer to ties. This is especially true for those that indicate ownership or rental status: "owner of the dwelling" (wlasciciel domu), "primary tenant" (glowny lokator), and "subtenant" (sublokator). It is possible to identify some ties from these statuses, for example those of tenancy between the owner of a dwelling and tenants or subtenants residing in the same dwelling; the tenants and subtenants will then be associated with the owner's identifier. However, there are many rental relationships that cannot be identified in this way from the register, especially when there were tenants and/or subtenants residing in the dwelling but no owner; this was the case in 711 of the 1,739 dwellings in Lubartów at the end of 1932.

Similarly, the professional ties that can be identified from residential status alone remain limited. Apart from the one case where the residential status indicates a relatively precise professional tie, that of "servant" (sluzaca), there remain 235 individuals whose status was simply recorded as "worker" (pracownik, pracownica); given the polysemy of the term, we do not know the exact professional tie they had with the other inhabitants of their dwelling. For example, how many were servants? Cross-referencing this residential status with the occupation recorded in the register provides the first elements of an answer. Out of the 235 "workers" in Lubartów, the occupation for 40 of them is "servant" (sluzacy, sluzaca); we can thus assume they were servants of the owner or the primary tenant of the dwelling in which they resided, and associate them with the identifier of this owner or tenant. Similarly, at 11 Rynek II, it was by crossing Wincenty's residential status as "worker" with his occupation as "baker" that we deduced he was the employee of the dwelling's owner, who also owned a bakery.

Identifying ties from marital status

Marital status is a second kind of information that can be used to identify ties within dwellings. While information on residential status already mentions "wives" (zona) and "husbands" (maz), marital status directly identifies conjugal ties: the Lubartów register indicates the order number of the person on the double page to whom this person is married. All that is required next is to associate their identifiers: 3,690 of the town's inhabitants can be linked to their spouses in this way.

Information on conjugal ties is not limited to marriage. Marital status also indicates the presence of some unmarried couples (nieslubna zona, niesl. maz), a divorced person (rozwiedziona), and widowed individuals (wdowa, wdowiec). However, the spouse's order number is rarely given in the case of unmarried conjugal relationships. This is the case for only 2 of the 157 persons listed as "unmarried" in the marital status column. Moreover, in the case of divorced or widowed persons, since these individuals no longer had an administrative relationship, the marital status alone does not allow for associating them with their former spouse.

Nevertheless, we already have, with this first identification based on residential and marital statuses, several types of ties within dwellings. Its application to 11 Rynek II provides an overview (see the table below).4

However, identifying ties within dwellings in this way does not enable us to link individuals together if they did not reside together. For instance, while Wincenty Jezior had an economic and residential tie with the Bajgielmans, with whom did he have matrimonial, filiation, or sibling ties? Were the Bajgielmans of Niecala Street and 11 Rynek II related? In short, how can we identify the ties without restricting ourselves to the limits of the dwelling?

Identifying ties outside the dwelling: using first and last names

Another way of determining ties using identifiers is to venture outside dwellings and associate individuals on the basis of their first and last names. The latter are frequently used to reconstruct kinship networks, from both historical5 and contemporary6 sources.

Since the Lubartów register in most cases shows the last name, maiden name (if any), and first name of the mother, as well as the last name and first name of the father, it is possible to at least search for the father, mother, and siblings of each person listed in the register. To find an individual's father with this technique, we search the entire database for a resident with the same last name (or whose last name corresponds to the maiden name in the case of married women), and whose first name corresponds to the father's first name. To locate an individual's mother, we look for a resident with the same last name in the case of married mothers (or whose name matches the maiden name for married daughters), whose maiden name matches the mother's maiden name (or the mother's last name if the mother is not married), and whose first name matches the mother's first name. Finally, associating the identifiers of individuals with the same last name (or maiden name in the case of married women), along with the same first and last names of the father and mother, allows for identifying siblings. Combining individuals who share only their father's or mother's first and last names also makes it possible to identify half-siblings.

By associating the names of the inhabitants without being restricted to the boundaries of dwellings, these dwellings can be linked together. To take our two previous examples, since Mordko Bajgielman of Niecala Street had-like Chana Bajgielman-a father whose first name was Kielman, and a mother whose first name was Etla and whose maiden name was Zandmer (which corresponds to the maiden name of Etla Bajgielman of 11 Rynek II), identification of ties via names allows for coding Kielman and Etla respectively as the father and mother of Mordko, and Mordko and Chana as brothers and sisters. The eldest son, born in 1903, therefore probably left the parental home before the register was opened, while his sister, who was 4 years younger, remained there until she moved out on her own in 1937. Other relationships arise from these parent/child and brother/sister relationships: for example, the children of two siblings identified by name may be considered cousins. For instance, Chaim-Alter Kawa of 11 Rynek II will be associated with the identifier of his cousin Pesa-Cywia Bajgielman of Niecala Street, and Pesa-Cywia with that of Chaim-Alter.

The identification of ties via names also allows us to complete the ties previously identified within dwellings. For example, while the residential status column for 11 Rynek II only indicates that Moszek-Hersz Kawa was a "family member" of Kielman Bajgielman, the search for ties based on names reveals that the identifier of Chaim-Alter Kawa's mother is that of Chana Bajgielman. Therefore, we can assume that Moszek-Hersz was probably Chana's (unmarried) companion, and that Kielman was Chaim-Alter's grandfather. In sum, this technique of identifying via names clearly fleshes out the ties found between the inhabitants (see the table below).

However, this technique of identifying ties through names has its limitations. The first is homonyms. For example, there are 7 Czeslaw Jeziors in the Lubartów register, and 1,991 individuals have the same first and last name as at least one other individual in the register. The identifiers found are therefore frequently multiple for the same person, which does not allow us to isolate an identifier, and therefore a tie recipient. The identifier found can also be that of an unrelated homonym. Moreover, the spellings of the same name vary within the register, with two equivalent names potentially appearing to be different. In the case of the Lubartów register, the use of the soundex algorithm developed as part of the Lubartworld project partially solves this difficulty: it can code ties by taking these spelling variations into account, namely by identifying ties from their soundex rather than from names. This makes it possible to identify more relatives, even if the multiple identifiers are also more numerous (see table below). A final limitation of identifying ties by name is that it mainly accounts for modal kinship configurations; for instance, it does not account for "non-legitimate" children or siblings, who do not bear the same last name.

Once all these ties have been found both inside and outside dwellings, there are still cases where we can identify the existence of a tie, but where its recipient remains unknown or uncertain. What to do in such cases?

Identifying ties when information is missing or uncertain

The failure to isolate an identifier for a given tie may either be because the recipient of that tie did not reside in the dwelling and/or in Lubartów, or because the various tracing techniques were unable to identify the tie. In the latter case, knowing the existence of the tie may still be useful, as it suggests that a recipient of the tie (e.g., a son or daughter if a father was identified) was probably residing in the town when the individual in question (in this case, the father) was entered into the register.

Identifying recipients of supposed ties'

In a number of cases, various clues can identify at least one supposed recipient of the tie; it is then possible, for instance, to associate the identifier of a "supposed spouse" or "supposed mother" with an individual. This is how direct ties (spouse, parent/child, brother/sister) can be identified from the numerous indirect ties appearing in the register, especially in the residential status column. For example, this can be used to identify the father of a "son of the husband" (pasierb) or the husband of a "wife of the brother" (synowa). For instance, in apartment no.1 at 9 Browarna Street (see photograph below), Piotr-Ryszard Krasuski is listed as the "daughter's husband" (ziec); he is probably the husband of Eleonora Krasuska née Koziarska, the only "daughter" (corka) of the owner residing in the apartment. This tie is "assumed" because even when a "daughter" is in the same dwelling as a "daughter's husband," the latter could also be the husband of another daughter of the reference person of the dwelling, who is not residing in this dwelling or even in Lubartów.

Leaving some ties without a recipient

There are still cases where we do not have enough clues to identify or to assume the recipient of the ties; the ties subsequently remain without recipient, at least initially.

In fact, even the ties already entered in the register, especially in the residential status column, do not always allow us to isolate the supposed recipients of these ties. This is especially true when we seek to identify direct ties from the indirect ties indicated, but this time we find no possible recipient within the dwelling, or on the contrary we find several. For example, a "brother's son" (brataniec) cannot be associated with his father if no "brother" (brat) is found in the dwelling; nevertheless, it is still possible to code this "brother's son" as the supposed child of someone present in the town at the time of the child's entry in the register. In other cases, there are several possible recipients of the tie in the dwelling, such as several "sons" (syn) where there is a "son's wife" (synowa) whom we are trying to link to her husband. For example, at 9 Browarna Street, the "son's wife" Anna Koziarska (line 6) could be the wife of either Stanislaw (line 3) or Jozef (line 5) Koziarski, both "sons" of the owner. We can initially code her as the supposed wife of someone living in Lubartów, without associating a husband with her. The other techniques for identifying ties then make it possible, in a number of cases, to complete these ties without a recipient. Here again, it is Anna Koziarska's marital status ("married to No. 5", malz. patrz Nr. 5) that ties her to her husband Jozef.

Similarly, while it is often insufficient on its own to associate individuals with each other, the "profession" entered in the register nevertheless can identify numerous ties. Some "occupations" thus refer directly to family ties, especially when inhabitants are designated as "with" (przy) or "dependent on" (na utrzymaniu) someone. For example, a person listed as "with husband" (przy mezu) will be coded as married (to a person living in the town at the time the information was entered in the register). Similarly, a resident whose "occupation" is "with son-in-law" (przy zieciu) will be coded as both a step-parent (in the sense of a relative of the son's or daughter's husband) and a parent.

Completing ties with more specific clues from the source

When the previous techniques for identifying ties cannot complete these supposed ties or ties without recipient, there is sometimes more specific information in the register that can provide clues for specifying these ties.

For instance, in the Lubartów register the names (of spouses in particular) occasionally appear in the various columns, constituting as many possible recipients for the identified ties. With regard to matrimonial ties, the spouse's name appears in the marital status column for 181 of the 3,898 "married" persons, and in the residential status column for 2 of the 156 "unmarried women" and the only "unmarried man." In addition, there is a single husband's name in the "occupation" column (Ruchla Wajc is said to be "with the unmarried man Biegancu", przy niesl. mezu Biegancu). 7 When these names are extracted from the database and cross-referenced with other people with this name (or the corresponding soundex), the identifiers of the two relatives can be associated. For example, for the 181 "married women" for whom a name appears in the residential status column, 26 potential spouses can be found by searching the database for the soundex of the mentioned names, and 11 are found by searching the names themselves. In this case, this tracing technique allows us to link spouses across dwellings, 8 and in a few cases to extend the identification of conjugal ties beyond marriage alone.

This technique can also capture types of relationships that are relatively uncommon in the town. For instance, Adela and Helena Dolak, 9 who were aged 10 and 8 when they arrived in Lubartów in 1934, and whose residential status indicates they were "wards" (sierota na wychowaniu)the only ones in the register-are said to be "dependent on Jaskowiaka" (na utrzymaniu Jaskowiaka) by virtue of their "profession." While no individual in the register has this name, a single inhabitant has a name whose soundex coincides with that of "Jaskowiaka": Michal Jaskowiak, an "unskilled workerday laborer" (robotnik niewykwalifik. wyrobnik) registered as a "subtenant" (residential status) in the same dwelling on Lubelska Street. 10 A recipient was thus found for this unique tie of "dependence."

In sum, a historical source such as a population register allows for identifying a wide range of ties between inhabitants (co-residence, kinship, ownership/rental, professional ties). These ties can be identified both inside (via the statuses entered in the source) and outside dwellings (especially through the names), whether or not we manage at first to pair the individuals caught in these ties. To identify ties beyond the closed space of Lubartów-to which a local source 8 In fact, it is especially in cases where the spouse is not in the same dwelling that a name is specified next to the mention of the marriage tie. It replaces the serial number of the spouse who resides in the dwelling. 9 Lublin State Archives (APwL), 35/43/0/7/43/124-125, Lubartów population register, vol.43, pages for residents of apartment no. 2 at 147 Lubelska Street. 10 The form "Jaskowiaka" comes from the declension of the name "Jaskowiak".

such as the register gives access-it would have to be cross-referenced with the many complementary sources that can provide information on individuals and their ties.

Nevertheless, identifying ties within the town already opens up many avenues for research. It makes it possible to describe the configuration of ties at the dwelling or individual level with regard to the social properties that characterize them, religious ones in particular. It also paves the way for networks analyses at the group, street, or town level, thereby incorporating the logic of local belonging, and its role in social and migratory trajectories.11

7

  Lublin State Archives (APwL), 35/43/0/7/47/144-145, Lubartów population register, vol.47, pages for residents of apartment no.2 at 4 Zabia Street.

We possess the register entry, in the form of a database, as well as the original digitized source.

In French, "le partage du quotidien": Florence Weber, Penser la parenté aujourd'hui. La force du quotidien, Paris, Éditions Rue d'Ulm, 2013.

The Lubartów register was completed from forms filled in directly by the residents of a dwelling, and then forwarded to the local government by its owner.

Columns for "name" and "name of the tie recipient" have been added here to facilitate reading; it is not necessary for them to appear in the final database.

Karine Karila-Cohen, "Le graphe, la trace et les fragments. L'apport des méthodes quantitatives et des outils numériques à l'étude des élites civiques athéniennes," Annales.Histoire, Sciences Sociales 73, no.4 (2018): 785-815.

 6 Catherine Capron, "Essai de reconstruction automatique des parentés à partir du registre de population belge," Revue informatique et statistique dans les sciences humaines 24, no.1-4 (1998): 9-49.

On the implementation of this approach in the case of the Jews of Lens (France), see for example Pierre Mercklé and Claire Zalc, "Peut-on modéliser la persécution? Apports et limites des approches quantifiées sur terrain de la Shoah", Annales. Histoire, SciencesSociales, vol.73, no. 4, 2018, p. 923-957.


Huu-Thinh Do
email: huu-thinh.do@lcis.grenoble-inp.fr

Ionela Prodan
email: ionela.prodan@lcis.grenoble-inp.fr

Florin Stoican
email: florin.stoican@upb.ro

Analysis of alternative flat representations of a UAV for trajectory generation and tracking

Keywords: Differential flatness, B-spline, Fixed-wing UAV (Unmanned Aerial Vehicle), Trajectory generation and tracking

Motion planning problems benefit greatly from the properties of differential flatness, which are widely employed for trajectory generation and controller design. This paper aims to highlight the fact that various flat representations of a system can have different implications in the trajectory generation and tracking objectives. In particular, we consider a fixedwing UAV (Unmanned Aerial Vehicle) and analyze various flat representations. We reformulate the trajectory generation and tracking problem in terms of different flat outputs and analyze the optimal cost, constraints satisfaction, tracking error and computational complexity. Insights on the future work complete the analysis. The research shows promising directions, especially in the area of disturbance rejection and robust control.

I. INTRODUCTION

Flatness theory [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF] has proven useful in several stages of designing a control architecture for motion planning problems, i.e., in computing a reference trajectory [START_REF] Stoican | Flat trajectory generation for way-points relaxations and obstacle avoidance[END_REF], [START_REF] Martin | Flat systems, equivalence and trajectory generation[END_REF], [START_REF] Prodan | Predictive Control for Path-Following. From Trajectory Generation to the Parametrization of the Discrete Tracking Sequences[END_REF] or in providing a feedback linearizable controller [START_REF] Nguyen | Stability guarantees for translational thrust-propelled vehicles dynamics through nmpc designs[END_REF], [START_REF]Flat trajectory design and tracking with saturation guarantees: a nano-drone application[END_REF]. Having a flat representation of a dynamical system is equivalent with carrying a model inversion (the nonlinear input is mapped in terms of the flat output and linearizes the nonlinear dynamics), which in turn, provides the foundation for both the planning and controlling stages. While there are many formal results for flat output representations [START_REF] Nicolau | Flatness of multi-input control-affine systems linearizable via one-fold prolongation[END_REF], [START_REF] Bekcheva | Control of differentially flat linear delay systems with constraints[END_REF], to our knowledge, there is no generic algorithm capable to obtain a flat representation for arbitrary dynamics which admit one. An attempt in this direction is provided in [START_REF] Franke | On the computation of flat outputs for nonlinear control systems[END_REF], however the results is not generic. Furthermore, even when a flat representation is obtained, it may quickly become difficult to handle. Usually the inputs have convoluted expressions in terms of the system's flat output, leading to difficulties when handling constraints and cost formulations.

In [START_REF] Stoican | Flat trajectory generation for way-points relaxations and obstacle avoidance[END_REF] and [START_REF]Flat trajectory design and tracking with saturation guarantees: a nano-drone application[END_REF] a flat representation and subsequent Bspline parameterization [START_REF] Prodan | Necessary and sufficient lmi conditions for constraints satisfaction within a b-spline framework[END_REF] are exploited to construct an optimization problem which, simultaneously, minimizes trajectory length and satisfies various constraints.

However, none of the works aforementioned have addressed the problem of changing the flat output and analyzing the implications of this change for trajectory generation and tracking. Hence, our main contribution is the analysis of alternative flat representations within the same framework of planning and controlling a fixed-wing UAV (Unmanned Aerial Vehicle) and to point out the differences in the performance of the control scheme. More specifically, we reformulate the trajectory generation and tracking problem in terms of different flat outputs and analyze the optimal cost, the constraint satisfaction, tracking error and computational complexity. Note that when the flat output is changed, the construction of constraints and cost function in the trajectory generation process is also altered. Similarly, the feedback linearization control law derived from the model inversion induced by the flat representation will also transform remarkably.

Through this work we want to bring to the community's attention that choosing a certain flat output cascades throughout the rest of the control design procedure (for better or worse). Not in the least, we believe that the research carried here shows promising directions, especially in the area of disturbance rejection and robust control.

The paper is organized as follows. In Section II, we present the main issue of the trajectory generation and the subsequent tracking controller design. This problem will then be implemented with different flat outputs and then discussed in Section III. Section IV provides the numerical simulations. Finally, the conclusions are drawn in Section V.

II. PROBLEM FORMULATION

We first briefly introduce the 2D UAV model used hereinafter and recall the standard problem formulation of trajectory generation and of control design.

A. Fixed-wing UAV model

The UAV kinematic model can be described by the differential equations [START_REF] Stoican | Flat trajectory generation for way-points relaxations and obstacle avoidance[END_REF]:

ẋ = V a cos ψ + W x , ẏ = V a sin ψ + W y , ψ = g tan φ V a , (1)
where (x, y) and ψ are the position and the heading (yaw) angle. W x and W y are the wind velocity components on the x and y axis. Finally, the input signal are the airspeed velocity V a and the bank (roll) angle φ, respectively. The equations [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF] are rewritten in a more compact form as:

ξ = f (ξ, u) , (2)
where

ξ = [x y ψ] ∈ R 3 , u = [V a φ] ∈ R 2 and f (•, •) : R 3 × R 2 → R 3 is the state vector field.
Indeed, the nonlinear system (2) is differentially flat when we consider the fact that there exists a flat output vector, z = Φ(ξ, u, u, ü, ...) ∈ R 2 , such that the states and inputs can be algebraically expressed in terms of z and a finite number of its derivatives [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF].

B. Trajectory generation problem

Further, we recall the B-spline framework used in [START_REF] Stoican | Flat trajectory generation for way-points relaxations and obstacle avoidance[END_REF] where the flat output is parameterized in a time interval [t 0 , t N ] by a set of basis functions B d (t) = [B 0,d (t), ..., B n,d ] through the control points P ∈ R 2×n = [P 0 , ..., P n ] such that:

z(t) = PB d (t), ∀t ∈ [t 0 , t N ], (3)
where z(t) ∈ R 2 is the parameterized flat output, n is the number of control points and (d -1) the polynomial order of the B-spline functions, respectively.

Assuming the parameterization in (3) we can now formulate the trajectory generation problem in terms of splines and their associated control points. First, we consider a collection o way-points and their associated time stamps:

W = {w k } and T W = {t k }, (4)
through which the trajectory has to pass, i.e.,

z(t k ) = ω k =⇒ PB d (t k ) = ω k .
(5) The final objective of this process is to generate profiles, equivalently stated, to find the control points which meet the constraints [START_REF] Nguyen | Stability guarantees for translational thrust-propelled vehicles dynamics through nmpc designs[END_REF] and minimize the cost function as in [START_REF] Prodan | Necessary and sufficient lmi conditions for constraints satisfaction within a b-spline framework[END_REF]:

P = arg min P t N t0 Ξi (B d (t), P)dt, (6)
where Ξi (B d (t), P) is the energy function at the time instant t defined with the control points as decision variable, known basis function B d (t) and the flat output i which will be introduced in Section III.

C. Feedback linearization control design

As proposed in [START_REF] Martin | Flat systems, equivalence and trajectory generation[END_REF], many flat systems can be linearized in closed-loop. In here, using the flat representation we arrive at a controller which, for the nominal dynamics, cancels out the nonlinearities of the dynamics and arrives at a desired linear dynamic [START_REF] Slotine | Applied nonlinear control[END_REF] in closed-loop.

More specifically, the investigated system is in companion [START_REF] Slotine | Applied nonlinear control[END_REF] form represented as:

ξ = f (ξ, u) (7
) and an input transformation must be found:

z = z(ξ, ξ, ...), u = u(v, ξ, ξ, ...), (8)
so that the nonlinear system dynamics is converted to an equivalent linear time-invariant dynamics:

z (n) = v, v = z (n) ref + K n-1 e (n-1) + ... + K 0 e, (9)
where v is the virtual control input induced from the flat representation, e = z ref -z is the difference between the reference z ref and the feedback value z, and the coefficients K i with i ∈ {0, 1, ..., n-1} will be chosen such that the poles of the characteristic equation will ensure the stability of the model in closed-loop. Therefore, equation [START_REF] Franke | On the computation of flat outputs for nonlinear control systems[END_REF] becomes:

e (n) + K n-1 e (n-1) + ... + K 0 e = 0. (10)

III. ANALYSIS OF ALTERNATIVE FLAT OUTPUTS

As stated earlier, most results from the literature skip relatively quickly over the choice of a flat output and proceed to the control design steps. In here, we plan to investigate several flat representations and observe how they affect (for better or worse) the procedures of trajectory generation and feedback control design. In Subsection III-A we introduce the four available flat outputs together with the flat representations associated with them. In Subsections B and C, the problem formulation for the trajectory generation and the control design is given when considering the variation in the flat outputs.

A. Various flat representations

In this section we propose different flat representations computed from the chosen flat outputs with no wind perturbation (W x = W y = 0) and with perfect state measurement.

1) Flat output 1: Consider the following flat output in terms of the system's position:

z = z 1 z 2 = x y . (11)
Next, the remaining states and inputs are expressed as in the equation ( 12) which is detailed in Table I:

ξ u = Φ 1 (z, ż, z) . (12)
2) Flat output 2: The same process is applied for the second flat representation. In this selection, cylindrical coordinate is used to describe the system:

z = z 1 z 2 = x 2 + y 2 arctan y x . ( 13
)
The states and inputs then are expressed as in the equation ( 14) which is detailed in the third column of Table I:

ξ u = Φ 2 (z, ż, z) . (14)
where

f 1 (z, ż, z) = z2 ż1 z 1 -z1 ż2 z 1 g ż2 1 + z 2 1 ż2 2 , f 2 (z, ż) = 2 ż2 1 ż2 + ż3 2 z 2 1 g ż2 1 + z 2 1 ż2 2 .
3) Flat output 3: In here, we take into account the fact that when [z 1 z 2 ] is a flat output, [z 1 z 2 + ż1 ] is also a flat output. Hence, we have that:

z = z 1 z 2 = x y + ẋ . ( 15
)
The remaining states and inputs are expressed as in equation ( 16) which is detailed in the forth column of Table I:

ξ u = Φ 3 (z, ż, z, ... z ) . (16)
(•) Φ 2 (•) Φ 3 (•) Φ 4 (•) x z 1 z 1 cos z 2 z 1 z 2 -ż1 y z 2 z 1 sin z 2 z 2 -ż1 z 1 + z1 -ż2 tan ψ ż2 ż1 ż1 sin z 2 + ż2 z 1 cos z 2 ż1 cos z 2 -ż2 z 1 sin z 2 ż2 -z1 ż1 ... z 1 + ż1 -z2 ż2 -z1 Va ż2 1 + ż2 2 ż2 1 + z 2 1 ż2 2 ż2 1 + ( ż2 -z1 ) 2 ( ... z 1 + ż1 -z2 ) 2 + ( ż2 -z1 ) 2 tan φ 1 g z2 ż1 -z1 ż2 ż2 1 + ż2 2 f 1 (z, ż, z) + f 2 (z, ż) ( ż2 -z1 )z 1 -(z 2 - ... z 1 ) ż1 g ż2 1 + ż2 2 g 1 (z, ż, z, ... z , z (4) ) + g 2 (z, ż, z, ... z )

4) Flat output 4:

The following flat representation is a mix among the system's position, velocity and the acceleration:

z = z 1 z 2 = ẋ + y x + ẍ + ẏ . ( 17
)
The remaining states and inputs are expressed as in equation ( 18) which is detailed in the last column of Table I:

ξ u = Φ 4 z, ż, z, ... z , z (4) . ( 18
)
where

g 1 (z, ż, z, ... z , z (4) ) = ( ż2 -z1 )(z (4) 1 + z1 - ... z 2 ) g ( ż2 -z1 ) 2 + ( ... z 1 + ż1 -z2 ) 2 , g 2 (z, ż, z, ... z ) = (z 2 - ... z 1 )( ... z 1 + ż1 -z2 ) g ( ż2 -z1 ) 2 + ( ... z 1 + ż1 -z2 ) 2 .
Remark 1: These flat outputs are inspired from various sources: the flat output 1 in [START_REF] Slotine | Applied nonlinear control[END_REF] is the one most employed in the literature while the second flat output in (13) exploits a change of coordinates and the two last flat outputs are derive from the fact that if [z 1 z 2 ] is a flat output, then [z 1 z 2 + ż1 ] T is also a flat output.

Remark 2: All the flat representations above are found knowing the complete description of the system (inputs, outputs, actuators and the like). There is, however, some works related to "flat input" representation [START_REF] Nicolau | Constructing flat inputs for two-output systems[END_REF]. In this study, proper actuators or inputs are chosen such that a priorly given output becomes a flat output.

B. Trajectory generation using flatness

In equations ( 4) and ( 5) we provided a generic motion planning problem and in Section III-A we provided several flat representations. In what follows, we show how the construction of both constraints and costs change depending on the particular flat representation employed.

More specifically, we propose the constraints w k concerning the position of the UAV which associate with the same amount of time-stamps t k :

W = {w 1 , w 2 , ..., w N } , T W = {t 0 , t 1 , ..., t N }. (19)
In other words, at the time instant t k ∈ T W , the vector [x(t k ) y(t k )] must satisfy:

x(t k ) y(t k ) = w k ∈ W. (20)
Furthermore, the general optimization mentioned in (6) will be replaced by the integral:

P = arg min P t N t0 ẋ2 + ẏ2 dt (21)
Initially, the constraints (20) and the cost (21) have been given in the position components (x, y) and now, for each flat representation, we rewrite them in terms of the flat output.

1) Cost function with flat output 1: With the representation of flat output 1, we can rewrite the problem (6) with the energy function as:

Ξ1 (B d (t), P) = ż2 1 + ż2 2 = ż ż. (22)
Similarly, the constraints in (20) are rewritten as:

x(t k ) y(t k ) = z 1 (t k ) z 2 (t k ) = w k . (23)
2) Cost function with flat output 2: Similarly, the cost function ( 21) can be written with the second flat output as:

Ξ2 (B d (t), P) = ż2 1 + z 2 1 ż2 2 .
(24) The corresponding form of (20) with the flat output (13) is characterized by:

x(t k ) y(t k ) = z 1 (t k ) cos z 2 (t k ) z 1 (t k ) sin z 2 (t k ) = w k . (25)
However, it is trivial to show that equation (25) does not give unique solution for [z 1 (t k ) z 2 (t k )] due to the periodicity of the trigonometric functions. Hence, another way to specify the desired value for the flat output is introduced as:

             x(t k ) y(t k ) = w k , z 1 (t k ) = x(t k ) 2 + y(t k ) 2 , z 2 (t k ) = arctan y(t k ) x(t k ) + 2kπ. (26)
where k can be chosen so that the continuity of z 2 (t) will not break the desired direction of the trajectory.

3) Cost function with flat output 3: With the representation of the flat output 3 in (15), we rewrite the problem (21) as:

Ξ3 (B d (t), P) = ż2 1 + ( ż2 -z1 ) 2 .
(27) Similarly, the constraints in (20) are indicated as:

x(t k ) y(t k ) = z 1 (t k ) z 2 (t k ) -ż1 (t k ) = w k . ( 28
)
4) Cost function with flat output 4: Equivalently, with flat output 4, we can rewrite the problem (21) as:

Ξ4 (B d (t), P) = ( ...

z 1 + ż1 -z2 ) 2 + ( ż2 -z1 ) 2 . ( 29
)
Similarly, the constraints in (20) are indicated as:

x(t k ) y(t k ) = z 2 (t k ) -ż1 (t k ) z 1 (t k ) + z1 (t k ) -ż2 (t k ) = w k . (30)

C. Tracking control with feedback linearization

Let us recall the process mentioned in Section II-C, we consider the new input of the system as:

u = u 1 u 2 = Va tan φ = u(z, ż, z). (31)
Then, the virtual input is chosen as: v = z.

(32) With such transformation, virtual input (32) will be computed with respect to the linear error dynamics:

v = zref + K 1 ėz + K 2 e z dt + K 3 e z , (33)
with e z = z ref (t) -z(t) where z ref (t) is the reference signal and z(t) is the feedback flat output. However, with the flat representations from ( 16) and ( 18), there will exist the derivative of the flat output with the order greater than 2 in the representation of Va and tan φ. Hence, those two flat output do not meet the condition in (31).

Therefore, using only the representation in ( 12) and ( 14), we have the new input:

-for flat output 1 given in ( 12) :

Va = z1 ż1 + z2 ż2 ż2 1 + ż2 2 , tan φ = z2 ż1 -z1 ż2 g ż2 1 + ż2 2 . (34)
-for flat output 2 given in ( 14):

Va = z2 ż2 z 2 1 + z1 ż1 + ż2 2 ż1 z 1 ż2 1 + z 2 1 ż2 2 , tan φ = z2 ż1 z 1 -z1 ż2 z 1 + 2 ż2 1 ż2 + ż3 2 z 2 1 g ż2 1 + z 2 1 ż2 2 . ( 35
)
Hence, the virtual input v can be formulated as in (36) and computed with the linear dynamics proposed in (33).

v = v 1 v 2 = z = z1 z2 , (36)
Ultimately, with the flat output ( 11) and (13), we obtain the two different control laws (37) and (38), respectively:

       Va = v 1 ż1 + v 2 ż2 ż2 1 + ż2 2 , tan φ = v 2 ż1 -v 1 ż2 g ż2 1 + ż2 2 , (37)
       Va = v 2 ż2 z 2 1 + v 1 ż1 + ż2 2 ż1 z 1 ż2 1 + z 2 1 ż2 2 , tan φ = v 2 ż1 z 1 -v 1 ż2 z 1 + 2 ż2 1 ż2 + ż3 2 z 2 1 g ż2 1 + z 2 1 ż2 2 , ( 38
)
where v 1 and v 2 are computed from feedback as in (33).

IV. SIMULATION RESULT A. Trajectory generation

For illustration, we use the list of 6 way-points and associated time-stamps taken equidistantly between t 0 and t N = 45s for the trajectory generation problem:

W = 10 2 × -1.5 0.1 , -1.3 1 , -0.1 1.2 , 0.8 0.1 , 1.6 -0.9 , 0.2 -1.2
T W = {0, 9, 18, 27, 36, 45}.

(39) Next, we require that all generated states and inputs are continuous. Since the flat output is a weighted sum of spline functions, this means in fact that we have to choose the splines to be of sufficient degree, as detailed in Table II.

Note that each further increase in continuity (e.g., velocity or acceleration are continuous themselves) translates into an incrementation of the B-spline functions' degree. Fig. 1 depicts the profiles of the state ξ(t) which are generated with 4 proposed flat representations ( 12), ( 14), ( 16) and (18) respectively. Hereinafter, for simplicity, the 4 aforementioned representations with their corresponding trajectories will be labeled representation as r (rep. r) with r ∈ {1, 2, 3, 4}. Similarly, the reference for the inputs is shown in Fig. 2 and the corresponding position is in Fig. 3.

All the reference signals are computed such that they satisfy the optimization problem (21), the constraints in (39) and the continuity constraints, as given in Table II.

B. Feedback linearization control

On the other hand, in the control simulation process, we reuse the model ( 1) and verify it with wind perturbation.

The reference signal is the profile generated with flat output 1 (the curve denoted as rep. 1 in Fig. 3) and simulation data specification is detailed in Table III where the parameters K 1 , K 2 and K 3 are chosen from the characteristic equation and its poles p 1 , p 2 and p 3 such that:

(s -p 1 )(s -p 2 )(s -p 3 ) = s 3 -K 3 s 2 + K 1 s + K 2 . (40)
For the sake of illustration, the poles are chosen identically as p 1 = p 2 = p 3 = -2. In this simulation, the tracking errors will be labeled correspondingly with the control laws proposed in the transformation (34) and (35) as control law 1 (ctrl. law 1) and control law 2 (ctrl. law 2). Details for the errors are given in Fig. 5 for the position x, y and the heading angle ψ.

Besides, in this simulation, we provide 4 different scenarios of wind perturbation vector (W ) to experiment on the dependence of the tracking error on the wind direction. Details are provided in Table IV where RM S(e ij ) is the root mean square value of the tracking error in the i axis (i ∈ {x, y}) implemented with the control law j (j ∈ {1, 2}).

C. Discussions

Let us provide some discussions and insights related to the UAV trajectory generation and tracking when considering various flat representations.

Trajectory generation.

• As expected, from Table II, all the representations can be used to satisfy the way-points constraints. It is also notable that the number of control points needed is not affected by the variation of flat outputs, confirming the unique dependence of the control points on the number of way-points. • From the parameterization point of view, it can also be seen in the table that the polynomial order of the trajectories are dissimilar, which can be explained with the degree of derivatives existing within each flat output. More specifically, in representation 1 in (12) and 2 in (14), the highest derivative order is 2 while that of representation 3 (16) and 4 (18) are 3 and 4, respectively, resulting in different requirement on the B-spline functions' degree to satisfy the continuity constraints. • Moreover, when the number of control points is considered, the variation of flat representations brings about the differences in not only the curves' length but also the value of the cost function (Fig. 4). Nonetheless, these differences fades when this number gets larger. This can be explained by the convergence of the trajectory to the control polygon, suggesting that after a certain number of control points, there is no use in increasing the number. • Finally, of interest is also that the cost and path length do not greatly depend on which flat representation is chosen. Still, the complexity of the representation plays a role, by making the problem less (or more so) complex. More precisely, if we look at the representation 1, 3 and 4, the two values increase along with the number of derivative involving in the representations. On the other hand, the second flat output proves more complex, since it contains trigonometric functions, whose periodicity affects the optimization procedure. • In conclusion, since the curves are constrained in space and time, it comes as no surprise that all the trajectories are relatively similar. Yet, depending on the mathematical expressions inside each representation, each trajectory will have a particular representation. Trajectory tracking: While the trajectories are distinguishable, the tracking problems seem to have similar behaviors:

• As depicted in Fig. 5, there are differences between the error dynamics of the two control system. These dissimilarities come from the variable where the error dynamic is applied. In detail, the first control law implement the linear dynamic in the first states which are x(t) and y(t). Meanwhile, with the second law, the linearity is applied in the flat output which is a combination of the two first states. However, this property may not have a strong enough impact on error, thus, the results present little meaningful information. • However, the results given in Table IV suggest the reliance of the system's performance on both the direction of the wind and the control laws. More particularly, the tracking of the UAV shows better results for the first two scenarios (W = [+5; +5] and W = [-5; +5]m/s). On the contrary, when W = [-5; -5], only the control law 2 can stabilize the system and when W = [+5; -5] both of the controllers cannot handle the perturbation.

V. CONCLUSIONS

This paper presented the trajectory generation and tracking problem for a fixed-wing UAV. We have shown how multiple flat representations affect various elements of the control architecture. Within the former problem, it is notable that changing the flat output leads to significant changes of the cost function and constraint definitions. The control action is also changed in the latter issue, with implications in what regards closed-loop robustness and performance. As future work, it remains to further analyze complexity of the trajectory generation procedure on one hand and disturbance rejection for the tracking mechanism on the other hand.

Fig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. x(t), y(t) and ψ(t) with 4 flat outputs

Fig. 4 .

 4 Fig. 4. The the curves' length and objective function values in terms of the number of control points

Fig. 5 .

 5 Fig. 5. Tracking error of x(t) and y(t)


Gourab Bhattacharya

Bhattacharya Gourab

THE YAMABE INVARIANT OF THE GRAVITATIONAL MONOPOLE

We show the Yamabe invariant is non-positive due to the presence of the Gravitational Monopole equations.

Introduction

In this short note we announce some results on the Yamabe invariant of a Gravitational monopole, the notion of the Gravitational monopole was first introduced in [cf .1]. We shall give details of the proofs in a different paper.

Let (M n , g) a smooth, compact-oriented Riemannian n-manifold. As we know the Einstein-Hilbert action (1.1) g → s g dv g =: I g with s g the scalar curvature and dv g is the volume form with respect to the metric g gives the Einstein equations at the critical points of the action I g with a volume constraint. The Yamabe problem is to minimize I g in a fixed conformal class of metric. Due to such a constraint, critical points of I g have constant scalar curvature. Let us define now the Yamabe invariant Y (g),

(1.2) Y (g) := inf g=e 2w g vol( g) -n-2 n s g dv g .
Due to the extensive work of Yamabe, Aubin, Trudinger, and Schoen, we now know that every conformal class on a compact manifold admits a metric that achieves Y (g) and therefore has constant scalar curvature. Now in particular for n = 4, M 4 is an oriented four-dimensional manifold. The Hodge * -operator induces a splitting of the space of two-forms

(1.3) 2 = 2 + 2 -
into the subspace of self-dual 2-forms 2 + and anti-self-dual 2-forms 2 -. This decomposition induces a splitting of the Weyl curvature (defined as a trace-free endomorphism of the 2 ) into its self-dual and anti-self-dual components W ± . We have the Hirzebruch signature formula

(1.4) 12π 2 σ(M 4 ) = M 4 (|W + | 2 -|W -| 2 )dv g . Let us define the Weyl functionals W[g], W ± [g] by (1.5) W[g] := |W g | 2 dv g , W ± [g] := |W ± g | 2 dv g .
Due to the Hirzebruch signature formula (1.4), the study of the Weyl functional W[g] is equivalent to the study of the self-dual functional

W + [g].
The signature formula implies the following fact, Proposition 1.

(1.6) W + [g] ≥ max{12π 2 σ(M 4 ), 0}.
The condition

(1.7) W + [g] = max{12π 2 σ(M 4 ), 0}.
is true if and only if

W + ≡ 0 or W -≡ 0, that is g is half-conformally flat.
The following formula is useful

(1.8) |W | 2 = |W + | 2 + |W -| 2 ,
also since W is conformally invariant, W ± are conformally invariant pieces of W .

The following result is due to Taubes: for any (closed, compact, orientable) four-dimensional manifold N 4 , the manifold M 4 = N 4 #kCP 2 admits anti-self-dual metrics (i.e. W -= 0) for all k sufficiently large.

It is a classical result that using the conformal invariance of the Bach tensor, one concludes that any metric which is (locally) conformally Einstein is critical for W + .

The Gravitational Monopole Equations

In [START_REF] Bhattacharya | Gourab Gravitational Monopoles[END_REF], the following equations are introduced (sometimes we omit the mapping c, and denote by " • " the Clifford multiplication (or composition) and the dimension 4 for the convenience of computations):

/ ∇ψ = (d + d * )ψ = 0, c(W + g ) = 1 4 ⟨e i • e j ψ, ψ⟩e i ∧ e j , (2.1)
One can rewrite (2.1) in the following form

/ ∇ψ = 0, c(W + g ) = ψ * ⊗ ψ - |ψ| 2 2 Id, (2.2)
therefore we have the following proposition Proposition 2.

(2.3)

|W + | 2 = |ψ| 4 8 .

Constraint on the Yamambe invariant due to the Gravitational Monopole Equations.

We have the following propositions.

Proposition 3. Let (M 4 , g 0 ) be a compact, oriented, four-dimensional manifold with a Spin C -structure. Let E the trace-free part of the Ricci tensor. We also assume the validity of (2.2) on M 4 . We further assume

(3.1) 3|ψ| 4 dv g0 = 32π 2 (2χ(M 4 ) + 3σ(M 4 )),
then there is a metric g = e 2w g 0 such that (3.2) 6∆s g + 3|ψ| 4 + 12|E| 2 = s 2 g . It was also shown in [START_REF] Gursky | The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics[END_REF] without the existence of the Gravitational monopole hypothesis that Proposition 4. Let (M 4 , g 0 ) be a compact, oriented, four-dimensional manifold. Let E the trace-free part of the Ricci tensor. Assume

(3.3) |W + | 2 dv g0 = 4 3 π 2 (2χ(M 4 ) + 3σ(M 4 )),
then there is a metric g = e 2w g 0 such that

(3.4) ∆s g = -4|W + | 2 -2|E| 2 + 1 6 s 2 g .
Also,

(1) if Y (g 0 ) > 0, then s g > 0.

(2) if Y (g 0 ) = 0, then g is Ricci-flat anti-self-dual metric.

Since the Gravitational monopole puts the restriction on the scalar curvature s g0 ≤ 0 of M 4 [cf. 1], and since Proposition (4) implies whenever Y (g 0 ) > 0, then s g > 0, we therefore conclude, the following theorem: Theorem 3.1. Let (M 4 , g 0 ) be a compact, oriented, four-dimensional manifold with a Spin C -structure. Let E the trace-free part of the Ricci tensor. We also assume the validity of (2.2) on M 4 . We further assume

(3.5) 3|ψ| 4 dv g0 = 32π 2 (2χ(M 4 ) + 3σ(M 4 )),
then there is a metric g = e 2w g 0 such that

(3.6) Y (g 0 ) ≤ 0.
Since Y (g 0 ) = 0 implies Ricci-flatness and anti-self-duality, it requires a special attention.

Proposition 5. Let (M 4 , g 0 ) be a compact, oriented, four-dimensional manifold with a Spin C -structure. Let E the trace-free part of the Ricci tensor. We also assume the validity of (2.2) on M 4 . We further assume

(3.7) 0 < 3|ψ| 4 dv g0 < 32π 2 (2χ(M 4 ) + 3σ(M 4 )),
then there is a metric g = e 2w g 0 such that (1) Y (g 0 ) ≤ 0. (2) There is no metric g 0 satisfying (3.7) with Y (g 0 ) = 0, therefore, Y (g 0 ) < 0.

(3) 24∆s g + 3(4 + ϵ 1 )|ψ| 4 + 48|E| 2 + 4s 2 g = 0, for some ϵ 1 > 0. Proposition 6. Let (M 4 , g 0 ) be a compact, oriented, four-dimensional manifold with a Spin C -structure. Let E the trace-free part of the Ricci tensor. We also assume the validity of (2.2) on M 4 . We further assume

(3.8) |ψ| 4 dv g0 = 16π 2 (2χ(M 4 ) + 3σ(M 4 )), then (1)
Y (g 0 ) ≤ 0. (2) there is a metric g = e 2w g 0 such that [cf. 2]

∆s g = - 3 2 |E| 2 + 1 8 s 2 g .
Proposition 7. Let (M 4 , g 0 ) be a compact, oriented, four-dimensional manifold with a Spin C -structure. Let E the trace-free part of the Ricci tensor. We also assume the validity of (2.2) on M 4 . We further assume (3.9) 0 < |ψ| 4 dv g0 = 16π 2 (2χ(M 4 ) + 3σ(M 4 )), then

Y (g 0 ) ≤ 0. (2) there is a metric g = e 2w g 0 such that [cf. 2] ∆s g + ϵ 2 |ψ| 4 + 12|E| 2 = s 2 g , for some ϵ 2 > 0.


Romain Pic

Zeina Al Masry

Clément Dombry

Christine Devalland

Prediction of Oncotype DX using DRFs

Introduction

Oncotype DX (ODX) is a multi-gene expression signature for breast cancer that provides prognostic and predictive breast cancer recurrence information for estrogen receptor (ER)-positive and HER2 negative patients. This test can predict if a chemotherapy treatment would be beneficial. However, this test is expensive and several studies have shown its link with certain clinico-pathological data. The result of the test is a score between 0 and 100 and guides clinicians regarding prescription of chemotherapy (e.g. for score higher than 25). Aim: Predict the ODX score based on histopathological variables. Proposed methodology: Probabilistic forecast using Distributional Regression Forest (DRF; [START_REF] Athey | Generalized random forests[END_REF])

Presentation of the cohort

The cohort is a retrospective study between 2012 and 2019 with 334 cases that underwent ODX assay from three hospitals: Besançon, Belfort and Dijon. All patients have ER-positive and HER2-negative early breast cancer. We used 9 variables for the prediction of the ODX: age, tumor size, SBR grade, Nottingham grade, ER status, PR status, Ki67 index proliferation, P53 protein and lymph node status. In order to remove missing values of ODX, we had to remove one patient and we finally keep 333 patients.

Flowchart of Distributional Regression Forests

Outputs of DRF

DRF is a powerful tool and is able to cover numerous methods such as classification, mean regression and quantile regression. Moreover, DRFs assess the uncertainty of a prediction.

Evaluation of the predictive distribution

The Continuous Ranked Probability Score (CRPS; [START_REF] Matheson | Scoring Rules for Continuous Probability Distributions[END_REF]) is a scoring rule used to compare a predictive distribution F and the outcome y. The CRPS is used in Figure 3 to evaluate the performances of DRF for each patient in an out-of-bag prediction. The CRPS evaluates the dispersion and the bias; thus, a higher CRPS correspond to higher bias and/or higher dispersion of the predictive distribution.

Figure 3. We highlight three cases : a "good" prediction, a "medium" prediction and a "bad" prediction associated with low, medium and high values of CRPS respectively. The red dashed line shows the true ODX, the regions in green and red corresponds to ODX ≤ 25 and ODX > 25 respectively and the numbers at the center of these regions are the predicted probabilities of each class.

Conclusion

Novel approach: distribution prediction; Various outputs providing information for practitioners (e.g. similar profiles); Performance similar to state-of-the-art techniques in terms of classification; Explainable and transparent method; Operational clinical RShiny app for practitioners.

Perspectives: robustness to missing values and noise.

Limitation: predictions are only as good as the training dataset they are based on.

Figure 1 .

 1 Figure 1. Flowchart of distributional random forest. Starting from the training data, a large number of subsamples are chosen and binary trees are constructed on each subsamples; the neighborhoods and weights at the point to predict (cross) are computed in each tree and then averaged so as to give the forest weights; the predictive distribution corresponds to the weighted sample of the original training data with these forest weights.

Histogram

  of the predictive distribution; Probability of classes of interest; Mean or median prediction; Uncertainty associated to the prediction (standard deviation or confidence interval); Similar/influential patients for meaningful and informative comparison. (very important for practitioners)

Figure 2 .

 2 Figure 2. Applications of DRF.

Comparison with state-of-the-art techniques

					Accuracy	79.3%
			Predicted	Sensitivity	92.0%
			ODX ≤ 25	ODX > 25	Specificity Positive Predictive Value 82.5% 40.3%
	True	ODX ≤ 25 ODX > 25	231 49	20 33	Negative Predictive Value 62.3% F1-score 0.870
					Area Under Curve	0.759

Table 1 .

 1 Evaluation of DRF with classification diagnostics. Confusion matrix (left) and standard metrics (right).

	Study	Data set (n train , n test , p)	Accuracy (%)	Sensitivity (%)	Specificity (%)	AUC

Table 2 .

 2 Comparison (on different datasets) of the DRF method with 3 other methods for the 2-class classification problem (ODX ≤ 25 and ODX > 25). The methods are taken from Kim et al. (2019), Orucevic et al. (2019) and Pawloski et al. (2021) [4, 6, 7]. The high class is taken as positive.

	Study	Acc.	Sensi.	Speci.	PPV	NPV	AUC
	Hou et al. (2017)	68.7	85.7	41.4	72.6	61.5	NA
	Baltres et al. (2017)	46.3	55.0	78.0	58.3	62.4	0.63
	DRF	57.7	58.5	83.6	87.0	51.9	0.748

Table 3 .

 3

Comparison (on different datasets) of the DRF method with 2 other methods (Hou et al. (2017) and Baltres et al. (2017) [3, 2]) for the 3-class classification problem: ODX ≤ 18 (low), 18 < ODX ≤ 30 (medium) and ODX > 30 (high). The sensitivity, specificity, PPV, NPV and AUC are given only for the lower class.


Lisa Chabrier

Anton Crombach

Sergio Peignier

Christophe Rigotti

Semi-supervised learning for tree-based regressors to improve the prediction of the interactions between genes

1-INFERRING GENE REGULATORY NETWORKS

2-SEMI-SUPERVISED LEARNING FOR TREES

Semi-supervised learning for tree-based regressors to improve the prediction of the interactions between genes

...

Single-cell RNA-sequencing measures the expression of genes cell by cell. A major challenge in the analysis of these data is so-called dropouts, that are indistinguishable from real zeros. And zeros can represent up to 95% of the data.

Single cell RNA sequencing

QUESTIONS

How to efficiently build the tree regression models in order to use the algorithm in the Arboreto pipeline?

How to build tree-based models that integrate measured values and handle the uncertain origins of zeros?

Can we find a unique optimal value of the parameter for all single-cell datasets?

Unlabeled Examples

Labeled Examples

We trained 100 random forest using the ssl criterion to choose the splits over the dataset, with different ratio of unlabeled values and different values for the parameter w.. As expected, the R² decreases when the number of labeled examples increases, but performance remains extremely high across a wide range of % unlabeled data.

Kmeans clustering

?

Arboreto framework [1] ? ?

? ?

?

Regression models built from single-cell data, predicting the expression of one target gene from the expression of transcription factors (TFs).

Feature ranking is computed for each model and used to score the interaction between the TF (the feature) and target gene.

[

NEW PARAMETER

In this impurity definition, a new parameter is introduced: w. Tuning this parameter might be an issue, but according to the original paper [4] and the early results we have, it is possible that the value of the parameter will not have such a big influence in our experimental setting. This is still an open question.

COMPUTATION TIME

Scikit-learn implements a so-called impurity proxy that has the same monotony as the real impurity. This reduces the time needed to find the optimal split at each node. For now, our implementation in scikit-learn [3] does not come with such a proxy, but finding and implementing one could reduce the computational power needed.

Reducing the number of features on which to compute the second term is also a possibility by using a random subset of features.

REAL SETTING

The implementation of this impurity measure still needs to be tested in a realistic situation, to reproduce the experiment presented in Section 2. This is time-consuming but necessary to answer the initial questions.

MSE

  3] F. Pedregosa, G. Varoquaux, A. Gramfort,et al., "Scikit-learn: Machine learningin Python", Journal of Machine Learning Research, 2011. [1] T. Moerman, S. Aibar Santos, C. Bravo González-Blas, et al., "GRNBoost2 andArboreto: Efficient and scalable inference of gene regulatory networks", Bioinformatics, Jun. 2019, issn: 1367-4803,1460-2059, doi:10.1093/bioinformatics/bty916. D. Marbach, J. C. Costello, R. Küffner, et al., "Wisdom of crowds for robust gene network inference", Nature methods, Jul. 2012, issn:1548-7091. doi:10.1038/nmeth.2016.[4] J. Levatić, M. Ceci, T. Stepišnik, et al., "Semi-supervised regression trees with application to QSAR modelling", Expert Systems with Applications, Nov. 2020, doi: 10.1016/j.eswa.2020.113569.

  SSLCriterionsExperiment on the DREAM5 dataset[2]. For each curve, dropouts were added to a different part of the data: TFs, non-TF genes, and the whole dataset. Splits are composed of two elements: the feature and the threshold with max reduction of the impurity in child nodes. In scikit-learn [3], the Python library used in Arboreto [1], the impurity of a node is based on the mean squared error. Levatic et al. [4] proposed to implement semi-supervised learning for regression trees by replacing the impurity computation with one that takes into account unlabeled examples. R² R² -4.41 For this experiment, we choose one feature to be the target function, and we trained 100 regression forest for each experimental setting decribed by the couple (Impurity measurement, training data). On the figure is represented the mean R² score for each experimental setting Real expression SSL predictions MSE predictions