Keywords: Consumer, semi-directive interview, dairy & plant-based hybrid product, non-dairy, substitution

Background -The global demand for animal protein sources increases fast. In a perspective of reducing the proteins from animal sources of our diets to the benefit of plant proteins, known to have better health and environment-friendly properties, create some hybrid products associating dairy and plant ingredients could help the transition to diets higher in plant proteins. Indeed, the transition remains slow and non-dairy products are still perceived to be for vegetarians and lactoseintolerant people.

Objectives -The purpose of this project is to explore consumers' attitudes and representations towards innovative hybrid products associating dairy and plant ingredients.

Design -An explorative study has been conducted using semi-directive interviews as a qualitative technique for data collection.

Setting and participants -Semi-directive interviews were held with 20 participants aged between 18 and 58 in April and May 2020 conducted via videoconference.

Results -Participants did not reject hybrid innovative products. Attitudes and representations were different as function of several characteristics of the ingredients used and individual appreciation of dairy and non-dairy products.

Introduction 1.General context

The global demand for animal protein sources such as meat, milk, and eggs is increasing fast [START_REF] Steinfeld | laitiers ? De quelle façon combinez-vous les produits laitiers avec d'autres ingrédients ou d'autres aliments ? Relance : Vous pouvez réfléchir à cela en partant de vos repas d'hier ou en pensant à d'autres produits ou recettes que vous avez l'habitude de consommer Si on se concentre maintenant uniquement sur les associations que vous faites de produits laitiers avec des ingrédients ou des aliments d'origine végétale[END_REF]. Livestock farming and activities affect the environment and are responsible for the production of greenhouse contributing significantly to climate change. Besides, meat and animal product consumption has substantial effects on long-term health [START_REF] Godfray | Meat consumption, health, and the environment[END_REF].

Therefore, it is needed to feed the growing population with more sustainable food.

Animal protein intake is commonly recommended to be reduced in favor of plant proteins known to have better health effects and lower environmental impacts [START_REF] Rizkalla | Health benefits of low glycaemic index foods, such as pulses, in diabetic patients and healthy individuals[END_REF][START_REF] Stehfest | Climate benefits changing diet[END_REF].

There is a growing demand for dairy products alternatives driven by the rising environmental and ethical issues as well as reducing proteins from animal sources [START_REF] Haas | Cow Milk versus Plant-Based Milk Substitutes: A Comparison of Product Image and Motivational Structure of Consumption[END_REF]. However, the substitution of dairy products by plant-based substitutes is an interesting option that is possible only when consumers accept these new products. Consumers expect plant-based substitutes to be similar as the animal product. Moreover, one way to introduce unfamiliar foods is to combine them with existing foods (Wansink, 2002).

Therefore, propose some new products to help consumers transitioning towards more plant-based diets appears as a new horizon. However, understanding consumers' attitudes and representations towards potential innovative products is the first step to succeed in this initiative.

STLO's project

The trend to substitute meat has been initiated for several years, but the interest in substituting dairy products does not yet seem very present among consumers. Most of the time, consumers who substitute dairy products do so for very specific reasons such as lactose intolerance. Even when consumers are convinced that it is necessary to substitute, they tend to be attached to the taste of products from animal origin and they sometimes have negative opinions about plant-based substitutes.

The creation of innovative transition products, based on milk and plant-based ingredients, could help consumers to gradually reduce their consumption of dairy products towards higher consumption of plant-based diet.

Hybrid is defined by the Cambridge dictionary as 'something that is a mixture of two very different things'. Herein, within the framework of this project, hybrid products refer to any product made of both dairy and plant-based ingredients. If the consumers do not see any interest to consume such hybrid products, then the interest to spend time and money on their creation is relatively restricted. This is the reason why the STLO lab (Science et Technologie du Lait et de l'Oeuf) contacted the CSGA. Their goal was to ensure if those kinds of products would be favorably received by consumers, as upstream research to a broader project.

Objectives and hypothesis of my internship

The objective of my internship was to study the representations and attitudes of consumers towards hybrid products.

I tried to find out if consumers would see an interest of the use and consumption of innovative hybrid products, compared to dairy and plant-based products that they probably already associate in their diet.

My internship was organized in two phases:

-Qualitative part: semi-directive interviews in order to identify the main components involved in the consumers representations and attitudes towards hybrid products.

-Quantitative part: creation of a questionnaire that will be submitted to a high number of people and will allow us to determine the importance of each component.

This report will present the qualitative study only: the second part (quantitative) is still under progress.

In order to understand how consumers considered hybrid products, we needed to explore consumers' representations and attitudes towards dairy products and non-dairy products separately, to finally look into the consumers' representations of hybrid products.

Our hypothesis is that, in consumers mind, the advantages and drawbacks of dairy products on the one hand, and plant-based products on the other hand, would compensate one another within hybrid products.

A second objective was to investigate whether the level of transformation of the hybrid products would influence the representations that consumers have, and their attitudes towards hybrid products. The hypothesis associated to this objective was that the interest of consumers for hybrid products may be different as function of the level of transformation and their possible use.

2 Moreover, we tried to know whether such products would meet all types of consumers' expectations or only those already engaged in a reduction of the part of animal proteins in their diet, such as vegetarian or flexitarian consumers.

Material and methods

For this study, we decided to investigate consumers' attitudes and representations through a qualitative investigation. Semi-directed interviews and focus groups lend themselves well to those approaches. However, semi-directed interviews have been chosen for convenient reasons due to the Covid-19 lockdown.

Recruitment of the participants to interviews

Participants were recruited via e-mailing via social networks and through word of mouth.

The recruitment ad specified that a student was seeking participants in the context of his internship to take part in a 30 minutes discussion about food via videoconference. The real topic of the study was purposely not specified. The ad mentioned that participants would be rewarded for their participation in this discussion, but didn't specify what the reward consisted in, to avoid people only interested in earning money. The potential participants were invited to send an e-mail to the main investigator if they were interested to take part in a session.

Once participants made contact, they were asked to fill out a recruitment questionnaire (appendix 1), for us to collect specific information such as their gender, age, and their job. Then, they had to fill out some information related to their usual consumption frequency of some food product categories such as meat products, seafood, dairy products, non-dairy products, plant-based meat alternatives. Furthermore, respondents were asked to rate on a 5-point-scale how much they felt concerned about the influence of their diet on the environment and their health. The last question was related to the specific diet they were currently observing. A unique code was communicated to each participant in the e-mail with the questionnaire link, to keep their personal information anonymous.

That information allowed us to select participants according to defined inclusion criteria. The participants chosen had to be aged between 18 and 60 years old, and to consume dairy products as part of their usual diet (vegan and non lacto-vegetarian were excluded). People working in the dairy or food industries or food research were excluded.

The answers to the online questionnaire allowed us to sort people into two different groups:

-A group of people 'engaged in the food transition', this group was composed of people that selfdeclared as vegetarian, flexitarian, but also people declaring to eat meat and seafood less than once a week and declaring to eat plant-based alternatives and to try to reduce their animal products consumption.

-The second group gathered people we defined as 'omnivorous', composed by people who didn't declare any specific diet restriction and self-declared to eat meat at least 'once a week and more', and declared to eat less than once a month plant-based alternatives.

We selected people from these two groups in order to obtain a sample of people with heterogeneous profiles, represented by different food behaviors.

Participants

Overall, twenty participants living in France (16 females, 5 males) aged between 18 and 58 were recruited and interviewed. Ten of them belong to the group 'omnivorous' and ten to the group 'engaged in the food transition' (appendix 2). The group of people we defined as 'engaged in the food transition' is composed of nine females, one male, mean age of 32.5 years old (median: 33.5, min: 19, max: 54). The second group we defined as 'omnivorous people' was composed of 6 females, 4 males, mean age of 36.5 (median: 39.5, min: 18, max: 58).

The interview guide

We built an interview guide divided into five parts (appendix 3), each part containing some pre-defined questions we wanted to address to the participant. The parts were designed according to our research question and the project goal. Each part was approached in a precise and logical order.

Some pre-defined questions were systematically asked every participant. Additional questions were prepared to rephrase each initial question in case the participant did not answer spontaneously. They were used only when the moderator felt that the interviewee was running out of ideas and were adjusted based on the answers given to the previous questions.

The first part is a general introduction to the topic of the discussion. We first asked the participants to describe their diet by listing the food they consumed during the meals of the two previous days, including breakfast, lunch, diner, and any snacks they could have. This task was meant to put interviewees at ease and to serve as a basis for discussion. Once they had described the food they took, the interviewer wrapped up and took care to mention each family of food products the participant had cited (fruits and vegetables, carbs…), and finished by highlighting the presence of dairy products. Then he proposed the interviewee to talk about those dairy products.

The second part deals with dairy products and the reasons to consume them. We wanted to know the attitudes and opinions of the participants towards dairy products. To do so, the moderator first asked the interviewee to complete the following sentence: « Je consomme des produits laitiers pour ou parce que…». This was followed by a few questions meant to know if the interviewee was currently trying to reduce, increase, or maintain his/her dairy product consumption level. After that,

participants were asked to complete another sentence to know his/her representations and beliefs on dairy products: « les produits laitiers sont bons pour…» ; « les produits laitiers sont mauvais pour… ».

The third part is about the usual associations of dairy products with ingredients and products from other food families. We wanted to obtain information about respondents' habits concerning those associations, then their attitudes and knowledge towards associations, and if they found an interest in these associations.

The fourth part is about non-dairy substitutes. The moderator first proposed the interviewee, based on his/her food habits, to imagine a situation where they would no longer be able to eat any kind of dairy product. We first asked them which dairy-free products they would use to replace the ones they usually consume. Afterward, we asked further questions to understand their attitudes and habits towards dairy products substitution. Then, as function of their previous answers, we asked them what were the motivations and barriers to use non-dairy alternatives in their diet. Finally, we wondered about the impact on social norms on their attitudes towards non-dairy products.

The last part deals with innovative hybrid products. The word 'hybrid' will be used in this report to refer to this kind of products but was never used during the interviews.

Based on a method used by Clarkson et al. (2018) we first asked respondents to imagine a kind of product that would associate dairy and plant ingredients. They were free to write or draw and were given some time to think about this hybrid product they would like to find in a grocery store for instance. After the interviewee presented her/his product, we showed them nine images of hybrid products we had previously created (figure 1). The products on these images were chosen to represent a wide variety of dairy products and plant ingredients. In some of the products, the plant ingredient was selected so that it could go unnoticed for some participants. In other products, on the contrary, it was chosen to provoke a reaction. For each image, respondents were asked to freely

express their feelings about it.

After being done with the nine product images, the moderator asked the participants whether he/she had additional comments and concluded.

figure 1: hybrid product images presented to the participants during the interviews

Procedure

The interviews were held in April and May 2020 using a secured videoconference system.

Their mean duration was about 30 minutes (minimum: 20 minutes; maximum 45 minutes). Due to COVID-19 lockdown in France, the entire protocol was adapted to allow interaction between participant and moderator, at distance. All respondents took part to the study from home.

Previously to a session, participants were asked to read an online document explaining the conditions of the study, that they had to sign for informed consent. They were sent by e-mail the date of the discussion and a link to connect to the videoconference.

I was the moderator of all the interviews.

At the beginning of the videoconference, I introduced myself and the purpose of the discussion, in order to 'break the ice' and let the interviewee feel comfortable (still no mention of the topic of the interview). Then, I reminded the participant that the interview would be audio taped from this moment. Then the interview was held, following the guide previously created. At the end of the session, the respondent was informed that the recording was stopped. Therefore, I explained the general purpose of the interview and asked the respondent's address to send her/him a €10 voucher once the lockdown ended. The participants were finally thanked and invited to leave the videoconference.

Interviews transcription and analysis

Audiotapes were used to transcribe accurately the speech of both respondent and moderator.

After reading all of the interview transcripts, two people analyzed the verbatim transcripts separately on their own: G. Arvisenet and myself, We draw the main themes and sub-themes that arose from the transcripts, then we compiled the separate analyses to keep the common ideas and verbatims. Finally, for all these subthemes we picked-up some sentences we found the most representative and relevant of the ideas highlighted during the interviews.

Results

The analysis of the semi-directive interviews conducted with all the participants allowed us to draw and organize three main axes: how dairy products are part of the daily life for participants, how they consider dairy products substitution in the daily food, and finally, how hybrid (dairy-plant based) products would meet their expectations. The ideas and opinions of the participants on these three points were either shared by many participants or some of them. These ideas and opinions will be presented and supported by some excerpts from the interviews. We chose to let in French the verbatim extracted from the interviews transcripts, in order not to lose any part of their meaning.

The sentences of participants will be in quotes, the parts in brackets are the ones pronounced by the interviewer.

Dairy products as part of the daily life

The participants to our study usually consumed dairy products: it was an inclusion criteria.

Many of them considered dairy products to play an important role in their diet.

Overall appreciation of dairy products

First of all, the participants globally appreciated dairy products' taste. When we asked them why they consume dairy products, the taste appeared as an obvious reason: « parce que bah… il y a le goût ». Dairy products are mostly described as fresh: « c'est une source de fraîcheur » and they are appreciated for their texture: « (le lait) donne une onctuosité, ça change la texture, ça fait un truc un peu plus fondant ». Dairy products are also popular for their great variety, as a Spanish respondent, having lived in France for a while, said: « en France il y a beaucoup de produits laitiers, donc on s'ennuie jamais ; on peut changer le type de lait, les parfums… ».

Dairy products as a part of the daily routine

The analysis of the transcripts indicates that for many of the participants, eating dairy products is necessary. They usually eat milk products: « parce que les produits laitiers sont essentiels » ; because « il faut en consommer un petit peu ». « parce que les produits laitiers sont essentiels ».Nevertheless, the reasons behind this necessity to eat dairy products remain not very well defined and unclear. Yet, the speech of the respondents shows that using dairy products is rooted in their habits, and reveal some kind of automatism: « c'est une habitude : depuis tout petit on a mangé des produits laitiers » ; « on m'a élevée un peu dans cette idée ». It's also The analysis of the discussions with the participants shows that they seem to be aware of the health benefits and drawbacks of dairy product consumption. Nevertheless, the reasons behind their ideas, rather positives or negatives, remain unclear.

3.1.4. Dairy products production has a negative impact on the environment Several participants feel concerned by the impact of livestock production industry on climate change: « les produits laitiers c'est lié à la production animale et en terme de CO2, d'émissions de gaz à effet de serre, ça a un impact très, très important, notamment sur le réchauffement climatique » ; « (-manger des produits laitiers n'est pas bon…) -pour la planète: je sais que produire un litre de lait ça consomme énormément d'eau ».

This first part allowed us to observe that dairy products are well known and usually consumed among the respondents. They are appreciated for their taste and health-related properties. But they can also be considered bad for health and the environment.

Dairy products substitution in the daily food

The second part of our study was meant to understand the attitudes of the participants towards the substitution of dairy products in their diet.

What kind of food could be used to remplace milk products

The semi-directive interviews shows that most of the participants know non-dairy products substitutes currently existing, such as plant-based "milk" for instance. Some of them are used to consume those kinds of products: « j'ai l'habitude d'utiliser du lait végétal aussi ». But a majority, which know non-dairy products, and have already used or tasted them, rarely use them to cook or to eat: "éventuellement, si vraiment j'étais obligé, je me tournerais vers ce genre de produits » ; « Je vois pas spécialement l'intérêt de se tourner vers ça pour l'instant, si ce n'est par curiosité ».

Even if every interviewee knew non-dairy substitutes when it came out to how they would replace dairy products in their food, a few of them first said they would use some other products that do not belong to non-dairy products: « donc je pense que je mettrais de l'huile à la place du beurre ».

Non-dairy substitutes properties in favor of the substitution

Some of the respondents are used to have dairy products in their diet although they usually consume some non-dairy alternatives. First for the taste of those alternatives, in comparison with milk products: « je préfère le goût ». For some participants, the taste is not a barrier for the use of such kind of products, instead of dairy products when cooking: « je l'ai déjà fait et ça change pas réellement au final la recette ». These participants have adopted such kind of non-dairy products as new food, for new recipes: « par choix en fait, de varier au maximum les produits de base dans l'alimentation […] aussi pour les côtés créatif de la cuisine ».

The speech of respondents shows that they see some health-related benefits to consume non-dairy alternatives compared to dairy products: « le lait de soja […] pour moi c'est un petit peu plus digeste ». They also see an interest of those products for milk intolerant people: « je trouve que ça se justifie pleinement pour les personnes qui ont toutes ces intolérances aux produits laitiers ».

Animal welfare and environmental issues appear as another reason to use more non-dairy alternatives in the daily diet: « petit à petit la volonté de diversifier l'alimentation et de manger moins carné, moins animal, plus végétal » ; « c'est bien de trouver des alternatives […] pour des raisons d'environnement ».

Despite all these positive opinions about non-dairy alternatives, for many of the respondents, the substitution of dairy products stays unfamiliar and sometimes challenging.

A challenging dairy products substitution

For several respondents, non-dairy substitutes are easy and common to use but for the major part of them, their use seems to be a constraint.

Non-dairy products properties limit their adoption

The taste of non-dairy products is the major brake to their use, especially because respondents are used to the taste of dairy products they know and use for a long time: « c'est plus le goût, parce que je pense que je suis pas trop habituée donc du coup je suis plus exigeante ». It also depends on the plant-based product source: « j'ai testé […] de remplacer le lait par le lait d'amande, j'ai encore un peu du mal » ; « j'ai déjà goûté le lait de soja et … pas très très fan ». It seems that non-dairy products are unusual for them and do not arise from a demand or a need: « c'est pas un type de nourriture qui m'attire hein, si je peux m'en passer, je m'en passe ».

Accessibility of non-dairy substitutes came up as an obstacle for several respondents, first for their availability in grocery stores: « c'est pas si évident que ça d'en trouver » ; « ça peut être aussi un frein, l'offre qui est pas forcément suffisante », but also for their affordability. Non-dairy substitutes are deemed to be way more expensive than their dairy equivalent: « c'est plus cher que les produits normaux, du coup c'est sûr que je regarde moins, c'est plus une question de prix ».

It emerges from the discussions that non-dairy alternatives are not necessarily better considered than dairy products: « le lait de riz, même si c'est fabriqué en France aujourd'hui, ça pousse pas en France ». There is a particular distrust in soy, both for its production conditions « Le soja est importé de je n'sais pas d'où au Brésil et c'est super mauvais pour la planète » ; « c'est pas forcément respectueux de l'environnement, je crois la culture du soja » , and its health-related properties: « le soja a pas une très bonne réputation au final, apparement ça serait cancérigène ».

Practical aspects making dairy products substitution harder

The first reason retrieved from the discussion transcripts explaining the challenges met to substitute dairy products is the lack of knowledge in the culinary practices of non-dairy products: « il faudrait que je me mette à cuisiner beaucoup plus de substituts végétaux qui existent mais dont je ne maîtrise pas les recettes ». In a few case, non-dairy substitutes are barely known: « je peux pas te donner des produits spécifiques parce que honnêtement j'en ai aucune idée, mais je sais qu'il en existe […] mais après quoi exactement j'en ai aucune idée ».

Habits are firmly rooted in the daily life, and that is another reason which may explain why respondents find it hard to replace dairy products: « la volonté de diversifier l'alimentation et de manger […] plus végétal, euh tout en maintenant je dirais, tout en restant dans nos dans nos habitudes alimentaires, de ce qu'on est habitué un petit peu à faire, à cuisiner ». Traditions seem to refrain the participants to use non-dairy products: « qu'est-ce qui me gène ou qu'est-ce qui me freine? Bah c'est les traditions », especially for a specific kind of product very popular for French people. The best example of this is cheese. For some respondents, cheese is impossible to replace: « pour tout ce qui est fromage et tout ça, bah je pense qu'à part arrêter il y a pas trop d'autres solutions ». Its unique taste is the most frequent explanation « pour le fromage c'est plus dur de remplacer parce que j'aime ça », but also its convenience to cook: « je ne vois pas comment j'arriverais à faire gratiner sans fromage ».

Something really interesting that came out is the determinant role of information to help people to better know dairy products substitutes and reduce dairy products in their diet. Information provided on internet seem to be privileged for our participants, no matter their age and the media they use: « des articles dans la presse ou alors je vais chercher sur internet aussi » said a 58-year-old women ; « j'ai regardé sur internet des avis de plusieurs personnes qui disaient que le lait d'amande s'était quand même mieux pour la santé », « ça m'est venu par des avis sur internet, sur instagram, sur des comptes 'healthy', […] par le biais des réseaux sociaux » claimed a young student that began to take an interest in non-dairy substitutes.

The questions that meant to understand the influence of the social norm on the participants' attitudes towards plant-based products did not allow us to extract some relevant information to conclude about it.

We can conclude from this discussion that non-dairy substitutes are pretty well known but only used by a part of the participants. Participants had quite contradictory opinions about those products.

Would a hybrid product meet expectations?

Usual associations of dairy and plant ingredients or products

We asked our interviewees what kind of food they usually combine with dairy products.

Many of the respondents needed some time to think about it. Unsurprisingly, the association they most usually do is cheese and bread: « le fromage avec une tranche de pain parce que je crois que ça se complémente bien ». Associations of dairy products with fruits and vegetables are also frequent, and the use of dairy products in recipes was cited quite often: « je mets du lait dans le gratin » ; « je le fais dans tout ce qui est […] gâteaux, crêpes, gaufres… ». The reasons behind those associations are commonly the habits and cooking rules: « c'est des recettes de famille souvent et du coup c'est juste parce que j'ai l'habitude de les manger comme ça et que c'est pas facile de changer » ; « ça peut être aussi la base de, comment dire, de l'élaboration d'un plat ».

Innovating hybrid products

This third ax of analysis deals with the concepts of new hybrid products associating milk and non-dairy we introduced the participants during the discussion.

Two main parts of our interview guide were meant to know how the participants may consider mixing milk and plant-based ingredients in the same product: the first was by asking them to imagine a product associating dairy and non-dairy ingredients, the second way was to show them pictures of fictive hybrid products.

3.3.2.1. Hybrid products might be transition products, with ingredients offsetting one another Some participants observed an interest in mixing both kinds of products. Hybrid products of such a kind were seen as a transition product helping them to consume more plant-based products but still have the products with the taste they like and they are used to: « une personne par exemple qui a du mal à, comme moi, à se décider entre du lait d'amande ou du lait de vache » ; « je trouve que ce sont des trucs qui sont bien quand tu veux tendre vers un truc végétal et tout mais que t'as encore besoin de ta part de lait parce que tu peux pas transitionner ».

For some participants, such a kind of products may have mutual and complementary properties:

« peut-être que pour des gens qui aiment bien le goût de la crème et qui souhaiteraient en diminuer la consommation là c'est peut-être intéressant » ; « c'était peut être intéressant d'avoir le côté gras et onctueux du yaourt […] et un peu plus de légèreté et de goût avec l'amende, ça peut être un mix un peu complémentaire ».

Health-related benefits have also been underlined, where both plant and dairy ingredients would share their benefits and hide their drawbacks when mixed: « je sais tous les bénéfices qu'il y a dans le lait de soja et dans le lait de vache […] donc bah si les deux sont ensemble c'est un peu comme dans une pub à la télé […] genre ils ont trouvé un truc nouveau pour qu'elle le digère mieux » ; « il y a quand même beaucoup de personnes intolérantes et par exemple on peut répondre vite à des personnes qui veulent changer de régime, tu vois, en se rapprochant toujours d'un goût qu'ils connaissent ».

Hybrid products may have no interest for some of the participants

During the interview, we asked the participants to imagine their ideal hybrid product. Only a few of them succeeded in proposing some concepts of hybrid products: « ce serait une boisson lactée, style comme du Yop, avec du lait d'amande, et du lait de, ben, de vache » ; « je pense à des produits genre des produits de base pour cuisiner et en fait je pense que si il y avait une gamme de crème mixte qui est associent à la fois lait animal lait végétal ».

For most of the interviewees, imagine a hybrid product was quite difficult, they took some time to think about it but they could not complete this task. Moreover, most of the respondents that proposed a concept of product, actually proposed something already existing, or plant-based products: « j'ai pensé à une vache qui rit ou un kiri, mais cette fois ci complètement végétal ». The fact that they did not succeed imagining this product suggests that they see no interest in it. Indeed, some interviewees seemed to be questioning themselves but also the moderator about the possible interest of such a kind of product. Many of them were not perceiving any interest and expressed some reluctance towards those associations they considered unnecessary or useless: « j'ai du mal à saisir l'intérêt du mixte » ; « je ne trouve pas le mélange utile » ; « je me pose la question:

pourquoi ? (rire) je me demande pourquoi faire ». Some of the participants even considered the idea of an association as something impossible or even contradictory: « moi la première réaction ça serait de me dire: en fait c'est bizarre quoi, c'est pas logique » ; « c'est complètement incompatible dans ma tête, pour moi c'est comme une contradiction en fait ». For a part of the respondents, you may only have distinct products : a dairy one or its non-dairy alternative: « je comprends pas trop l'intérêt de faire un mixte des deux, genre vraiment j'ai du mal, je me dis … je sais pas, soit l'un, soit l'autre » ; « je vois pas forcément l'intérêt de mettre un mix donc soit on met du lait, soit on met du soja ».

Thanks to the use of pictures of hybrid products, we could see that this contradiction was perceived more important for certain kinds of products, well anchored in French traditions and habits, such as

Discussion

The different results described above allowed us to have some response elements to our hypothesis and objectives. The following section provides an argued interpretation of the results obtained. Our study allowed us to draw the attitudes and representations of participants towards innovative hybrid products in order to understand the possible brake to their consumption.

Our first hypothesis dealing with the compensation of the advantages and drawbacks of dairy products and plant-based products with one another within hybrid products doesn't seem to be confirmed. Indeed, one of the ingredients often guided the reactions towards the hybrid product.

The acceptance of the products depends on the characteristics of its individual components (dairy and plant ingredients). First, if one of the ingredients is not well known or considered bad, this ingredient will take the lead compared to the other one in the overall appreciation of the product.

When the plant used was unknown or not well known by the respondent, the declared willingness to try was lower. For instance, the hybrid cream packaging only mentioned 'ingrédients végétaux', and this product got comments such as: « ça fait un peu ingrédient inconnu euh tu sais pas trop ce qu'il y a dedans ». Another product containing lupine, an ingredient not familiar for participants, who considered it is not something we can eat: « lupin pour moi c'est une fleur donc euh je mange pas forcément ». Soy, that had a bad reputation, also induced quite negative attitudes towards the hybrid products it composed, no matter if the "equivalent" milk product (same form, same use) had good or bad taste related properties.

The congruency of the ingredients used in each product also appears as a key determinant in the attitudes towards the products. For example, the chocolate ice cream with coconut milk had a lot of positive reactions: « J'adore le lait de coco, j'adore le chocolat ». As well as the 'camemvert' product: « pourquoi pas gouter parce que j'aime bien […] quand il y a par exemple du fromage avec des noix dedans ». This positive attitudes towards those products may be explained by the perceived congruency of the ingredients used: chocolat and coconut are already known and appreciated associations in recipes, as well as nuts in cheese.

On the contrary, other products had quite bad appreciations because of the perceived uncongruency of the ingredients used, such as peas in 'vache qui rit' product: « Avec du lait et des pois ! Ça veut dire que la vache qui rit elle serait verte ? ».

The price of such hybrid products was a concern for the participants: « je pourrais aussi aisément essayer sous réserve, comme d'habitude avec ce type de produits, que le tarif ne soit pas euh totalement excessif ». Non-dairy products had the reputation among participants to be more expensive than dairy ones, and participants inferred this property to hybrid products. It resulted in high concerns about the price hybrids would have.

Our second hypothesis relative to the level of transformation of the final product on the consumers' attitudes and representations towards it seems to be confirmed. Indeed, reputed highly processed products such as 'vache qui rit' had quite negative consumers' attitudes, contrary to some basic food ingredients such as the product 'cream with plant ingredients' because the cream was not associated with being highly processed.

Hybrid 'margarine' and 'vache qui rit' provoked quite strong rejection: « la margarine j'ai une image hyper négative de ce produit j'ai l'impression que c'est bourré de chimique » ; «(la vache qui rit) un produit historique mais pour moi totalement inintéressant ».

We also noticed that generally, the sweet products received quite more positive comments than the savory ones, as a respondent expressed after seeing all the pictures: «j'ai l'impression à voir vos images que je vais plus me diriger vers […] des en-cas ou des choses sucrées ».

Limitations of this study

Due to the COVID-19 global crisis and lockdown in France, we had to use semi-directive interviews via videoconference for this study. This technique lends itself well to the objectives of this study. However, some parts of our guide interview would have been more relevant to apply into a focus group session. For instance, the ideal hybrid product design would have probably raised more reactions and ideas because participants would have interacted all together.

The questions about social norms did not provoke exploitable reactions from participants, who only mentioned they would/would not cook with plant-based products for guests, and generally that they would taste plant-based products if they were invited. But they did rarely comment this. Once again, focus groups would probably have given more conclusive results.

It would have been better to have more participants in this study; we would have expected five more persons in each group. It would have also been better to have more gender-balanced groups.

Conclusion

This study showed that participants did not reject hybrid innovative products. However, the approach of each product was different as function of the congruency of the ingredients used, the reputations of the ingredients composing the products, the possible use of each product, and of course the individual appreciation of dairy products and non-dairy substitutes.

The need of further information came out of the interviews, information about the plant source as well as the benefits of such an association would be necessary to encourage the adoption of such products, as a respondent said: « si ça devient plus connu et que je peux avoir je sais pas une info comme quoi ça pourrait avoir […] un bénéfice santé et qu'on le trouve pas dans d'autres choses assez facilement ».

As far as we do know, this study is the first of her kind about investigating consumers' attitudes and representations towards food products before to formulate and create them.

This part of the study allowed us to bring out several properties about dairy, plant-based and hybrid products we will include in the second step of our study: the quantitative one. This second part will use the results of the present study to build a questionnaire that will be submitted online to hundreds of participants. For this, we will use the same images we showed during the semi-directive interviews. This quantitative study will allow us to obtain more precise elements about interindividual differences in representation towards hybrid products. Si quelque chose n'est pas clair, vous pouvez me demander de répéter ou de préciser.

Avant qu'on commence, vous avez des questions ?

Alors pour commencer notre discussion, je voudrais que l'on discute ensemble de l'alimentation en général et comme point de départ je vous propose de se baser sur les repas que vous avez pris dans la journée d'hier.

On va commencer par le petit-déjeuner, qu'avez-vous l'habitude de consommer au cours de ce repas ? Au cours de votre dernier déjeuner ou dîner ?

Relance: quels ingrédients constituent/avez-vous utilisé dans ce plat ?

  ......................................................................................................................... 1.2. STLO's project 1 ............................................................................................................................. 1.3. Objectives and hypothesis of my internship 2 ................................................................................ 2. Material and methods 4 ........................................................................................................................ 2.1. Recruitment of the participants to interviews 4 .............................................................................. 2.2. Participants 5 ................................................................................................................................... 2.3. The interview guide 5 ...................................................................................................................... 2.4. Procedure 7 ..................................................................................................................................... 2.5. Interviews transcription and analysis 8 ........................................................................................... 3. Results 9 ................................................................................................................................................. 3.1. Dairy products as part of the daily life 9 ......................................................................................... 3.1.1. Overall appreciation of dairy products 9 3.1.2. Dairy products as a part of the daily routine 9 3.1.3. Dairy products and health-related properties 3.1.4. Dairy products production has a negative impact on the environment 3.2. Dairy products substitution in the daily food ............................................................................... 3.2.1. What kind of food could be used to remplace milk products 3.2.2. Non-dairy substitutes properties in favor of the substitution 3.2.3. A challenging dairy products substitution 3.3. Would a hybrid product meet expectations? ................................................................................ 3.3.1. Usual associations of dairy and plant ingredients or products 3.3.2. Innovating hybrid products 4. Discussion ............................................................................................................................................ Limitations of this study ..................................................................................................................... 5. Conclusion ........................................................................................................................................

  related to the education received from the parents: « c'est plus par rapport à l'éducation, [...] mes parents ils m'ont toujours dit 'ouais, tu prends un laitage'[…]. Même à la cantine et tout[...] tu dois avoir un laitage ». Dairy products consumption is also associated with structuring the meals and being as a full part of it: « c'est bon parce que ça diversifie mon alimentation » ; « parce que ça complète bien le repas […] pour fermer un repas » ; « pour finir mon repas avec une petite touche sucrée ».3.1.3. Dairy products and health-related propertiesSome of the participants considered dairy products as good for health and mentioned that their consumption is needed to be in good health: « je consomme des produits laitiers pour euh, la santé […] parce que il faut des produits laitiers pour le corps ». Eating those products is perceived as a crucial part in the healthy growth of children: « pour la croissance, pour le bon développement », as well as a key part for maintaining a good bones health: « les produits laitiers sont bons pour mes os, je sais qu'il y a du calcium et que, genre, ça va fortifier mes os ».However, some other participants surveyed also raised some negative aspects of dairy products consumption relative to health. The major concern is about their digestibility: dairy products are deemed to be hard to digest and to cause abdominal pain during their digestion: « c'est pas forcément bon pour l'organisme, ça peut créer des espèces de douleurs intestinales […] ça peut être mal digéré aussi ». Likewise, dairy products are considered to be rich in fat, to be responsible for weight gain: « ça fait grossir, c'est un peu de la graisse […] c'est un peu lourd », and for provoke a raise of harmful cholesterol in the body: « à forte dose, […] on peut avoir euh... du cholestérol, je crois». Hence, a frequent question raised by the participants is that adults do not need as many dairy products as children do: « les produits laitiers c'est bien pour les enfants mais à un certain âge il fallait quand même diminuer parce qu'on en avait plus autant besoin ». The consequence is that dairy products daily intake must remain low or at least in limited quantities: « je sais qu'il faut pas trop en prendre […] c'est comme tout, il faut pas en abuser » ; « ils sont mauvais si on en prend effectivement à forte dose ».

  vous souhaite la bienvenue et je vous remercie de votre participation, votre rôle est très important et je vous remercie de la confiance que vous avez décidé de m'accorder. Je me présente brièvement, je suis Valentin, je suis étudiant en deuxième année de master et je suis actuellement en stage dans le but de réaliser mon projet de fin d'étude. Notre objectif d'aujourd'hui sera de discuter ensemble sur des questions précises pour recueillir vos réactions, vos expériences personnelles et votre opinion sur différents sujets liés à l'alimentation. Vous êtes libre de partager ce que vous souhaitez. Tout ce que vous avez à dire est intéressant, il n'y a ni bonne ni mauvaise réponse. Il est très important que vous vous sentiez en pleine liberté de parole. Cette discussion va durer entre 3à et 45 minutes maximum et sera enregistrée afin d'être retranscrite fidèlement à vos propos pour être analysée, bien entendu tout cela restera confidentiel, votre anonymat sera préservé et aucune de vos données personnelles ne seront collectées et les bandes audios seront détruites après analyse.

  Alors ce que j'ai relevé dans tout ce que vous venez de me dire, c'est que vous citez différents types d'aliments. Par exemple, *exemple autre que produits laitiers*, mais aussi *exemple autre que produits laitiers*… -Si des produits laitiers sont cités : ainsi que les produits laitiers qui sont présent dans votre alimentation quotidienne, sous différentes formes comme *citer les produits laitiers cités par les participants*. -Sinon : Je vois par contre qu'il y a des types d'aliments que vous ne citez relativement pas ou peu qui sont *exemple* ou encore les produits laitiers Partie 2: les produits laitiers Je vous propose de commencer notre discussion par parler de ces produits laitiers et des raisons pour lesquelles vous les consommez (peu) Je vous donner une phrase et je vous laisserai la compléter, voici la phrase : « Je consomme des produits laitiers pour ou parce que… » *Participant complète la phrase* Relance : est-ce que vous avez d'autres raisons qui vous viennent à l'esprit ? Est-ce que vous avez essayé dernièrement d'augmenter votre consommation de produits laitier ? -Si oui : pour quelles raisons ? -Si non ou pas spécialement : Alors au contraire, est-ce que vous avez tenté de diminuer? Pour quelles raisons le faites-vous ? Je vais vous donner une nouvelle phrase que je vous laisserais continuer, voici la phrase: « Les produits laitiers sont bons pour… « *Participant complète la phrase* Pourquoi considérez-vous que c'est bon pour… ? Maintenant à l'inverse, « les produits laitiers sont mauvais pour… » *Participant complète la phrase* Pourquoi considérez-vous que c'est mauvais pour… ? Relances : Quels autres avantages voyez-vous à leur utilisation et à leur consommation ? Au contraire quels autres inconvénients ? Avez-vous d'autres idées ?

  the figure of cheese: « c'est une valeur de la France profonde qui tombe, […] dans l'inconscient collectif le camembert c'est quand même la France » ; « de la vache qui rit, avec du lait et des pois, ah ouais non mais là moi j'aime pas […] là c'est pas possible de changer la recette, là on est sur le fromage donc il y a des choses sacrées on peut pas toucher ». One the one hand, adding some plant milk to a traditionally animal-ingredient based product seems conflicting for some respondents, and on the other hand, adding milk to a product traditionally considered as belonging to vegetarian diet provoques the same reactions in other respondents: « je me demande ce que le lait vient faire là dedans, mais c'est marrant parce que dans l'autre sens ça me paraissait moins bizarre, de faire du lait avec du soja plutôt que là rajouter du lait dans du tofu » ; « c'est peut-être l'aliment emblématique des végétariens ou des personnes qui ont des difficultés à consommer des produits laitiers et je trouve que là pour le coup c'est pas très sympa de dénaturer le tofu ».

Acknowledgment

I would like to thank everyone that helped me in achieving this project.

Firstly, I would like to express my deepest thanks and appreciation to my internship supervisor Gaelle Arvisenet. Her constant guidance and advice played a vital role in making the execution of this project. She always gave me her advice that was crucial in making this report as flawless as possible. Many thanks for her trust and kindness with me.

I express my deepest thanks to my co-supervisors, Valérie Gagnaire and Fanny Guyomarc'h, for taking part in useful decisions & giving necessary advice and guidance during my internship. I choose this moment to acknowledge their contribution gratefully. Many thanks for their support, kindness, and trust.

My deepest thanks to everyone in STLO lab that helped me with the interview transcriptions.

Many thanks to everyone in CSGA, team 9 for their help and kindness.

Appendix

Je me sens concerné.e par l'impact de mon alimentation sur l'environnement * Avez-vous déjà tenté de réduire l'impact de votre alimentation sur l'environnement ? Si oui, que faitesvous ?

Votre réponse

Je me sens concerné.e par l'impact de mon alimentation sur ma santé * Avez-vous déjà tenté de réduire l'impact de votre alimentation sur votre santé ? Si oui, que faites-vous ?


Florentina Nicolau
email: florentina.nicolau@ensea.fr

Witold Respondek
email: witold.respondek@insa-rouen.fr

Jean-Pierre Barbot
email: barbot@ensea.fr

Construction of flat inputs for mechanical systems

Keywords: Mechanical systems, flatness, flat inputs, mechanical flat inputs

Introduction

In this paper, we consider observed mechanical systems, with n degrees of freedom, of the form

M:    ẋi =v i vi =-Γ i jk (x)v j v k + d i j (x)v j + e i (x), 1 ≤ i ≤ n, y = h(x), y ∈ R m , (1)
where (x, v) = (x 1 , . . . , x n , v 1 , . . . , v n ) are local coordinates on the tangent bundle T X of the configuration manifold X and y = (h 1 (x), . . . , h m (x)) are (everywhere independent) outputs of the system. We use the Einstein summation convention, i.e., any expression containing a repeated index (in general, upper and lower) implies summation over that index up to n (whenever the summation is taken over another indexing set, we use the summation symbol). The expressions Γ i jk (x)v j v k correspond to Coriolis and centrifugal terms. The terms d i j (x)v j correspond to forces linear with respect to velocities, like dissipative forces, and e(x) represents an uncontrolled force (that can be potential or not). It is typically possible to control a mechanical system through external forces and mechanical control systems form an important class of control systems that has attracted a lot of attention because of their numerous applications. They form a natural bridge between mechanics and control theory and are studied, for instance, in [START_REF] Ortega | Passivity-based control of Euler-Lagrange systems[END_REF][START_REF] Bloch | Nonholonomic mechanics and control[END_REF][START_REF] Bullo | Geometric control of mechanical systems[END_REF][START_REF] Ricardo | When is a control system mechanical?[END_REF]. An important question is how and where controlled external forces should act in order to achieve a desired property for the resulting input-state-output system. A property that is very useful in applications (for instance, for trajectory tracking, constructive controllability or trajectory generation) is that of flatness (see, e.g., [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF][START_REF] Fliess | A Lie-Bäcklund approach equivalence and flatness of nonlinear systems[END_REF][START_REF] Lévine | Analysis and Control of Nonlinear Systems: A Flatness-Based Approach[END_REF] and references therein). The problem of placing the actuators or the inputs in order to render an observed dynamical system flat has been introduced by [START_REF] Waldherr | Conditions for the existence of a flat input[END_REF][START_REF] Waldherr | Flat inputs in the mimo case[END_REF] who call those inputs flat inputs.

Flat inputs are objects dual to flat outputs and their construction can been seen as a dual problem to that of constructing a flat output. One of the motivations to construct a flat input for a given output is that with such an input, the tracking problem for that output can be solved with no need to calculate the zero dynamics [START_REF] Graichen | A new approach to inversion-based feedforward control design for nonlinear systems[END_REF][START_REF] Stumper | Trajectory tracking control with flat inputs and a dynamic compensator[END_REF], but constructing flat inputs may be useful for other problems as well, like parameter identification [START_REF] Schenkendorf | Parameter identification for ordinary and delay differential equations by using flat inputs[END_REF] or private communication [START_REF] Nicolau | Secure communication with the help of flat inputs for chaotic systems[END_REF]. Similarly to the construction of a flat output (that can be seen as a problem of sensors placement in order to achieve flatness of the resulting input-state-output system) for which, from a technological point of view, it is not always possible to place the sensors exactly where we want them, for constructing flat inputs it may also be technologically difficult to place the actuators at the right place. This is often the case for mechanical systems for which the states variables are positions and velocities (denoted, resp., by x i and v i ), and thus the derivative of a position always equals the corresponding velocity, i.e., we always have ẋi = v i and we cannot modify those equations by adding inputs. Therefore from a physical point of view, it is impossible to build an actuator that acts directly on the position.

The problem that we are studying in this paper is the construction of flat inputs that are consistent with the mechanical structure of the original dynamical system. More precisely, given the observed mechanical system (M, h), we want to find mechanical control vector fields g 1 , . . . , g m (or equivalently, to place the actuators or the inputs) such that the mechanical control-affine system

M c : ẋi =v i vi =-Γ i jk (x)v j v k + d i j (x)v j + e i (x) + m r=1 g i r (x)u r (2)
associated to M, is flat with the original measurements (h 1 , . . . , h m ) being a flat output. By mechanical control vector fields g r we mean that they are consistent with the mechanical structure of M, i.e., are of the form g r = n i=1 g i r (x) ∂ ∂v i , 1 ≤ r ≤ m. The inputs u r multiplying the mechanical control vector fields will be called mechanical flat inputs, and the vector fields g 1 , . . . , g m will be called mechanical flat-input control vector fields.

The problem of constructing flat inputs consistent with the mechanical structure has been first considered in [START_REF] Waldherr | Flat inputs in the mimo case[END_REF] where they are called physically realizable flat inputs. In [START_REF] Waldherr | Flat inputs in the mimo case[END_REF], it is shown that a necessary condition for the flat inputs to be physically realizable (according to our definition, to be mechanical flat inputs) is that the distribution spanned by their associated flat-input control vector fields is contained in the vertical distribution of the mechanical system. A related question has then been studied in [START_REF] Fritzsche | On a generalized flat input definition and physical realizability[END_REF], where the output expression is redesigned, by allowing affine injections of the input and its derivatives, in order to compute physically realizable flat inputs for observable implicit nonlinear systems. In this paper, we work with outputs depending on positions only and we do not modify them to achieve flatness for the resulting mechanical control system. We notice first that if n (independent) functions h i (x), 1 ≤ i ≤ m = n, are observed (as many as the degrees of freedom of the system), then we can always construct mechanical flatinput control vector fields, but if m < n, then even in the simplest possible case (n, m) = (2, 1), structural conditions have to be satisfied for the existence of mechanical flat-input control vector fields. Therefore, the goal of the paper is to give necessary and sufficient conditions for the existence of mechanical flat-input control vector fields for observed mechanical systems with n degrees of freedom, with n arbitrary, and n -1 measurements (i.e., m = n -1). The paper is organized as follows. In Section 2, we recall the definition of flatness, formalize that of flat inputs and mechanical flat inputs, and present the first results of the paper. In Section 3, we give our main results and illustrate them by an example in Section 4.

Definitions and motivations

Consider the control system Ξ : ξ = f (ξ) + m r=1 g r (ξ)u r , where ξ ∈ R N , u ∈ R m and g 1 , . . . , g m are independent. Definition 2.1. Ξ is flat at (ξ 0 , u 0 ) ∈ R N × R m if there exist a neighborhood O of (ξ 0 , u 0 ) and m smooth functions ϕ r = ϕ r (ξ), 1 ≤ r ≤ m, defined in a neighborhood of ξ 0 , with the following property: there exist an integer s ≥ 1 and smooth maps (γ, δ) : O×R m(s-1) → R N ×R m such that ξ = γ(ϕ, φ, . . . , ϕ (s-1) ) and u = δ(ϕ, φ, . . . , ϕ (s) )

(3) for any C s-1 -control u(t) and corresponding trajectory ξ(t) that satisfy (ξ(t), u(t)) ∈ O, where the m-tuple ϕ = (ϕ 1 , . . . , ϕ m ) is called a flat output.

There exists a more general notion of flatness for which the functions ϕ r may depend on the control and its successive time-derivatives up to a certain order q, i.e., ϕ r = ϕ r (ξ, u, u, . . . , u (q) ). We do not need this general notion since, in our study, all ϕ r depend on the state ξ only and singularities depend on ξ and/or u (but never on derivatives of u). We send the reader to [START_REF] Fritzsche | On a generalized flat input definition and physical realizability[END_REF] for a generalized definition of flat inputs that allows dependence on the input and its derivatives.

Consider the dynamical system

Σ : ξ = f (ξ), y = h(ξ),
whose state ξ ∈ R N (or more generally, ξ belongs on an N -dimensional manifold), together with the output y = h(ξ) ∈ R m . In order to emphasize the fact that the system is observed, we will use the notation (Σ, h). When we say that the dynamical system Σ is observed, this does not mean that Σ is necessarily observable with respect to the output h. The problem of constructing flat inputs consists of finding independent control vector fields g 1 , . . . , g m such that the control-affine system Σ c , associated to Σ, and given by

Σ c : ξ = f (ξ) + m r=1 g r (ξ)u r ,
is flat with respect to the original output (h 1 , . . . , h m ) . In that case, we will say that the pair (Σ c , h) is flat. The inputs u 1 , . . . , u m multiplying, resp., g 1 , . . . , g m , are called in [START_REF] Waldherr | Conditions for the existence of a flat input[END_REF][START_REF] Waldherr | Flat inputs in the mimo case[END_REF] flat inputs. The vector fields g 1 , . . . , g m that render Σ flat will be called flat-input control vector fields. For the single-output case, according to [START_REF] Waldherr | Conditions for the existence of a flat input[END_REF], a flat input can be constructed if and only if the system Σ together with its output h is observable 1 (see [START_REF] Hermann | Nonlinear controllability and observability[END_REF][START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF] for different notions of observability). The observable multi-output case has been discussed in [START_REF] Waldherr | Flat inputs in the mimo case[END_REF], see also [START_REF] Fritzsche | On the computation of differentially flat inputs[END_REF] for another approach based on the notion of unimodularity. In [START_REF] Nicolau | Constructing flat inputs for two-output systems[END_REF][START_REF] Nicolau | Flat inputs: Theory and applications[END_REF] (see also [START_REF] Fritzsche | Nonlinear control of non-observable non-flat mimo state space systems using flat inputs[END_REF]), the authors solved the unobservable case for which locally, around any point of an open and dense subset of the state space, we constructed flat-input control vector fields g 1 , . . . , g m such that the controlaffine system Σ c is flat with h being a flat output.

When comparing the problem of existence of flat inputs with that of verifying flatness for control systems, an interesting phenomenon can be noted: contrary to flat control systems that are very rare (the class of flat control systems is of codimension infinity among all control systems), any dynamical system (Σ, h) can be rendered flat (on an open and dense set) by adding suitable flat-input control vector fields (or equivalently, suitable flat inputs), see [START_REF] Nicolau | Flat inputs: Theory and applications[END_REF]. As explained in the introduction, similarly to the construction of a flat output and the corresponding sensor placement, from a technological point of view, for constructing flat inputs it is not always possible to place the actuators exactly where we want them. This is often the case for mechanical systems, where in general, the controls correspond to an action of forces or torques on the mechanical system that act on velocities only, enter the system in an affine way and multiply terms that depend on positions only. Therefore (and keeping in mind that a dynamical observed system (Σ, h) can always be rendered flat by adding suitable flat inputs), a natural question arises: for a given observed mechanical system (M, h), is it always possible to construct g 1 , . . . , g m that render (M, h) flat and, additionally, are consistent with the mechanical structure of the system, that is, are of the form g r = n i=1 g i r (x) ∂ ∂v i , 1 ≤ r ≤ m? Such g r 's will be called mechanical flatinput control vector fields and the inputs u 1 , . . . , u m multiplying them will be called mechanical flat inputs.

We start with the simplest situation: the answer to the above question is always positive for observed mechanical systems with n degrees of freedom, for which n (independent) functions

h i (x), 1 ≤ i ≤ n, are measured.
In what follows, we denote by f the drift of M, i.e., we have

M : ż = f (z), where z = (x, v). The value of a differential form ω = ω x i (x, v)dx i + ω v i (x, v)dv i on T X on a vector field η = η i x (x, v) ∂ ∂x i + η i v (x, v) ∂ ∂v i on T X is ω, η = ω x i η i x + ω v i η i v 2
. For a distribution G on T X denote by G ⊥ its annihilator, i.e., the codistribution given by G ⊥ = {ω ∈ Λ 1 (T X) : ω, g = 0, ∀g ∈ G}, where Λ 1 (T X) is the space of smooth differentials one-forms on T X.

Proposition 2.1. Consider the observed mechanical system (M, h) with m = n. Then (M, h) admits mechanical flat inputs around any (x 0 , v 0 ) ∈ T X and all choices of mechanical flat-input control vector fields g 1 , . . . , g n that render (M, h) flat are given by (i)

dh i , g j = 0 and dL f h i , g j = D ij (x), 1 ≤ i, j ≤ n, where (D ij (x)
) is any smooth invertible (n × n)-matrix depending on positions only or, equivalently, by (ii) vector fields

g i = g i (x), 1 ≤ i ≤ n, that generate the distribution G = (span{dh i , 1 ≤ i ≤ n}) ⊥ .
The left-hand side of dL f h i , g j = D ij (x) in (i) may a priori depend on (x, v) but the right-hand side implies, since D ij = D ij (x), that it depends on x only. Notice also the condition g i = g i (x), for 1 ≤ i ≤ n, in (ii); without that assumption g i 's need not be mechanical. According to Proposition 2.1, if all n positions (or independent functions of them) are measured, then similarly to the results of [START_REF] Waldherr | Flat inputs in the mimo case[END_REF][START_REF] Nicolau | Flat inputs: Theory and applications[END_REF] for general systems, the observed mechanical system (M, h) always admits mechanical flat inputs and we can always construct mechanical control vector fields without any structural conditions. The above theorem reminds very much Theorem 3.3 of [START_REF] Nicolau | Flat inputs: Theory and applications[END_REF] and Theorem 4 of [START_REF] Waldherr | Flat inputs in the mimo case[END_REF] treating the case when the considered dynamical system is observable with respect to h. That was to be expected since in the case when m = n, the mechanical system M is obviously observable with respect to h. It is however important to emphasize that the existence of mechanical flat-input control vector fields is not due to observability, but to the fact that we have as many measurements as degrees of freedom. This is illustrated by Proposition 2.2 below treating the case of mechanical systems with n = 2 and m = 1 (which is the simplest non trivial case). Before stating it, let us introduce some notations that will be used throughout.

Consider the observed mechanical system (M, h), given by (1), and suppose m = n -1, that is, the system has n degrees of freedom and n -1 independent functions h 1 (x), . . . , h n-1 (x) of the positions are measured. For 1 ≤ i ≤ n, we denote by a i the functions

a i (x, v) = -Γ i jk (x)v j v k + d i j (x)v j + e i (x), (4)
i.e., we have vi = a i (x, v), and recall that f denotes the drift of M. To the output (h 1 (x), . . . , h n-1 (x)) we associate the following sequence of codistributions

H j = span{dL q-1 f h i (x), 1 ≤ q ≤ j, 1 ≤ i ≤ n -1}, (5)
for j ≥ 1. So, we have

H 1 = span{dh i (x), 1 ≤ i ≤ n -1} and H 2 = span{dh i (x), dL f h i (x, v), 1 ≤ i ≤ n -1}
which are of constant rank n -1 and 2n -2, resp.. We denote by V the vertical distribution associated to M, that is,

V = span{ ∂ ∂v 1 , . . . , ∂ ∂v n }.
Since the functions h 1 (x), . . . , h n-1 (x) are supposed everywhere independent, we can always complete them by another function ζ(x) such that the extended point transformation Φ = (φ, Dφ(x)v), where

φ = (h 1 (x), . . . , h n-1 (x), ζ(x)) ,
defines a local (triangular) diffeomorphism that gives new coordinates

x = φ(x), ṽ = Dφ(x)v, (6)
in which M takes the form ẋi = ṽi , vi = ãi (x, ṽ), 1 ≤ i ≤ n, with ãi quadratic with respect to the new velocities ṽi , and which, dropping the "tildes", gives

M :    ẋi =v i vi =a i (x, v), 1 ≤ i ≤ n, with h i = x i , 1 ≤ i ≤ n -1. (7)
We call (x, v) of ( 7) to be M-coordinates compatible with h. Consider now the simplest non trivial case (n, m) = (2, 1).

Proposition 2.2. Consider the observed mechanical system (M, h), with n = 2 and m = 1. We have (FI) (M, h) admits a flat input at (x 0 , v 0 ) if and only if rk H 4 (x 0 , v 0 ) = 4, that is, M is locally observable with respect to h.

(MFI) The system (M, h) admits a mechanical flat input at (x 0 , v 0 ) if and only if

(C) (H 3 ) ⊥ ⊂ V and rk H 3 (x 0 , v 0 ) = 3;
or, equivalently, (C)' In any M-coordinates (x,v) compatible with h, the function a 1 satisfies:

∂a 1 ∂v 2 ≡ 0, ∂a 1 ∂x 2 (x 0 , v 0 ) = 0, (8)
and a 2 is any.

Moreover, any mechanical flat-input control vector field, for systems satisfying (C), is given by

L g L j f h = 0, for 0 ≤ j ≤ 2, and L g L 3 f h = γ(x),
where γ = γ(x) is any smooth function such that γ(x 0 ) = 0, and, in particular, by

g(x) = g 2 (x) ∂ ∂v 2 , g 2 (x 0 ) = 0,
for any system of the form (7) satisfying (C)'.

Under (H 3 ) ⊥ ⊂ V, the condition rk H 3 (x 0 , v 0 ) = 3 of (MFI) is actually equivalent to rk H 4 (x 0 , v 0 ) = 4, showing that the observability rank conditions for (FI) and (MFI) are the same. The structural condition ∂a 1 ∂v 2 ≡ 0 means

a 1 = Γ 1 11 (x)v 1 v 1 + d 1 1 (x)v 1 + e 1 (x)
. Condition (FI) simply says that in the single-output case, a flat input (there is only one since we have only one output) exists if and only if (M, h) is observable. So, if a system is observable, then it always admits a (not necessarily mechanical) flat input and for the flat input to be mechanical, the equivalent conditions (C) or (C)' of (MFI) have to be satisfied. Thus the above proposition shows that, even in the simplest case (mechanical systems with two degrees of freedom and one measurement), the problem of constructing mechanical flat-input control vector fields has non trivial solutions and that an additional structural condition is needed. Finally, observe that Proposition 2.2 completely describes the case (n, m) = (2, 1). The goal of the paper is thus to give necessary and sufficient conditions for the existence of mechanical flat-input control vector fields for observed mechanical systems with n degrees of freedom, where n ≥ 3 is arbitrary, and n -1 measurements (i.e., m = n -1).

Main results

Consider the observed mechanical system (M, h), given by ( 1), and suppose that n ≥ 3 and m = n -1. We denote by G n-1 the distribution

G n-1 = (H 2 ) ⊥ ∩ V. (9)
The distribution G n-1 is of constant rank one and we denote it by G n-1 because, as we will see below, it actually gives the only candidate for one of the mechanical flat-input control vector fields (that will be denoted by g n-1 ). Theorem 3.1 provides necessary and sufficient conditions for the (local) existence of mechanical flat-input control vector fields g 1 , . . . , g n-1 such that (M c , h) is locally flat.

Theorem 3.1. Consider the observed mechanical system (M, h), given by (1), together with (h 1 (x), . . . , h n-1 (x)) .

(i) There exist mechanical flat-input control vector fields g 1 , . . . , g n-1 such that (M c , h) is locally flat at ((x 0 , v 0 ), u 0 ) if and only if there exists a differential one-form δ on T X of the form

δ = n-1 i=1 δ i (x)dL f h i such that δ (x 0 ) = 0, for a certain 1 ≤ ≤ n -1, satisfying (C) δ, ad f g n-1 = 0 and L g n-1 µ (x 0 , v 0 ) = 0, where µ = L 2 f h + i = δ i
δ u i0 , and g n-1 = g i n-1 (x) ∂ ∂v i spans the distribution G n-1 , given by (9), and is the vertical lift of a nonzero vector field g n-1 = g i n-1 (x) ∂ ∂x i on X. (ii) For system [START_REF] Fritzsche | On a generalized flat input definition and physical realizability[END_REF], condition (C) reads as

n-1 i=1 δ i (x) ∂a i ∂v n ≡ 0, and (10)
∂ν ∂x n (x 0 , v 0 ) = 0, where ν = a + i = b i u i0 and b i (x) = -δ i δ , for 1 ≤ i ≤ n -1, i = .
(iii) For system M given by (7) and any δ satisfying (10), mechanical flat-input control vector fields g 1 , . . . , g n-1 can be constructed algebraically by formula (12) below, and lead to the following control system (M c , h) which is flat at ((x 0 , v 0 ), u 0 ), and given, in M-coordinates compatible with h, by

M c :        ẋi =v i ẋn-1 =v n-1 vi =a i (x, v) + u i vn-1 =a n-1 (x, v) + n-2 i=1 b i (x)u i 1 ≤ i ≤ n -2, ẋn =v n vn =a n (x, v) + u n-1 , (11)
with

∂(a n-1 -n-2 i=1 b i a i ) ∂v n ≡ 0 and ∂ν n-1
∂x n (x 0 , v 0 ) = 0, where ν n-1 is defined as in (ii) and is (after a permutation) supposed to be

= n -1, b i = -δ i δ n-1 , 1 ≤ i ≤ n -2, h = (x 1 , . . . , x n-1
) , and andg

g i = ∂ ∂v i + b i (x) ∂ ∂v n-1 , 1 ≤ i ≤ n -2,
n-1 = ∂ ∂v n . ( 12
)
Theorem 3.1 states that, in the case m = n -1, a necessary and sufficient condition for the existence of mechanical flat inputs is that the derivatives of the functions a i (associated to the measured positions) with respect to the velocity v n (whose corresponding position is not measured) are dependent over the ring of smooth functions depending on configurations only, see (ii) where condition (C) is given in M-coordinates (x, v) compatible with h. In (i) condition (C) is expressed in an invariant way. The distribution G n-1 gives the only candidate for the mechanical vector field g n-1 . The functions b i defining the mechanical flat-input control vector fields of M c , given by ( 11), and the functions δ i defining the one-form δ, are obviously related: indeed, we take b i as b

i (x) = -δ i (x)
δ n-1 (x) , with δ n-1 (x) being assumed nonzero. The problem of constructing mechanical flat inputs reduces to the problem of computing the functions δ i (x) for which a system of algebraic equations is provided in Subsection 3.1. Theorem 3.1 applies to both observable and non observable case (if (M, h) is observable, then introducing b i keep observability; if (M, h) is not observable, then introducing b i creates observability). The construction of the mechanical flat-input control vector fields g 1 , . . . , g n-1 leading to (M c , h) can be simplified for two particular cases treated in Propositions 3.1 and 3.2 below.

By the analysis of M in the form [START_REF] Fritzsche | On a generalized flat input definition and physical realizability[END_REF], we conclude that if we render it flat, then n -2 flat inputs, say u 1 , . . . , u n-2 , have to affect (permute h i 's if necessary) ẍ1 = v1 , . . . , ẍn-2 = vn-2 , where x i = h i , 1 ≤ i ≤ n -2, and the remaining output h n-1 = x n-1 or, more generally, a function hn-1 = Ψ(h 1 , . . . , h n-1 ), invertible with respect to h n-1 , has to provide information about v n-1 (with the help of L f hn-1 ), about x n (with the help of L 2 f hn-1 ) and about v n (with the help of L 3 f hn-1 ). In other words dx n ∈ H 3 , and an interesting property is whether there exists hn-1 such that dx n ∈ H 2 + span{dL 2 f hn-1 }, that is, in order to express x n , we use h i = x i and v i = ẋi , for 1 ≤ i ≤ n -2, as well as hn-1 = xn , ẋn , and ẍn , that means positions, velocities, and the second derivative of hn-1 only. It follows that expression (3) involves h (j) i , 0 ≤ j ≤ 3, 1 ≤ i ≤ n -2, and h(j) n-1 , 0 ≤ j ≤ 4. Proposition 3.1. Consider the observed mechanical system (M, h), given by (1), together with (h 1 (x), . . . , h n-1 (x)) . The following conditions are equivalent:

(i) There exist mechanical flat-input control vector fields g 1 , . . . , g n-1 such that (M c , h) is locally flat at (x 0 , v 0 ) and admits a flat output (ϕ 1 , . . . , ϕ n-1 ) verifying span{dϕ i , 1 ≤ i ≤ n-1} = span{dh i , 1 ≤ i ≤ n -1}, and for which expression (3) involves ϕ (j) , 0 ≤ j ≤ 4, for a certain

1 ≤ ≤ n -1, and ϕ (j)
i , 0 ≤ j ≤ 3 only, for i = . (ii) There exists a differential one-form form δ = n-1 i=1 δ i (x)dL f h i satisfying (C) in Theorem 3.1 and moreover, such that the differential one-form ω = n-1 i=1 δ i (x)dh i on X, associated to δ, fulfills ω ∧ dω = 0.

(iii) There exists a map Ψ = Ψ(h 1 , . . . , h n-1 ), invertible with respect to h , for a certain

1 ≤ ≤ n -1, such that ψ(x) = Ψ(h 1 (x), . . . , h n-1 (x)), in M-coordinates compatible with h, satisfies n-1 i=1 ∂ψ ∂x i ∂a i ∂v n ≡ 0 and ∂L 2 f ψ ∂x n (x 0 , v 0 ) = 0.
(iv) There exists a map Ψ = Ψ(h 1 , . . . , h n-1 ), invertible with respect to h , for a certain 1 ≤ ≤ n -1 (say, without loss of generality, = n -1), such that in the coordinates in which

x 1 = h 1 (x), . . . , x n-2 = h n-2 (x), x n-1 = ψ(x), where ψ(x) = Ψ(h 1 (x), . . . , h n-1 (x)), the system M takes the form        ẋi =v i ẋn-1 =v n-1 vi =a i (x, v) vn-1 =a n-1 (x, v), 1 ≤ i ≤ n -2, ẋn =v n vn =a n (x, v) (13)
where ∂a n-1 ∂v n ≡ 0, ∂a n-1 ∂x n (x 0 , v 0 ) = 0, and h = (x 1 , . . . , x n-2 , h n-1 (x 1 , . . . , x n-1 )) . Moreover, if M satisfies any of the equivalent conditions (i)-(iv), then mechanical flat-input control vector fields g 1 , . . . , g n-1 are given for [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF], by

g i = ∂ ∂v i , 1 ≤ i ≤ n -2, and g n-1 = ∂ ∂v n , (14)
i.e., modify only the equations for v i , 1 ≤ i ≤ n -2, as vi = a i (x, v) + u i , and for v n as vn = a n (x, v) + u n-1 .

Proposition 3.1 is analogous to Theorem 3.1, but its conditions are more restrictive than those of Theorem 3.1. To compute ψ we have to solve a system of PDE's (we actually have ∂ψ ∂x i = δ i (x)) and therefore integrability conditions have to be satisfied, see Subsection 3.2. The integrability conditions are expressed as ω ∧ dω = 0 and guarantee the existence of ψ and allow to avoid calculating ψ. The function ψ is, in fact, another candidate for one of the flat output components. Indeed, for (M c , h) obtained from ( 13) by adding the mechanical flat-input control vector fields given by ( 14), it is clear that (h 1 , . . . , h n-2 , ψ) = (x 1 , . . . , x n-2 , x n-1 ) is also a flat output, and, moreover, it is actually such that expression (3) uses derivatives up to order three of h i , for 1 ≤ i ≤ n -2, and up to order four of ψ (so (h 1 , . . . , h n-2 , ψ) satisfies the same conditions as the flat output ϕ of Proposition 3.1(i)). The mechanical flat-input control vector fields constructed with the help of Proposition 3.1 have only n -1 non zero components (which is the minimal possible). Proposition 3.1 covers also the case n = 2 and m = 1 (compare it to Proposition 2.2), and it provides a structural sufficient condition (far from being trivial) for the existence of mechanical flat inputs, but also a regularity condition. The flatness singularity ∂a n-1 ∂x n (x 0 , v 0 ) = 0 depends on the function ψ (we have a n-1 = L 2 f ψ), so different choices of ψ (if they exist) lead, in general, to different singularities. Finally, remark that the systems M, for which Proposition 3.1 applies, are necessarily observable with respect to h. Now we consider the unobservable case for which the defect of observability is maximal.

Proposition 3.2. Consider the observed mechanical system (M, h), given by (1), together with (h 1 (x), . . . , h n-1 (x)) , and suppose that the rank of the observability codistribution H = span{dL j f h i , 1 ≤ i ≤ n -1, j ≥ 0} is constant and equals 2n -2, i.e., H = H 2 . Then in M-coordinates compatible with h, we have ẋi = v i , vi = a i (x, v), with ∂a i ∂x n = ∂a i ∂v n ≡ 0, for all 1 ≤ i ≤ n -1. For that form, we construct mechanical g 1 , . . . , g n-1 such that (M c , h) is flat at any ((x 0 , v 0 ), u 0 ), with u 10 = 0, and is given by

M c :      ẋi =v i ẋn-1 =v n-1 vi =a i (x, v) + u i vn-1 =a n-1 (x, v) + x n u 1 , 1 ≤ i ≤ n -2, ẋn =v n vn =a n (x, v) + u n-1 , (15)
with ∂a i ∂x n = ∂a i ∂v n ≡ 0, 1 ≤ i ≤ n -1, h = (x 1 , . . . , x n-1 ) .

Proposition 3.2 applies for the unobservable case for which there are two unobserved variables x n and v n in the well chosen M-coordinates compatible with h. In that case, mechanical flat inputs always exist without structural conditions. Moreover, if rk H = 2n -2 everywhere, there is no singularity in the state space, i.e., h = (x 1 , . . . , x n-1 ) is a local flat output around any (x 0 , v 0 ) ∈ T X. Notice, however, that the system exhibits a singularity in the control space given by u 10 = 0 that is unavoidable.

3.1 Computation of the functions δ i . Consider the system M in M-coordinates compatible with h. Since all a i (x, v) are quadratic with respect to the velocities (see ( 4)), the condition n-1 i=1 δ i (x) ∂a i ∂v n = 0 translates into a polynomial of degree one with respect to v i , 1 ≤ i ≤ n, (whose coefficients are thus functions of x only) being identically zero. This leads to the following system of n + 1 algebraic equations with unknowns δ 1 (x), . . . , δ n-1 (x):

n-1 i=1 δ i (x)Γ i jn (x) = 0, 1 ≤ j ≤ n -1, n-1 i=1 δ i (x)Γ i nn (x) = 0, n-1 i=1 δ i (x)d i n (x
) = 0, which may or may not have nontrivial solutions (we used the symmetry of Christoffel symbols Γ i nj = Γ i jn ). 3.2 Computation of the function ψ. The existence of ψ is guaranteed by ω ∧ dω = 0. If we need to compute it, we deduce that ψ has to satisfy the following system of PDE's:

n-1 i=1 ∂ψ ∂x i (x)Γ i jn (x) = 0, 1 ≤ j ≤ n -1, n-1 i=1 ∂ψ ∂x i (x)Γ i nn (x) = 0, n-1 i=1 ∂ψ ∂x i (x)d i n (x) = 0
, which has a nontrivial solution if and only if ω ∧ dω = 0.

Example

Consider the following mechanical system with 4 degrees of freedom that can be seen as a combination of a two-link manipulator and a single link manipulator with joint elasticity [START_REF] Kozlowski | The adaptive control algorithm for manipulators with joint flexibility[END_REF][START_REF] Nowicki | Feedback Linearization of Mechanical Control Systems[END_REF]:

M :    ẋi =v i ẋi =v i vi =-Γ i jk (x)v j v k + e i (x), vi =e i (x), 1 ≤ i ≤ 2, 3 ≤ i ≤ 4, (16)
with x ∈ S 1 × S 1 × S 1 × S 1 = T 4 , where x 1 , x 2 denote the angles of the first and the second link, resp., and x 3 , x 4 denote the angles of the first and the second motor shaft. The only non-zero Christoffel symbols are Γ i jk , for 1 ≤ i, j, k ≤ 2; the functions e 1 (x) and e 2 (x) depend explicitly on all positions x i , for 1 ≤ i ≤ 4, while e 3 (x) (resp., e 4 (x)) is a function of x 1 and x 3 (resp., x 2 and x 4 ) only, see [START_REF] Nowicki | Feedback Linearization of Mechanical Control Systems[END_REF] for their exact expressions. The controlled two-link manipulator with joint elasticity (MS) : ż = f (z) + u 1 g 1 (x) + u 2 g 2 (x), with z = (x, v), f the drift of (16), and control vector fields g 1 = 1 J 1 ∂ ∂v 3 and g 2 = 1 J 2 ∂ ∂v 4 , has been considered in [START_REF] Nowicki | Feedback Linearization of Mechanical Control Systems[END_REF] from a point of view of static feedback linearization (see [START_REF] Jakubczyk | On linearization of control systems[END_REF][START_REF] Hunt | Linear equivalents of nonlinear time varying systems[END_REF]); it has been shown that (MS) is static feedback linearizable with linearizing outputs x 1 , x 2 , but the transformed linear system is not mechanical since the linearizing diffeomorphism is not mechanical and, moreover, that there are no linearizing outputs defining a mechanical diffeomorphism (i.e., (MS) is not mechanical static feedback linearizable, see [START_REF] Nowicki | Feedback Linearization of Mechanical Control Systems[END_REF] for a formal definition). It is thus natural to investigate whether there exist other mechanical control vector fields (equivalently, mechanical flat inputs) that render M flat and, in particular, mechanical static feedback linearizable. For the system to fall into the class considered in this paper (that is, n degrees of freedom and n -1 outputs), we need 3 outputs. We study two cases: we first consider both linearizing outputs of [START_REF] Nowicki | Feedback Linearization of Mechanical Control Systems[END_REF], i.e., h 1 = x 1 , h 2 = x 2 , and complete them by h 3 = x 3 , and then we study the case h

1 = x 1 , h 2 = x 3 and h 3 = x 4 .
Suppose first that h 1 = x 1 , h 2 = x 2 , h 3 = x 3 . Notice that the functions a i are such that: a i = a i (x, v 1 , v 2 ), for 1 ≤ i ≤ 2, and a i = e i (x), for 3 ≤ i ≤ 4. Proposition 3.1 is obviously verified (simply take ψ = h 1 = x 1 ) and we define the mechanical g 1 = ∂ ∂v 2 , g 2 = ∂ ∂v 3 , and g 3 = ∂ ∂v 4 that lead to the mechanical control system

M c :      ẋ2 =v 2 ẋ3 =v 3 ẋ1 =v 1 v2 =a 2 (x, v 1 , v 2 ) + u 1 ,
v3 =e 3 (x) + u 2 , v1 =a 1 (x, v 1 , v 2 ) ẋ4 =v 4 v4 =e 4 (x) + u 3 , with h = (x 1 , x 2 , x 3 ) a flat output at any (x 0 , v 0 ) such that ∂a 1 ∂x 4 (x 0 , v 0 ) = 0. Other constructions are possible based on Theorem 3.1. For instance, if we work around a point such that ∂a 1 ∂x 4 (x 0 , v 0 ) = 0 (that is a flatness singularity for the above form), we can avoid that singularity by constructing the mechanical control system as follows (which is of the form (11) with b 1 = x 4 , b 2 = 0, equivalently, with δ 1 = x 4 , δ 2 = 0, δ 3 = 1):

M c :       
ẋi =v i ẋ3 =v 3 vi =a i (x, v 1 , v 2 ) + u i , v3 =e 3 (x 1 , x 3 ) + x 4 u 1 , 1 ≤ i ≤ 2, ẋ4 =v 4 v4 =e 4 (x) + u 3 , which is flat with h = (x 1 , x 2 , x 3 ) a flat output around any (x 0 , v 0 ) ∈ T X and u 10 = 0 (notice the new singularity in the control space). For both mechanical control systems M c and M c proposed above, a simple calculus shows that their linearizability distribution D 1 = span{g i , ad f g i , 1 ≤ i ≤ 3} is not involutive, hence none of them is static feedback linearizable (and in particular, not mechanical static feedback linearizable), although flat. Finally, observe that since no function a i depends on v 4 , we can construct a mechanical control system (M c , h), with h = (x 1 , x 2 , x 3 ) , by choosing any functions b i (x) (or equivalently, any one-form δ) in Theorem 3.1.

Consider now the case when h 1 = x 1 , h 2 = x 3 and h 3 = x 4 . We apply Proposition 3.1 with ψ = h 3 = x 4 , and define the mechanical g 1 = ∂ ∂v 1 , g 2 = ∂ ∂v 3 , and g 3 = ∂ ∂v 2 that lead to the mechanical control system

M c :        ẋ1 =v 1
ẋ3 =v 3 ẋ4 =v 4 v1 =a 1 (x, v 1 , v 2 ) + u 1 , v3 =e 3 (x) + u 2 , v4 =e 4 (x 2 , x 4 ), ẋ2 =v 2 v2 =a 2 (x, v 1 , v 2 ) + u 3 , with h = (x 1 , x 3 , x 4 ) a flat output at any (x 0 , v 0 ) ∈ T X (since we always have ∂e 2 ∂x 4 = 0, [START_REF] Nowicki | Feedback Linearization of Mechanical Control Systems[END_REF]). It can be easily verified that M c is now static feedback linearizable and moreover, that the linearizing diffeomorphism is compatible with the mechanical structure yielding a linear mechanical control system, the measurements h = (x 1 , x 3 , x 4 ) being also the mechanical linearizing outputs.

Actually, according to[START_REF] Waldherr | Conditions for the existence of a flat input[END_REF], if and only if the codistribution H N -1 = span{dL j-1 f h(x), 0 ≤ j ≤ N -1} is of full rank, thus implying local observability of (Σ, h), see[START_REF] Hermann | Nonlinear controllability and observability[END_REF]. We use that notion of observability in this paper.

The summation convention is used for ω, η, and ω, η .