domain: shs sde

text:
Muriel Deparis
email: muriel.deparis@insa-cvl.fr

Nicolas Legay

Francis Isselin-Nondedeu

Francis Isselin- Nondedeu

Sébastien Bonthoux

How managers and city dwellers relate to spontaneous vegetation in cities: towards an integrative approach

Keywords: Conceptualization, Methodology, Formal analysis, Investigation, Writing -Original Draft, Visualization Conceptualization, Methodology, Investigation, Writing -Review & Editing, Supervision Conceptualization, Methodology, Investigation, Writing -Original Draft, Visualization, Supervision communication campaign, focus groups, operational management, street maintenance, vegetation perception, weeding management

published or not. The documents may come

How managers and city dwellers relate to spontaneous vegetation in cities: Towards an integrative approach

Introduction

The current context of accelerating urbanization generates important issues in biodiversity conservation (McKinney, 2006), the well-being of city dwellers [START_REF] Soga | Extinction of experience: The loss of human-nature interactions[END_REF] and the relationships between human activities and ecological functioning [START_REF] Van Den Bosch | Urban natural environments as nature-based solutions for improved public health -A systematic review of reviews[END_REF]. In this context, it is imperative to find approaches that support the transition of urban systems towards sustainability. To that end, the promotion of vegetation that grows and colonizes the city freely (hereafter called spontaneous vegetation) is recognized as a means to enhance urban plant diversity at multiple scales [START_REF] Salisbury | Plant diversity increases in an urban wildland after four decades of unaided vegetation development in a post-industrial site[END_REF] as this vegetation provides breeding sites, refuges, and food resources for many animal species (see [START_REF] Bonthoux | How can wastelands promote biodiversity in cities? A review[END_REF] for a review; Twerd & Banaszak-Cibicka, 2019 for bees; Villaseñor et al., 2020 for birds). However, promoting spontaneous vegetation in highly anthropized spaces such as cities can lead to resistances from the various urban actors who use, plan, and manage the city. Planners and managers can anticipate negative reactions of the public associated with a feeling of abandonment, and an excessive difference between traditional management and spontaneous vegetation [START_REF] Hoyle | Not in their front yard‖ The opportunities and challenges of introducing perennial urban meadows: A local authority stakeholder perspective[END_REF]. They may also consider that they do not have enough time, money or equipment to change their practices, or they may be opposed to doing so because they believe that current practices are good and positive for biodiversity [START_REF] Lampinen | Reconciling road verge management with grassland conservation is met with positive attitudes among stakeholders, but faces implementation barriers related to resources and valuation[END_REF]. Indeed, supporting the transition of cities to more biodiversity-friendly management implies going beyond ecological knowledge and also investigating the urban socio-ecological system in depth [START_REF] Kowarik | Urban wilderness: Supply, demand, and access[END_REF], by linking ecological aspects with psychosocial, governance, and management approaches [START_REF] Frank | Urban systems: A socio-ecological system perspective[END_REF][START_REF] Bretagnolle | Actionorientated research and framework: Insights from the French long-term social-ecological research network[END_REF].

Recent studies have investigated the attitudes of city dwellers towards spontaneous vegetation in cities [START_REF] Brun | Residents' perceptions and valuations of urban wastelands are influenced by vegetation structure[END_REF][START_REF] Mathey | Public perception of spontaneous vegetation on brownfields in urban areas-Results from surveys in Dresden and Leipzig (Germany)[END_REF]. They show that perceptions are contrasted, ranging from negative to positive, depending on the contexts and respondents. City dwellers prefer vegetation with a diversity of structure and color [START_REF] Fischer | Beyond green: Broad support for biodiversity in multicultural European cities[END_REF] over hard landscape (Bonthoux et al., 2019a). Nevertheless, orderly vegetation that is perceived as being J o u r n a l P r e -p r o o f planted and maintained seems to be preferred against vegetation with messy aspects [START_REF] Weber | A walk on the wild side: Perceptions of roadside vegetation beyond trees[END_REF]Bonthoux, Voisin, et al., 2019;[START_REF] Hwang | It's ok to be wilder: Preference for natural growth in urban green spaces in a tropical city[END_REF]. Cues to care, which are immediately recognizable signs of human intention or presence (e.g., footpath, benches, plants in rows in urban green space), have been proposed as ways to facilitate relationships between the spontaneous characteristics of places and users [START_REF] J O U R N A L P R E -P R O O F Li | Cues to care: A systematic analytical review[END_REF]. They also can mitigate the feeling of insecurity and abandonment sometimes associated to the presence of spontaneous vegetation in streets [START_REF] Säumel | Toward livable and healthy urban streets: Roadside vegetation provides ecosystem services where people live and move[END_REF]. Other studies show that people have little perception of spontaneous vegetation, and this phenomenon, not limited to cities, is called ‗plant blindness' [START_REF] Balding | Plant blindness and the implications for plant conservation[END_REF]. It is explained by the fact that individual plants do not move like animals, grow in patches, tend to obscure each other, and are overall uniform in color [START_REF] Balding | Plant blindness and the implications for plant conservation[END_REF].

To promote positive city dwellers-spontaneous vegetation relationships, providing ecological information is a way to change the meaning and knowledge associated with plants, which could improve the acceptance of low management practices in urban areas [START_REF] Unterweger | Urban nature: Perception and acceptance of alternative green space management and the change of awareness after provision of environmental information. A chance for biodiversity protection[END_REF].

Promoting spontaneous vegetation in urban public places also requires considering the governance and management systems of municipalities, to analyze their operationalization level of ecological measures [START_REF] Dempsey | Defining place-keeping: The long-term management of public J o u r n a l P r e -p r o o f spaces[END_REF][START_REF] Hoyle | Not in their front yard‖ The opportunities and challenges of introducing perennial urban meadows: A local authority stakeholder perspective[END_REF]. A few studies have examined the propensity of public managers to shift their practices toward extensively managed urban spaces [START_REF] Duivenvoorden | Managing public space -A blind spot of urban planning and design[END_REF]. These inquiries have focused on perennial meadows [START_REF] Hoyle | Not in their front yard‖ The opportunities and challenges of introducing perennial urban meadows: A local authority stakeholder perspective[END_REF], urban plantings [START_REF] Muratet | Understanding stakeholder perceptions of acceptability and feasibility of formal and informal planting in sheffield's district parks[END_REF], low input turfgrasses [START_REF] Barnes | Public land manager discourses on barriers and opportunities for a transition to low input turfgrass in J o u r n a l P r e -p r o o f urban areas[END_REF] and urban forests [START_REF] Ordóñez | Urban forest governance and decision-making: A systematic review and synthesis of the perspectives of municipal managers[END_REF]. All these studies emphasized that the complexity of the management system must be analyzed in depth to move towards sustainable practices. Those studies showed that technical and scientific knowledge, the working conditions of public managers, the organization of municipal services, financial aspects, and user feedback are important determinants for better understanding managers' attitudes and practices vis-à-vis vegetation [START_REF] Hoyle | Not in their front yard‖ The opportunities and challenges of introducing perennial urban meadows: A local authority stakeholder perspective[END_REF][START_REF] Muratet | Understanding stakeholder perceptions of acceptability and feasibility of formal and informal planting in sheffield's district parks[END_REF][START_REF] Barnes | Public land manager discourses on barriers and opportunities for a transition to low input turfgrass in J o u r n a l P r e -p r o o f urban areas[END_REF]. Further, while the analysis of the decision-making processes of elected officials and municipal service managers is developing, this J o u r n a l P r e -p r o o f rarely includes surveying the field operatives who are daily maintaining the urban vegetation.

Linking these social and technical issues with ecological urban elements in an integrative framework can help to identify the relationships and feedbacks between urban actors and spontaneous vegetation, as well as promising opportunities and obstacles to develop positive people-biodiversity relationships. Two conceptual models are used in the field of landscape planning and management to address, in a systemic way, the governance and management of urban open places [START_REF] Randrup | Public green spaces in the Nordic countries: Development of a new strategic management regime[END_REF][START_REF] Dempsey | Defining place-keeping: The long-term management of public J o u r n a l P r e -p r o o f spaces[END_REF][START_REF] Jansson | The governance of landscape management: New approaches to urban open space development[END_REF]. Combining several previous theoretical frameworks, [START_REF] Jansson | The governance of landscape management: New approaches to urban open space development[END_REF] proposed a model with three interconnected elements: -users and city dwellers‖, -public actors and organization‖, and -urban open spaces‖. Users and public actors are connected through -discourses‖, and both actor groups are related to urban open spaces via -power and resources‖ that depend on the modes of governance [START_REF] Jansson | The governance of landscape management: New approaches to urban open space development[END_REF]. In their -Place keeping model‖, [START_REF] Dempsey | Defining place-keeping: The long-term management of public J o u r n a l P r e -p r o o f spaces[END_REF] argue for the need to think deeply about the management of urban projects to insure the long-term provisioning of social, environmental and economic benefits of urban spaces for future generations. This model is composed of interconnected management dimensions, namely maintenance, partnership, governance, funding, policy, and evaluation [START_REF] Dempsey | Defining place-keeping: The long-term management of public J o u r n a l P r e -p r o o f spaces[END_REF]. Both models have been conceptualized to better understand the organization and interconnection of actors who decide, manage and use public places in city, but as of yet without direct links to biodiversity conservation issues. In this paper, we used both models to analyze in an integrative way the social and management system related to spontaneous vegetation in streets.

Streets are ubiquitous urban spaces designed and managed by the municipality, traversed daily by its inhabitants, whose pavements harbor a diverse spontaneous vegetation (e.g. 307 plant species on pavements of a French city, Bonthoux et al., 2019a). This street vegetation provides ecological resources for various animals and represents an opportunity to understand and foster the relationships between city dwellers and nature [START_REF] Weber | A walk on the wild side: Perceptions of roadside vegetation beyond trees[END_REF]Bonthoux et al., 2019b).

Although this vegetation is associated with many services, a few disservices may reduce its J o u r n a l P r e -p r o o f acceptability [START_REF] Säumel | Toward livable and healthy urban streets: Roadside vegetation provides ecosystem services where people live and move[END_REF]. In the traditional view of the city, streets and pavements are kept bare and thought for the movement of city dwellers without any connection to nature. However, as recent national laws in several European countries banning pesticides in urban public spaces concern streets (www.pesticide-free-towns.info), municipalities are modifying their weeding practices, leading to a thorough rethinking of human relationships to spontaneous vegetation.

In this study, we analyzed how municipal service directors, field operatives and pedestrians relate to spontaneous vegetation in streets. Based on data collected in the medium-sized city of Blois, in central France, we combined different approaches. First, we investigated the history of the weeding in streets and the associated perception of spontaneous vegetation by municipal service directors. Then, we analyzed the parameters influencing the current attitudes and practices of municipal field operatives toward vegetation. Based on aforementioned studies of urban space management, we hypothesized that these attitudes are influenced by multiple determinants such as funding and technical parameters [START_REF] Dempsey | Defining place-keeping: The long-term management of public J o u r n a l P r e -p r o o f spaces[END_REF], how the governance of municipal services is organized [START_REF] Randrup | Public green spaces in the Nordic countries: Development of a new strategic management regime[END_REF], and relationships with city dwellers [START_REF] Jansson | The governance of landscape management: New approaches to urban open space development[END_REF]. Finally, in partnership with the municipal service in charge of weed control, we set up a field experiment to assess and modify pedestrians' perception of spontaneous vegetation. We hypothesized that providing ecological information in situ, on stickers placed on pavements near the plants, would improve the pedestrians' perception of plants.

Methods

City's context and history of the cleanliness service

Our study took place in Blois, France (47°34′59″N, 1°19′59″E). Agriculture land and two large forested massifs, to the west and to the south, dominate the surrounding landscape. The river Loire crosses the city from east to west (Fig. 1). Almost 50 000 people live in this city, whose urban area is ca. 22 km². The historical center dates from the medieval and Renaissance period, is mainly J o u r n a l P r e -p r o o f pedestrianized, with a high stake for the image of this area, considered as the showcase of the city by the municipality. Indeed, Blois is an important tourist city, with its castle and the Loire classified as a UNESCO heritage site. The rest of the city is mainly composed of residential pavilion areas. A large industrial area extends north of the city to the highway. Blois also has a very extensive neighborhood of collective residential buildings. The challenges of spontaneous vegetation management are different in these neighborhoods, which do not serve the same urban functions and are home to different socio-economic populations. Since 2015 our research team has studied the diversity of spontaneous vegetation of streets and city dwellers' perceptions of this vegetation (Bonthoux et al., 2019a;Bonthoux et al., 2019b).

From that time, our interactions with the municipal cleanliness service in charge of weed control have matured through several work meetings and informal conversations (about 3 meetings held per year). To deepen the information collected on the historical relationship between the municipality and spontaneous vegetation, we conducted two 1-h individual semi-structured interviews in 2020, one with the director of the cleanliness service and another with the elected official in charge of biodiversity for the municipality. Interview guides were made to understand the recent change in the municipality's orientations towards biodiversity, the change in the organization and practices of the cleanliness service, and the relations between this municipal service and others (Appendix 1).

Focus groups with field operatives 2.2.1. Set up and running

We analyzed the attitudes and practices of field operatives with respect to spontaneous vegetation on pavements. The municipal cleanliness service comprises five teams of field operatives -between 6 to 10 people per team (mean 7.8 people), including each team director. Four of the five teams are associated with a specific geographic sector (Fig. 1), the fifth being a -flying‖ team that steps in as back up for certain jobs. We interviewed each team individually in a focus group discussion [START_REF] Nyumba | The use of focus group discussion methodology: Insights from two decades of application in conservation[END_REF]. In total, because of absences, two-thirds of the service staff participated in As the facilitator asked the questions and led the interview, the moderator ensured that everyone had an opportunity to speak. In particular, the moderator would ask some participants by first name to get them more involved. The moderator also tried to notice physical gestures or attitudes, looking for signs of approval or disagreement. The moderator could then reopen the discussion in the group to confirm these physical attitudes, addressing the group in a general way (-Does everyone agree with what has just been said?‖), or by directly question field operatives. As the team director could be present during a focus group, we tried to mitigate possible dominance effects by encouraging field operatives to express any contrary views or thoughts, thereby ensuring that all opinions were freely shared [START_REF] Nyumba | The use of focus group discussion methodology: Insights from two decades of application in conservation[END_REF].

To build the interview grid, we made a qualitative content analysis [START_REF] Mayring | Qualitative content analysis-Theoretical foundation, basic procedures and software solution[END_REF] of literature about parameters structuring field operatives attitudes and perceptions regarding their work and the vegetation management. Based on that analysis, we identified four general themes used to guide our discussion with focus groups (Table 1): their work perception, the relation with the inhabitants, the organizational parameters and relations with hierarchy, their perception and opinion about spontaneous vegetation. We added introducing words in the guiding theme (Table 1)

to initiate the discussion in groups and all the questions were open-ended.

Data analysis

We recorded the contents of focus groups and then transcribed it in the four identified themes.

Contents were then coded according to new categories that were refinements of initial themes (Table 1). This refinement was based on inductive qualitative content analysis [START_REF] Cho | Reducing confusion about grounded theory and qualitative content analysis: Similarities and differences[END_REF].

These categories are those used in the results. Inspired by [START_REF] Lampinen | Reconciling road verge management with grassland conservation is met with positive attitudes among stakeholders, but faces implementation barriers related to resources and valuation[END_REF], each field operatives' statement was assigned to "positive", "negative" or "neutral" connotation. For example, statement as -weeding, I do not love that much‖ was assigned to negative opinion of -Vegetation J o u r n a l P r e -p r o o f management-weeding‖ category whereas -with [the parks and gardens field operatives], we work it out, we get on well‖ was assigned to positive opinion about -Organizational parameters-between services‖. Then we counted these positive, neutral or negative statements to know the majority opinion of the field operatives for the different categories. In the results, we illustrated each category with a given statement from a field operative, chosen to be representative of the common opinion held.

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What do you think of the communication action that has been put in place? Would you be in favor of generalization of action promoting vegetation in streets? Table 1 Guiding questions, classified by theme, for focus groups with cleanliness field operatives and categories used to analyze statements.

Communication about spontaneous vegetation and interviews with pedestrians 2.3.1. Set up and running

We designed a study with the implication of the cleanliness service, to understand the pedestrians' attitudes towards spontaneous vegetation and to evaluate the impact of in situ information about plants on these attitudes. We installed stickers on the pavement of 15 streets in Blois. Streets were chosen, in collaboration with the director of the municipal cleanliness service, because of their high levels of spontaneous vegetation irrespective of the weeding effort applied there. As the different neighborhoods in Blois are contrasted in term of amount of spontaneous vegetation, but also regarding socio-demographic profiles of inhabitants and pedestrian attendances, selected streets were in different neighborhoods (Fig. 1). These 15 streets were not weeded from January to September 2021. During April, stickers conveying information about spontaneous plants growing on these streets were stuck flat on the ground in the immediate vicinity of the plants. Stickers represented 13 plants among the most frequent in Blois (according to Bonthoux et al., 2019b, Appendix 2). Stickers were illustrated with hand-drawn pictures of the plants, mentioned both its French and Latin names, and some ecological information, such as their role as host plants for insects (information taken from [START_REF] Muratet | Understanding stakeholder perceptions of acceptability and feasibility of formal and informal planting in sheffield's district parks[END_REF], see Appendix 3). We pasted stickers in a 100m section on one side of each selected street. In addition, two signs (Appendix 2) that briefly presented the research approach were placed at the ends of the street sections (Fig. 1). The municipality funded the making of the stickers and signs.

Data analysis

We analyzed the data of open-ended question with deductive qualitative approach [START_REF] Cho | Reducing confusion about grounded theory and qualitative content analysis: Similarities and differences[END_REF]. Based on content of questionnaire, we identified the positive, negative or neutral opinion expressed for each theme of our questionnaire, and the underpinning thematic of answer. We then coded answers according to those thematic, presented in results.

Results

Change in the municipality-spontaneous vegetation relationship

The interviews with the elected official in charge of biodiversity and the director of the cleanliness service informed us about the change in municipal weeding strategies. Until 2012, the streets of Blois underwent regular chemical weeding by a company contracted by the municipality, to have very clean pavements with no vegetation. Since then, the municipality has progressively suppressed the use of chemical herbicides to weed out public places, including streets. This decision anticipated the French law of 2017 called -Labbé law‖, which forbidden public authorities to use chemical herbicide on public spaces (promenades, forest paths, roads, streets). Since 2012, the mechanical weeding of streets in Blois is the responsibility of the municipal cleanliness service whose role until that time was exclusively to collect waste and clean up public spaces. This weeding task was not assigned to the municipal service responsible for maintaining green spaces (the green spaces service), which do not deal with hard landscape such as streets. In lieu of chemical pesticides, the J o u r n a l P r e -p r o o f cleanliness service practiced frequent mechanical and thermal weeding. However, the service realized that mechanical weeding would never be as effective as chemical one. The progressive habituation to the presence, even modestly so, of spontaneous vegetation on street pavements has gradually triggered a change of vision of the service directors on the pavement vegetation and, accordingly, alternatives were pursued. Since 2017, the municipality of Blois developed the -Flowered pavements operation‖ which consisted in having some inhabitants take charge of the pavement in front of their homes. Under this operation, flower seeds are given to the residents for sowing in the gaps between the pavement and wall. In 2022, about 500 households participated in the operation. Since 2018, the new director of the cleanliness service wanted to continue to reduce the intensity of managing spontaneous vegetation in certain streets, to lessen the workload of its field operatives and to promote the flora in the city. This change in attitude towards spontaneous vegetation was reinforced by the arrival in 2020 of new elected officials who had an ecological agenda. Although the service and elected officials have learned that city inhabitants perceive spontaneous vegetation less negatively than they thought (Bonthoux et al. 2019b), they continue to search ways to change the attitudes of inhabitants towards spontaneous vegetation (hence this study).

Focus groups with field operatives of the cleanliness service

We met 27 field operatives of the cleanliness service (demographic information are in Appendix 5).

Field operatives' statements were organized in three structural themes: general conditions of their work; their interactions with inhabitants and other municipal services and officials; and their opinions about spontaneous street vegetation, the weeding task, and urban vegetation in general.

Meaning and conditions of their work

To describe their work, field operatives cited a multiplicity of tasks. Some tasks are considered as normal and part of their work role (e.g. clearing litterbins, sweeping, collecting rubbish, cleaning up graffiti and tags). In every group, field operatives expressed their pride in contributing to the J o u r n a l P r e -p r o o f cleanliness image of Blois and its inhabitants' appreciation of their city. Some other tasks, such as raking leaves and weeding, were presented as being rather sudden and abnormal, because they did not match their personal representation of their work:

"At the beginning, I thought it was illogical that we were the ones doing the weeding, our base is the sweeper, we turn into gardeners.‖ CS1 Some tasks are seasonal (raking leaves, weeding) and field operatives are free to decide when these tasks are required, based on their observation of streets. Some groups faced particular tasks (e.g. to deal with a lot of abandoned furniture in streets) not faced by others, and the need for weeding or clearing litterbins differs among geographic sectors. In the field operatives' mind, these geographical particularities were linked to the socio-economic characteristics of the inhabitants of different city sectors. Linked to the multiplicity of tasks, the field operatives explained that they use different tools. Some field operatives described and defined themselves in their work according to these tools:

"I am a sweeper driver, I also do the [thermal] burning [of weeds] and I drive the [truck]." N4

The type of tool used could determine whether some tasks are simpler or more complex: "Burning is less hard than hoeing, with hoeing sometimes it's hard to pull out plants, must weed them by hand, on your knees, even on all fours." CE5 Several field operatives cited the evolution of tools, particularly the arrival of lighter, self-propelled or electric tools that are quieter, as the main change they have observed in recent years in the service.

Interactions with inhabitants and organization between services

J o u r n a l P r e -p r o o f

The field operatives talked a lot about their interactions with inhabitants (Fig. 2). Inhabitants In addition, inhabitants express few complaints during the informal talks with them; just two field operatives in one group testified of direct complaints. Most complaints are expressed through the city's Rapid Response Service (RRS), which lets inhabitants report any problem to the municipal services, which are obliged to answer within 72 h. Some field operatives in different groups find that the use of RRS can be abusive, when used by some inhabitants to ask for weeding yet the condition of the streets does not warrant it. RSS is also seen as a potential constraint to change weeding management:

"It's too easy, phone call to weed […] calls for useless things." N4 "If there was more vegetation, we would have plenty of RRS." CS1

The field operatives did not talk about the other requests they received via RRS.

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Figure 2. The relationships among urban spontaneous vegetation in streets, the municipality, and its inhabitants (adapted from [START_REF] Jansson | The governance of landscape management: New approaches to urban open space development[END_REF]. Alongside the arrows are ways to improve these relationships and therefore the integration of spontaneous vegetation in the city.

When asked about it, most field operatives said they have few or no interactions with elected officials. This absence of a direct relationship creates a feeling of abandonment for some of them: "It would be nice to have [hierarchy's] opinion, compliments from the hierarchy." V2 "We don't have a relationship with them, we're at the bottom." N4

J o u r n a l P r e -p r o o f

A few field operatives, albeit in different groups, did tell us they have already received direct weeding requests from elected officials for specific areas of the city. Even if this behavior is unusual, the field operatives who shared it seemed angry, finding it at odds with the city's political display: "It's tiring, there is no ecological approach, we cut everything, we burn, we kill everything. Contradiction between ecological speech and actions. It's a pity, we say ecology, ecology...

[...] Reflex [to weed] is contradictory to what is asked to us." CE1

The field operatives talked much about a lack of interactions with the other municipal services of green spaces and urban facilities (Fig. 2). They all considered that weeding task should be borne by the green spaces service, because as plants are not rubbish, weeding is not a cleanliness task per se:

-Weeding is one job, cleaning is another. There is no relationship between the cleanliness service and the green spaces service." O5 However, they face many vegetation-related tasks because their service is in charge of asphalted and sandy areas, whereas the green spaces service is responsible for maintaining city areas covered with topsoil (e.g., parks). Many field operatives found this division unsuitable, forcing them to make gardening gestures or use gardening tools:

-There are places where you need a lawnmower, but it's not a cleaning service tool.‖ O5 Two groups have unofficial arrangements with the green spaces service, exchanging tasks with them, because of common facilities or personal relationships. These interactions had positive impacts on their work, and all field operatives were in favor of doing common work with the green spaces service:

J o u r n a l P r e -p r o o f "We arrange with green spaces service to split up, we see them every day in the field, we need more inter-service relationships, more dialogue.‖ N4 Two groups also explained precise difficulties caused by the absence of coordinated decisionmaking between the cleanliness and urban facilities services. Field operatives explained that some streets design attributes (i.e., positioning of trees, distance between lawns and building) made it difficult to use certain tools (particularly the sweeper) or can damage tools. Field operatives must then do the waste collection on foot, which is longer and more painful. If new arrangements for more plants are not positioned in collaboration with the cleanliness service, they can damage tools.

Therefore, that should be an obstacle to vegetated arrangements: "Lack of dialogue with urban development. They do not think about us when they do the development, you have to think about the volume to get the sweeper through. If we modify the base of the walls (add plants), we must also modify the entire sidewalk to think about the cleanliness." N2 "Need to anticipate management in sidewalk developments.‖ V4

Field operatives' attitudes towards spontaneous vegetation

Another theme we addressed during our focus groups was attitudes of field operatives towards spontaneous vegetation and the meaning of their weeding task (Table 1). Around a dozen of field operatives expressed themselves in favor of spontaneous vegetation, outlining that vegetation was neither rubbish nor pollution: However, all the field operatives agreed that a -white city‖ (-une ville à blanc‖ i.e., a city without spontaneous vegetation) is impossible. They surmised that the majority of inhabitants wanted little or no spontaneous vegetation and would carry out more weeding than the service does if that task relegated to them instead. Field operatives asked for more communication targeting inhabitants to explain the municipality's position street vegetation management. Thus, they supported the implemented communication campaign using stickers, believing it could make inhabitants more The field operatives appreciated the current political direction of the service being orientated to doing less weeding. As noted above ( §3.2.2.), some stated that weeding contradicts the current official position of the municipality about ecology and urban biodiversity. Further, all field operatives were in favor of more vegetation, supporting new flower arrangements or agreeing with current ones, like flowered pavements. They cited precise parts of the city where, in their opinion, there were under-exploited opportunities for vegetation establishment. But they stressed that such vegetation arrangements ought to be maintained and must not complicate the cleaning routine of pavements:

-People have to maintain, otherwise it's only weeds." CE8 "It is necessary to maintain these amenities." N1 They also said that the discontinuous aspect of vegetation on pavements was not aesthetic and complicated their weeding of streets. Thus, some field operatives asked for continuous flowered pavements.

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Communication about spontaneous vegetation and interviews of pedestrians

Due to the voluntary deterioration of some stickers, the low pedestrians attendance of some streets, the very rainy weather and a low response rate (49%), only 68 people in 8 streets responded (see Appendix 6 for demographic information). Only 11 respondents noticed the signs accompanied the stickers (Appendix 2) and just three had an opinion about it; hence, this part of the interview (use of signs) was omitted from the analysis.

Thirty pedestrians (44%) expressed a positive opinion about spontaneous vegetation and 17 (25%) pedestrians disliked it (Table 2 andAppendix 6). Thirty-six pedestrians (52%) declared there is a noticeable difference regarding presence and quantity of spontaneous vegetation between streets of the experiment (non-weeded) and the other in the city (weeded), of which 23 held a positive opinion about it.

Forty-six pedestrians (67%) noticed the presence of the stickers and 29 of them (63%) had a positive opinion about it (Table 2 andAppendix 6). Pedestrians mainly highlighted the educational aspect of stickers, citing almost exclusively the plants' names, ecological information, and drawings as retained elements. Of those who noticed the stickers, 17 told us that their opinion of these plants was changed by the stickers, mainly because of the new information they learned. Two questions dealt with the general opinion of pedestrians about leaving spontaneous vegetation in streets (Table 2 andAppendix 6). The ecological interest in plants was the first reason why pedestrians are likely to accept spontaneous vegetation on pavements (50% of answers), and facilitating the work of field operatives was the second motivation (27% of answers). The last question of the interview invited the pedestrians to share and elaborate on their opinion about the presence of spontaneous vegetation in streets, after a quick presentation of its benefits. Forty-nine pedestrians (74%) expressed a positive opinion, mostly because there were against the use of J o u r n a l P r e -p r o o f chemical herbicides, or in favor of biodiversity-friendly actions in the city. Nine pedestrians (14%) had a balanced opinion, in that they can support this kind of action but would prefer a more aesthetic approach to spontaneous vegetation in streets. Eight pedestrians (12%) were against the presence of spontaneous vegetation because they considered it not aesthetic.

Discussion

In this study, we tried to construct a socio-ecological model to analyze the relationships among the municipality, field operatives, inhabitants, and the spontaneous vegetation of streets in a mediumsized French city. Based on the interactions built since 2015 between our research team and the municipal service, we built two interconnected experiments, involving both field operatives and inhabitants. From our results, we can draw a portrait of field operatives' and inhabitants' perceptions of spontaneous vegetation, and of the parameters structuring these

Change in municipal strategy

Through the example of spontaneous street vegetation, this study analyzed the barriers and opportunities to biodiversity consideration and conservation at the local governance scale of a city [START_REF] Shih | Assessing governance challenges of local biodiversity and ecosystem services: Barriers identified by the expert community[END_REF]. Streets are widespread in the city and used by all publics, their study complements information on the acceptance of new management practices in green spaces such as parks [START_REF] Hwang | It's ok to be wilder: Preference for natural growth in urban green spaces in a tropical city[END_REF]. Caused by bottom-up (new staff with positive attitude towards spontaneous vegetation) and top-down (new national law banning pesticides) factors, the municipal weeding strategy shifted from clear-cutting pavements to allowing grown of spontaneous vegetation in less than 10 years. Those changes are supported by field operatives, who are at the forefront of technical constraints and physical hardship, but are decided by municipal council and services'

directors. Thus, we demonstrated the interests of analyzing the changes in relations to spontaneous vegetation at different hierarchical levels of the municipal organization [START_REF] Homsy | Multilevel governance: Framing the integration of top-down and bottom-up policymaking[END_REF].

Multiple factors mitigate field operatives-vegetation relationships

J o u r n a l P r e -p r o o f

We found that field operatives had mostly positive attitudes toward spontaneous vegetation on street pavements. Based on our results and literature, we believed these attitudes would be bolstered if this vegetation became integrated into design interventions (Bonthoux et al., 2019b;[START_REF] J O U R N A L P R E -P R O O F Li | Cues to care: A systematic analytical review[END_REF], and if such landscaping does not entail extra work time and does not impede the passage of cleaning tools. These relationships to vegetation are nevertheless directly and indirectly modulated by several psychosocial, organizational, and technical parameters. Those parameters must be considered to understand and transform the management system.

In our results, two kinds of psychosocial relationships are important for field operatives.

First, it is important for them to have clear objectives that accord with their perceptions of work.

Thus, they understand the meaning of what they do and can take pride and involvement in their activities [START_REF] Ellickson | Determinants of job satisfaction of municipal government employees[END_REF]. In the context of a changing management strategy, these elements seem to be of primary importance in facilitating the acceptance of new practices. Here, field operatives' perception of spontaneous vegetation is changing (Fig. 2), switching from a sign of insufficient management to an enrichment that should not be associated with dirt. Therefore, spontaneous vegetation is still not in conflict with their proud to make the city clean. Second, daily relationships with inhabitants are very important to field operatives, as found in another context [START_REF] Hoyle | Not in their front yard‖ The opportunities and challenges of introducing perennial urban meadows: A local authority stakeholder perspective[END_REF]. These relationships can enhance their work experience. However, they can also receive complaints about spontaneous vegetation, which is perceived as a lack of maintenance by some inhabitants. As our interviews with pedestrians show little opposition to spontaneous vegetation presence, maybe these complaints are expressed by a vocal minority of inhabitants who do not represents the overall attitude of urban population (Bonthoux et al., 2019b;[START_REF] Drijfhout | Mind the gap: Comparing expert and public opinions on managing overabundant koalas[END_REF]. These complaints may skew the municipality's understanding of inhabitants' opinion, impeding implementation of new practices. Thus, field operatives consider municipality communication of ecological objectives to inhabitants as a key factor in the implementation of alternative management practices [START_REF] Hoyle | Not in their front yard‖ The opportunities and challenges of introducing perennial urban meadows: A local authority stakeholder perspective[END_REF].

J o u r n a l P r e -p r o o f

The organization of the management system was also a structuring parameter for the wellbeing of field operatives and the effectiveness of their practices, as found elsewhere [START_REF] Jansson | The governance of landscape management: New approaches to urban open space development[END_REF][START_REF] Ordóñez | Urban forest governance and decision-making: A systematic review and synthesis of the perspectives of municipal managers[END_REF]. In our case, field operatives of the cleanliness service reclaimed more interactions with the green spaces service and between directors' service and field operatives, for a better consistency in task allocation. Some also mentioned a lack of dialogue with elected officials who are sometimes perceived as distant from practical and day-to-day concerns. Discussions should be engaged with the different levels of services to facilitate collective and individual appropriation of the changes in vision and practices. Collaboration and communication from elected officials and service director to field operatives and vice versa (Fig. 2), can facilitate the transfer of clear management policies while also drawing to attention to any technical constraints and ideas identified in the field [START_REF] Mathers | Place-keeping in action: Evaluating the capacity of green space partnerships in England[END_REF][START_REF] Hoyle | Not in their front yard‖ The opportunities and challenges of introducing perennial urban meadows: A local authority stakeholder perspective[END_REF][START_REF] Muratet | Understanding stakeholder perceptions of acceptability and feasibility of formal and informal planting in sheffield's district parks[END_REF].

Finally, the practical and technical aspects of work are also important for field operatives' well-being and acceptation of spontaneous vegetation. These elements must be considered along with management strategies. Tools are of particular importance: they determined how field operatives defined themselves, they provide a way to interact positively with inhabitants and they directly modulate the acceptance of vegetation by the agents by making weeding more or less difficult. This result is consistent with the findings of Nam & [START_REF] Muratet | Understanding stakeholder perceptions of acceptability and feasibility of formal and informal planting in sheffield's district parks[END_REF] showing that the acceptability of vegetated facilities by managers is linked to the perceived effort required to maintain it. Surprisingly, the funding parameter has never been discussed in focus groups, whereas in other studies the need to lower management-related costs is prevalent [START_REF] Barnes | Public land manager discourses on barriers and opportunities for a transition to low input turfgrass in J o u r n a l P r e -p r o o f urban areas[END_REF][START_REF] Ramer | Applying ‗action situation' concepts to public land managers' perceptions of flowering bee lawns in urban parks[END_REF]. In Blois, while the internalization of weed control in 2012 increased the costs of acquiring new equipment, the service director now believes that, in addition to the accrued ecological benefits, leaving spontaneous vegetation in place limits the costs of equipment and supplies (e.g., gas), a realization documented elsewhere [START_REF] Hoyle | Not in their front yard‖ The opportunities and challenges of introducing perennial urban meadows: A local authority stakeholder perspective[END_REF]. Accordingly, a portion of these saved funds could instead be used for public communication strategies, long-term ecological and social monitoring, and place-keeping of plant amenities [START_REF] Dempsey | Defining place-keeping: The long-term management of public J o u r n a l P r e -p r o o f spaces[END_REF].

J o u r n a l P r e -p r o o f

As amenities on pavement could complex vegetation management and increase vegetation acceptance by city dwellers, the long-term management of those amenities is on particular importance for acceptation of new practices.

Impact of communication on pedestrians' perceptions

More than half of the pedestrians held a positive to moderate opinion of spontaneous street vegetation and a third expressed preferences for non-weeded streets. These results confirm previous photomontage urban studies [START_REF] Fischer | Beyond green: Broad support for biodiversity in multicultural European cities[END_REF]Bonthoux et al., 2019b). Pedestrians mentioned that their main motive for retaining spontaneous vegetation in streets was for supplying ecological resources, as habitats or food for animals. Reducing the drudgery of work for field operatives was much less cited, likely because interactions between pedestrians and field operatives are modest.

Many pedestrians did not notice the spontaneous vegetation or did not express an opinion. This is consistent with the plant blindness concept, which posits that people do not notice plants in their environment. The fact that vegetation is not in any special amenities, and growing on pavement, which is an impervious element not normally considered as a space for plants' presence, could augment plant blindness outside traditional urban green spaces [START_REF] Balding | Plant blindness and the implications for plant conservation[END_REF][START_REF] Thomas | Plant blindness and sustainability[END_REF].

The presence of stickers as educational element near the vegetation was well received by pedestrians though its impact on changing perceptions was moderate. This in situ information enabled the differing species to be highlighted and underlines their ecological role in cities.

Nevertheless, few respondents said this information would change their attitudes toward plants.

These results echo the critics of the -knowledge deficit‖ approach [START_REF] Kidd | Messaging matters: A systematic review of the conservation messaging literature[END_REF]: providing city dwellers with knowledge about vegetation is only the first step in changing attitudes toward it.

Knowledge increase acceptability of biodiversity and biodiversity-friendly measures, but direct interactions with nature and plants, through time and various activities in urban green spaces for example, are also necessary [START_REF] Fischer | Public attitudes toward biodiversityfriendly greenspace management in Europe[END_REF][START_REF] Lumber | Beyond knowing nature: Contact, emotion, compassion, meaning, and beauty are pathways to nature connection[END_REF]. Then, we hypothesize that a J o u r n a l P r e -p r o o f diverse suite of actions in tandem, from installing local cues to care to designing new urban forms and precipitating changes in social norms on a larger scale, are needed to accompany changes in attitudes towards more acceptability of spontaneous vegetation and alternative managements.

Towards a place-based integrative approach to transform peoplenature relationships

This study was conducted in only one medium city and other future studies are required to evaluate the generality versus singularity of the results. Still, we think that the approach proposed here can be generalized to link ecological, social, and technical issues. This study shows that the municipal scale is relevant for developing integrated strategies for biodiversity conservation, linking national policies with local visions and applications. This has already been demonstrated as possible in the context of climate change adaptation [START_REF] Gustafsson | Municipalities as intermediaries for the design and local J o u r n a l P r e -p r o o f implementation of climate visions[END_REF]. Inspired by existing models [START_REF] Dempsey | Defining place-keeping: The long-term management of public J o u r n a l P r e -p r o o f spaces[END_REF][START_REF] Jansson | The governance of landscape management: New approaches to urban open space development[END_REF], we collected and brought together different pieces of information into a single framework linking spontaneous vegetation in streets with the management system and city dwellers (Fig. 2). The combination of different methods (semistructured interview with services managers, focus groups with field operatives, questionnaires with pedestrians) allows us to take in account the different reality of those groups, and to combine them to understand their interaction as a whole. In this system the ecological component, the consideration of city dwellers' well-being and workers' conditions are analyzed simultaneously, to move towards a biodiversity-friendly and a socially-valued future [START_REF] Rosenzweig | Win-win ecology: How the earth's species can survive in the midst of human enterprise[END_REF][START_REF] Francis | Urban reconciliation ecology: The potential of living roofs and walls[END_REF]. As integrated landscape approaches are growing up, we nevertheless emphasize the importance of finely qualifying the biodiversity of places to identify the paramount ecological factors and issues at play (Bonthoux et al., 2019b;[START_REF] Reed | Reintegrating ecology into integrated landscape approaches[END_REF]. As we found in this study, the relationships in this system are dynamic and can evolve quickly in response to internal and external factors (Fig. 2). Importantly, researchers can adopt different postures in the evolution of this system.

That is, they can be simple observers describing and analyzing the complexity of the relationships.

J o u r n a l P r e -p r o o f

On the other hand, maybe they should be more experimentally involved, furthering changes to the system's dynamics by designing actions whose socio-ecological repercussions will then be monitored.

Conclusion

In this study we used complementary methods to analyze the relationship between municipal managers, city dwellers, and spontaneous vegetation in streets. We show that even when municipal strategies move towards a biodiversity-friendly strategy, the organization of the management system at all hierarchical levels must be explicitly considered to engage in a shared and collaborative transition to a greener city. Indeed, while field operatives are rather well disposed to spontaneous vegetation, their well-being and practices and the drudgery of their daily work all depend heavily upon psychological, technical, social and organizational parameters that should be aligned with ecological objectives. The majority of city dwellers harbor a positive view of spontaneous vegetation, albeit a large proportion of them seem indifferent to it. The provision of ecological information in situ moderately improves their perceptions of these plants. This educational approach of providing knowledge must be supplemented by other approaches (e.g., testing of new ecological arrangements, regulations) to amplify the overall net effect on changes to attitudes of city dwellers. These different results are valuable, as they could inform the construction of an urban socio-ecological model and help in finding robust social and organizational levers to improve people-nature relationships in cities. The results need to be validated by other studies in towns in

France and around the world in order to try to change the mentality about vegetation on streets.

Figure 1 .

 1 Figure 1. On the left, photographs of the experiment: [clockwise from top-left] one of the A5

  the focus groups (27 field operatives). The five teams were interviewed during three consecutive days in May 2021. Each focus group lasted 2 h; the number of field operatives in each group varied from 3 to 8. Following the recommendation of Nyumba et al. (2018), we established dual moderator J o u r n a l P r e -p r o o f focus groups, with [Author 1] and [Author 4]. The focus groups took place in the streets or in field operatives' facilities, depending on weather condition.

  J o u r n a l P r e -p r o o f From 3 May through 1 June 2021, during the growing season of the vegetation, pedestrians on these streets were interviewed by [Author 1], [Author 4], and [Author 2]. Each interview lasted ca. 10 min, and included a mixture of closed-and open-ended questions. Different themes were addressed: the pedestrians' perception of spontaneous vegetation in the street, their opinion about the accompanying stickers, and their opinion on the reducing of weeding (Appendix 4). Interviews finished with socio-demographic questions.

  complimented and thanked them during their work in streets, contributing to the field operatives' well-being: "[We're] happy with the positive feedback from people, it's a personal reward. We like it when people express their satisfaction, thank us […] it's good for those who work." V2 Inhabitants are also curious about the field operatives' tools: "[Inhabitants] are attracted to the tools, [the thermal burner]. They look, are interested, always nice.‖ CS2

"

  Vegetation is not a pollution [whereas abandoned] papers are." O5 "Droppings and litter garbage cans are dirty, not weeds." CE6 "Flowers are an enrichment, they are not a neglect." N2 Five field operatives stated that spontaneous vegetation betrayed a lack of management: J o u r n a l P r e -p r o o f "Weeds, herbs don't look clean." V3 "It lacks maintenance." N3

  tolerant towards street vegetation by providing knowledge about plants: -When you know the plants, you change the opinions. Developing more communication will change opinions. I agree for more communication.‖ V4 "Have more information in the local newspaper, on social networks.‖ CE1

Table 2

 2 Yes, we can see that the herbs are present, it appeals but is not disturbing‖; -More beautiful, we must not destroy, more beautiful than other streets‖ For their ecological interest, to provide resources for animal species: 43 in agreement To reduce management by technical staff, improve their working condition: 23 in agreement Results of pedestrians' interviews (68 respondents). Answers were classified into different categories and illustrated by statements of respondents. Answers beyond the scope of the question are excluded from the count.

	Theme	Question	Answer	Example statement(s)
			categories and
			number of
			respondents*
	Perception of	What is your	Generally	-I love it, I wish there was more of it‖; -Very nice,
	spontaneous	opinion about	positive: 30	positive, but what effect on the surface in the long
	vegetation	this		term?‖
		vegetation?	Balanced	-It doesn't look super clean, but it looks green.
			opinion: 10	Between two‖
			Generally
			negative: 17

-It grows a little bit randomly, there is no maintenance‖; -I'd rather even plant hollyhocks‖

J o u r n a l P r e -p r o o f

No opinion: 6 -I don't pay too much attention to it, I never asked myself the question‖

Authors contribution

Visualization, Supervision

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Highlights

 We studied field operatives and pedestrians' attitudes about street spontaneous vegetation.

 Work conditions and interactions with inhabitants structure field operatives' practices.

 Majority of pedestrians are not against spontaneous vegetation in streets.  Ecological information moderately improves pedestrians' attitudes.

 Integrative approaches are needed to transform people-nature relationships.

J o u r n a l P r e -p r o o f


Influence of the solar wind dynamic pressure on the ion precipitation: MAVEN observations and simulation results.

Introduction

Although atmospheric sputtering is a minor component of atmospheric escape today, it is thought to have been much more important four billion years ago [3]. Heavy ion precipitation is the primary driver of atmospheric sputtering. At the present epoch, the efficiency of Mars' atmospheric sputtering by precipitating heavy ions to induce atmospheric escape is expected to be small compared to other mechanisms of atmospheric erosion. However, since the main driver of sputtering is ion precipitation, it is crucial to constrain the dependence of the precipitating ion flux on present solar wind conditions, before any extrapolation to past solar conditions. By comparing simulation results and MAVEN observations, we here investigate the mechanisms controlling the precipitation when the solar wind dynamic pressure change.

We will present how the precipitating ion flux, measured by MAVEN/SWIA, is influenced by the solar wind dynamic pressure and will analyze these observations by comparison with simulation results.

Observations and simulations results

We define two sets of different solar wind dynamic pressure from the set of MAVEN observations of the precipitating flux and simulate Mars' interaction with the solar wind for the average values of the solar parameters (Extreme Ultraviolet irradiance, Interplanetary magnetic field, solar wind density, solar wind speed...) for both sets. We then reconstruct map of the precipitating heavy ion flux at 250km in altitude and the simulated precipitating flux along each MAVEN trajectory used in our analysis.

Summary

Comparing MAVEN observations with models improves our understanding of the parameters that control the precipitating ion flux. By defining two sets, characterized by different solar wind dynamic pressure and modelling them, we present the comparison between models and observations.

Acknowledgements

This work was supported by the DIM ACAV and the ESA/ESTEC faculty. This work was also supported by CNES "Système Solaire" program and by the "Programme National de Planétologie" and "Programme National Soleil-Terre". This work is also part of HELIOSARES Project supported by the ANR (ANR-09-BLAN-0223), ANR MARMITE (ANR-13-BS05-0012-02) and ANR TEMPETE (ANR-17-CE31-0016). Spacecraft data used in this paper are archived and available in the Planetary Data System Archive (https://pds.nasa.gov/). Numerical simulation results used in this article can be found in the simulation database (http://impex.latmos.ipsl.fr).

[1] Leblanc F., R. Modolo and al. (2015), Geophys. Res. Lett, 42, 9135-9141, doi : 10.1002/2015GL066170.


Etienne Ménager

Alexandre Bilger

Wilson Jallet

Justin Carpentier

Christian Duriez

Condensed semi-implicit dynamics for trajectory optimization in soft robotics

Keywords: Optimization and Optimal Control. Modeling, Control, and Learning for Soft Robots. Soft Robot Applications

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

I. INTRODUCTION

In rigid robotics and humanoid robotics in particular, optimal control is an efficient way to formalize and solve multi-time step control problems [START_REF] Diehl | Fast direct multiple shooting algorithms for optimal robot control[END_REF], [START_REF] Schultz | Modeling and optimal control of human-like running[END_REF], [START_REF] Jallet | Implicit Differential Dynamic Programming[END_REF]. The general idea of this method is to minimize a cost function over a time horizon while satisfying the system dynamics and various constraints, such as actuation constraints. Many methods exist to solve this kind of problem. Among them, Differential Dynamic Programming-based methods [START_REF] Jacobson | Differential dynamic programming. modern analytic and computational methods in science and mathematics[END_REF], [START_REF] Tassa | Synthesis and stabilization of complex behaviors through online trajectory optimization[END_REF] are often used because they efficiently exploit the temporal structure of the problem and satisfy the dynamics of the robot for every iterate of the method. Recent variations of this method [START_REF] Lantoine | A hybrid differential dynamic programming algorithm for constrained optimal control problems. part 2: Application[END_REF], [START_REF] Xie | Differential dynamic programming with nonlinear constraints[END_REF], [START_REF] Pavlov | Interior point differential dynamic programming[END_REF], [START_REF] Jallet | Implicit Differential Dynamic Programming[END_REF], [START_REF] Jallet | Constrained differential dynamic programming: A primal-dual augmented lagrangian approach[END_REF] allow accounting for additional constraints related to the actuation or system state.

In soft robotics, Model Predictive Control (MPC) models have been studied in previous works and are used for multistep control. MPC is a model-based control method that predicts future trajectories and realizes online planning using this prediction horizon. However, in [START_REF] Lopez | Dynamic tube mpc for nonlinear systems[END_REF], [START_REF] Hewing | Recursively feasible stochastic model predictive control using indirect feedback[END_REF], [START_REF] Best | Model predictive control for pneumatically actuated soft robots[END_REF], the robots are controlled in the joint space, which limits their applications. In [START_REF] Spinelli | A unified and modular model predictive control framework for soft continuum manipulators under internal and external constraints[END_REF], the robot is controlled in the task space, but the problem formulation is based on the Piece-Wise Constant Curvature (PWCC) assumption, which limits the type of robot or actuation that can be used. Finite Element Methods (FEM) is a good candidate for accurate simulation 1 Univ. Lille, Inria, CNRS, Centrale Lille, UMR 9189 CRIStAL, F-59000 Lille, France. 2 Inria, Département d'informatique de l'ENS, Ecole normale supérieur, CNRS, PSL Research University, Paris, France. 3 LAAS-CNRS, 7 av. du Colonel Roche, 31400 Toulouse. * Corresponding author: etienne.menager@inria.fr of a wide range of geometries and actuations. However, FEM simulation is based on the discretization of continuous mechanics and represents the soft robot as a mesh. Such a representation is unsuitable for large optimization problems such as multistep predictive control, because it involves manipulating high-dimensional mathematical quantities. Some learned-based methods can be used to speed up simulation and write MPC problems for soft robots. In [START_REF] Alora | Data-driven spectral submanifold reduction for nonlinear optimal control of high-dimensional robots[END_REF], authors attempt to capture the dynamics of the system using a spectral submanifold. However, the method uses an approximation of the dynamics, and the applications are limited to robots with equilibrium points. Other works such as [START_REF] Tonkens | Soft robot optimal control via reduced order finite element models[END_REF] use Reduced Order Finite Element Models to implement optimal control methods. The model allows achieving computational efficiency but is based on the construction of a reduced order piecewise affine model to approximate the FEM. To the best of the authors' knowledge, there is no efficient optimal control method based on the FEM modeling of a soft robot.

Contributions: Inspired by the current one-step control in soft robotics and the trajectory optimization tools developed in rigid robotics, we propose a formulation of the trajectory optimization problem in soft robotics based on a condensed semi-implicit dynamics. The general idea of the method is to use the high-dimensional FEM model of the robot to simulate its behavior and a low-dimensional linearized condensed dynamics to optimize the actuation. This method is tested in simulation on three robots with different geometries and actuation.

II. BACKGROUND A. FEM simulation of soft robot

The behavior of soft robots can be described via the mechanical equations of continuous media. FEM simulation software is used to obtain an approximate solution of these equations [START_REF] Duriez | Control of elastic soft robots based on real-time finite element method[END_REF], [START_REF] Faure | Sofa: A multi-model framework for interactive physical simulation[END_REF]. They can be written as:

M qt = f int (q t , qt ) + f ext + H T a u ( 1
)
where M is the mass matrix, f int the internal forces, f ext the external forces, qt , qt , q t are respectively the acceleration, the velocity and position of the nodes of the FEM mesh, H a can be seen as the Jacobian matrix of the actuation and depends on the type of the actuator, and u is the Lagrange multiplier which represents the actuation constraints. The dynamics equation is discretized in time using an integration scheme. In soft robotics, an implicit Euler integration scheme is generally used because of its unconditional stability and its capacity to adapt to the "time-stepping" used in non-regular mechanics to deal with speed jumps on contact. In addition, internal forces are linearized using a first-order approximation. Using h the time step, q k = q kh and qk = qkh the discretized position and velocity, and d q = hq = qk+1 -qk , the discretized dynamics is written:

A = M -hD -h 2 K b(q k , qk ) = hf int (q k , qk ) + h 2 K qk + hf ext Ad q = b(q k , qk ) + hH T a u (2)
where K and D are respectively the compliance and the damping matrices.

The system Ad q = b + hH T a u is solved to find the next position and velocity of the FEM mesh points. First, the free configuration of the robot Ad qfree = b obtained with no actuation u = 0 is solved using a linear solver. Then, this value is corrected with the value of the actuation according to d q = d qfree + hA -1 H T a u. Finally, the position and velocity of all the mesh points are computed using the integration scheme.

B. One-step control in soft-robotics

Until now, the control of soft robots using FEM simulation is done on a time step using inverse modeling of the soft robot and a condensed description of its behavior [START_REF] Coevoet | Software toolkit for modeling, simulation, and control of soft robots[END_REF]. The movement of a soft robot is controlled using specific points called effectors. The concept of effector and actuator in soft robotics is illustrated in Figure 1, using the Diamond soft robot as an example (see Section IV)). Fig. 1. Example of the Diamond soft robot's actuation and end-effector, that will be developed in Section IV. The effector is a particular point on the robot that we want to control. Its position is modified by applying forces on the actuators.

To achieve position control of the soft robot, the idea is to minimize the distance between the effectors and goals:

δ e = p -p goal ( 3
)
where p is the position of the effectors. From this equation, the Jacobian matrix of the effector H e can be defined. It corresponds to the identity for the corresponding point in the mesh and 0 otherwise. As with the forward modeling described by equation ( 2), inverse modeling is solved in two steps. In the first step, the free configuration of the robot is computed. Then the violation of the constraints in the free configuration of the robot δ free i for i ∈ {a, e} is evaluated. Finally, this value is corrected using the actuation by making a linear approximation of the displacement of the nodes with a FEM interpolation. Using the condensed FEM modeling [START_REF] Coevoet | Optimization-based inverse model of soft robots, with contact handling[END_REF], this approximation is written:

δ e = W ea u + δ free e δ a = W aa u + δ free a (4) with W ij = H i A -1 H T j
for i, j ∈ {a, e} the projections of A in the constraint space. The matrices W ij represent the mechanical coupling between the effectors e and the actuators a. These quantities can be used in an optimization scheme to find the actuation that minimizes the distance δ e and respects some constraints on the actuator's displacement δ a .

C. Multi-step control in robotics

We consider the following numerical optimal control problem [START_REF] Gros | Numerical optimal control[END_REF]:

min x,u J(x, u) = N -1 k=0 l k (x k , u k ) + l N (x N ) s.t. x k+1 = f k (x k , u k ) x(0) = x0 (5)
with J the objective function, N the number of time steps,

x = [x 0 , ..., x N ] a sequence of states x k = [q k , qk ], u = [u 0 , ..., u N -1
] a sequence of control inputs, f k the discrete dynamics, l k the cost function at time k depending on the state and the actuation, and l N the terminal cost function depending on the final state.

There exist several methods [START_REF] Diehl | Fast direct multiple shooting algorithms for optimal robot control[END_REF] for solving this problem. In this article, we consider Differential Dynamic Programming (DDP) [START_REF] Mayne | Differential dynamic programming-a unified approach to the optimization of dynamic systems[END_REF]. DDP is based on the Bellman principle of optimality, which gives the following recursive formula:

V k (x) = min u l k (x, u) + V k+1 (f (x, u)) (6)
where V k is the value function at time k, with the terminal condition V N (x) = l N (x).

To solve this equation, DDP computes a Newton step δu in the control input u k + δu for a given variation x k + δx of the previous state x k during the backward pass. To do so, DDP relies on a quadratic approximation of the so-called Hamiltonian function, defined by:

Q k (x, u) = l k (x, u) + V k+1 (f (x, u))
This quadratic approximation around the nominal values (x, y) is given by:

Q k (x + δx, u + δu) -Q k (x, u) ≈ 1 2   1 δz δu   T   0 Q T x Q T u Q x Q xx Q xu Q u Q ux Q uu     1 δx δu   (7)
where the different quantities Q ij with i, j ∈ {x, u} are written as a function of the data of the problem, i.e., l, f and V k+1 and their first and second derivatives. As Q k is quadratic, V k is too, allowing the backward propagation of its value along the temporal structure of the problem starting from the end. We can add constraints to the equation [START_REF] Xie | Differential dynamic programming with nonlinear constraints[END_REF] and solve it thanks to augmented Lagrangianbased approaches [START_REF] Jallet | Constrained differential dynamic programming: A primal-dual augmented lagrangian approach[END_REF]. Once the backward step has been performed, i.e., once the δu sequence has been computed, it is possible to find the step size of the optimization process thanks to backtracking line-search procedure [START_REF] Tassa | Synthesis and stabilization of complex behaviors through online trajectory optimization[END_REF] and then perform the forward step. The forward step allows the calculation and evaluation of a new nominal trajectory from a given control sequence.

III. METHOD

A. Condensed dynamics in soft robotics

To solve multi-step control problem with DDP, we have to write the dynamics of the system using Equation ( 5) formalism. Equation ( 2) gives the general dynamics equation of a soft robot using an implicit Euler integration scheme with only one linearization of the internal forces per time step. By isolating d q and writing the time dependence of actuation u k , we can write a semi-implicit scheme:

d q = A -1 b(q k , qk ) + hA -1 H T a u k (8)
Equation ( 3) gives the relation between the effectors and the points of the mesh p = H e q. Equation ( 8) projected in the effector space can be written:

d ṗ = H e A -1 b(q k , qk ) + hH e A -1 H T a u k (9)
where we recognize the condensed FEM matrices in the right member of the equation. Using the implicit integration scheme, the condensed dynamics of the system can be written as:

ṗk+1 = ṗk + H e A -1 b(q k , qk ) + hW ea u k p k+1 = p k + h ṗk+1 (10)
The condensed dynamics of the soft robot can be written in matrix notation:

X k+1 = h 2 W ea,k hW ea,k u k + I hI 0 I X k + hI I H e A -1 b(q k , qk ) ⇔ X k+1 = f k (X k , u k ) (11)
where

X k = [p k , ṗk ] T .

B. Derivative of the condensed dynamic

As explained in section II-C, DDP uses the derivative of the dynamics to compute the Hamiltonian value. By differentiating Equation [START_REF] Hewing | Recursively feasible stochastic model predictive control using indirect feedback[END_REF] according to u k and X k we have for the control part:

∂f k ∂u = h 2 W ea,k hW ea,k (12)
and for the state using the chain rules and the fact that H e is its own pseudo-inverse:

∂f k ∂X = I hI 0 I ∂f k ∂X linear + H e A -1 0 0 H e A -1 h 2 K h 3 K + h 2 D hK h 2 K + hD H T e 0 0 H T e ∂f k ∂X non-linear ( 13
)
This quantity is composed of two terms: one corresponding to the derivative of the linear part of the dynamics (with respect to X k ) and the other one to the derivative of the non-linear part.

C. Constraint on the actuation

We can define additional mathematical constraints h k (X, u) ≤ 0 in the optimization problem. In this paper, these constraints correspond to actuator limits.

They can be expressed as:

h k (X, u) = h k (u) = -I I u + u min -u max ≤ 0 (14)

D. General method to solve the trajectory optimization problem

The function f k corresponds to the projection of the robot dynamics into the effectors/actuators space. As explained in section II-B, this projection has already been used for onetime step control. The idea is to use the condensed FEM model to find the actuation that minimizes the cost function around each step of the nominal trajectory.

The derivatives used for the optimization problem depend on the mechanical quantities W ea , A -1 , K, and D. Technically, these quantities depend on the position and velocity of all the mesh points of the robot. We assume that we are looking for an actuation around a nominal trajectory and that the mathematical quantities vary slightly around this trajectory. This means that these quantities are constant for one time step of the backward phase. Moreover, by improving the nominal trajectory with a succession of forward and backward steps, the error of the linear approximation is reduced (small displacement around the nominal trajectory) and there is little change in the values of W .

It is, therefore, necessary to compute a faithful nominal trajectory to obtain useful admittance, stiffness, and damping values for the optimization problem. To do so, the forward phase uses the complete dynamics of the robot, and the simulation is used to compute the next values of W ea , A -1 , K, and D for the backward phase. When only the derivative of the linear part of the dynamics is used in the backward phase, only W ea is recovered from the simulation.

The global method is summarized in Figure 2. Trajectory optimization is achieved through several forward and backward phases. At each iteration of the optimization process, the actuation sequence is tested with the complete FEM Fig. 2. Illustration of the trajectory optimization pipeline. Starting from an initial position, the forward phase computes the different positions and velocities of the complete mesh using the full dynamics (in orange). From the simulation data, a nominal trajectory is extracted, and the mechanical quantities necessary for the DDP are saved (in blue). The nominal trajectory (green) is used with the condensed trajectory (red) to compute the new action sequence that will be given to the forward pass. Several forward and backward steps may be necessary to converge to a solution. model during the forward phase and a new nominal trajectory is computed. Mechanical quantities are extracted from this nominal trajectory to create the condensed FEM model. This model is used in the backward phase to calculate a new actuation sequence that minimizes the cost function.

IV. MATERIALS AND RESULTS

A. Soft robots

We propose to illustrate the method using three different robots. These robots have different geometry and/or actuation and are simulated using the FEM model based on SOFA framework [START_REF] Faure | Sofa: A multi-model framework for interactive physical simulation[END_REF].

• The Diamond robot is a soft parallel silicone robot actuated by 4 cables. The FEM model of this robot has been validated in [START_REF] Duriez | Control of elastic soft robots based on real-time finite element method[END_REF]. • The Stiff-Flop robot is a slender soft robot actuated by pneumatic cavities. Each section of the two sections of the robot is composed of three cavities, allowing the bending in two directions. The FEM model of this robot has been validated in [START_REF] Chaillou | Reduced finite element modelling and closedloop control of pneumatic-driven soft continuum robots[END_REF]. • The Liandford robot is a soft parallel robot with 6 deformable legs developed for haptic applications. Each end of the legs is attached to a DC motor, whose rotation causes the legs to deform. The effector is attached to a central bar, whose movement is imposed by the deformation of the legs. Figure 3 shows these different robots in simulation.

B. Trajectory optimization

The trajectory optimization is performed using the Prox-DDP solver [START_REF] Jallet | Implicit Differential Dynamic Programming[END_REF], [START_REF] Jallet | Constrained differential dynamic programming: A primal-dual augmented lagrangian approach[END_REF] implemented within the ALIGATOR library [START_REF] Jallet | PROXDDP: Proximal Constrained Trajectory Optimization[END_REF]. This solver uses a primal-dual augmented Lagrangian approach to solve the optimization problem. The user interface allows for defining the underlying optimal control problem using a node-by-node basis, to write the dynamics and constraints, and to specify the differences between forward and backward stages. The forward step has been adapted in order to link with SOFA. Each time step in the simulation corresponds to 0.01 seconds. The different goals are selected according to the points that can be reached with one-time step-control approach. This allows to obtain scenarios with goals in the workspace of the robot. The code is available in Open Source on the GitHub page of the project. The results are presented for a fixed maximum number of iterations and tolerance.

1) Scenario 1: Use both derivatives of the linear and nonlinear part of the dynamic in the backward step: The first scenario compares the positioning performance obtained when the derivative of the non-linear part of the dynamics is used in the backward phase and when this is not the case. This comparison is illustrated with the Diamond robot and shown in Figure 4. The goal is to position the effector of the Diamond at two successive positions.

From these results, we can see that using condensed dynamics allows the control of the position and velocity of the robot, both with and without the derivative of the non-linear part of the condensed dynamics. The results in terms of positioning error are similar with and without the use of ∂fi ∂X non-linear . However, the number of iterations to reach the same performances is more critical with the nonlinear part (reach the maximum of 1000 iterations set for this experiment) rather than without (208 iterations). In addition, using the derivative of the non-linear part leads to additional calculations, as the resulting matrix is not calculated directly in the forward stage. More precisely, the matrix W is obtained with O(m 2 ) operations, where m is the number of constraints. The non-linear part of the derivative is obtained with O(m×n) operations, where n is the number of elements in the FEM mesh. n is a large number that depends on the accuracy of the FEM model. Therefore, considering only the linear part of the derivative of the dynamics according to X in the backward stage improves the calculation time.

This scenario shows that in the case of the Diamond robot, using the derivative of the linear part alone in the backward is sufficient to control the soft robot. One possible explanation of this result is that the full non-linear FEM model is used to find the nominal trajectory, and linearization around this nominal trajectory in the backward is sufficient to converge to a solution. According to this result and the computation time reduction, only the derivative of the linear part of the dynamics is used in the remainder of this work. The practical impact of this approximation for more general cases, such as highly non-linear materials, is left for future work.

2) Scenario 2: Constraint actuation with the Stiff-Flop robot: The second scenario shows that we can constrain the actuation during the trajectory. This scenario is illustrated using the Stiff-Flop robot. The results are shown in Figure 5. This robot illustrates the method with a different type of actuation. The objective is to position the end-effector attached to the center or the end-effector of the robot.

The results are shown for one axis and one actuator, but similar results can be plotted for the other axis and actuators. The framework enables to take into account constraints on the actuation and can integrate various kinds of geometry and actuation. Different deformations can be obtained depending on the position of the end-effector and the target. In practice, all the positions in the robot's workspace can be reached if enough time steps are available to solve the task.

C. Scenario 3: Multiple goals with the Liandford robot

The first two scenarios focus on positioning one effector with a maximum of two targets. In addition, the two previous robots are simulated with volumic FEM models, while the Liandford robot is simulated with a lattice of beam FEM elements. The third scenario involves optimizing a trajectory with multiple goals. We consider three different cases. The first is a spiral trajectory, where each time step corresponds to a new goal. The second is a pick-and-place task where the effector at the center of the robot has to move back and forth between two positions. The last is an orientation task using two effectors at the same time. The results are shown in Figure 6 and Figure 7.

The first task shows that the robot can be controlled over several time steps, with a goal that varies at each time step. The obtained trajectory corresponds to the one that minimizes the cost function and follows the constraints at the tolerance level set. There are a few differences between the target trajectory and the computed trajectory: the target trajectory only takes into account the positioning of the effector, whereas the computed trajectory minimizes an objective that includes not only the position but also the velocity of the effector and the actuation value. Changing the weight of these elements in the loss leads to different results. Here, the emphasis is on positioning. The second task shows that it is possible to control the robot's dynamics on more complex trajectories, such as tasks where it has to navigate between two different positions. Finally, the third task shows that it is possible to control several effectors simultaneously.

V. DISCUSSION AND CONCLUSION

We have proposed a method for optimizing the trajectory of soft robots using dynamic FEM models. This method is based on the joint use of a full FEM model to obtain an accurate nominal trajectory in the forward phase and the use of a condensed FEM model to find the actuation in the backward phase. This method can handle several types of robot geometry (parallel or slender), several types of FEM elements (tetra, hexa, beam), several types of actuation (cable, pneumatic, joint), and several types of task (one goal, several goals, one trajectory). We have used three different examples to illustrate all these features.

For the moment, we realize trajectory optimization and not an interactive control. The trajectory is optimized during an offline phase and then applied to the robot in an open loop. Future work will be to mix this method with a correction loop to obtain a robust controller of the soft robots. In addition, the method is based on the simulation of all the FEM models during the forward phases, which is a bottleneck for realtime simulation. The simulation time depends on the length of the trajectory and the size of the FEM mesh. One way to overcome this limitation is to use model reduction or a learned model in the forward phase. The advantage of this method over previous ones is that the optimization does not depend on the choice of model used for the simulation, the condensation being valid for FEM models, reduced models, or beam models. The only criterion that the forward model must satisfy is to provide the mechanical quantities needed to build the condensed model.

The fact that only the derivative of the linear part of the dynamics is used in the backward step means that no additional computations are required to be compared with the forward step and speed up the convergence of the method. The Diamond example showed that the two approaches lead to similar results. Using the derivative of the non-linear part of the dynamics with the other robots didn't yield the same results as those obtained with just the linear part, with the same number of iterations. To achieve the same performance, some additional work on algorithmic performance and computation is required. However, the linear part alone allows the algorithm to converge on an optimal solution to our problem. In addition, the dynamics used in the forward stage and the linearized condensed dynamics used in the backward stage are not the same. This leads to different gradients between those calculated from the position of the end-effectors and those calculated from the condensed model. This can lead to the framework's dual error not converging, although in practice, this does not affect the control of the robots.

It is also possible to extend the capabilities of the method, in particular by adding different constraints. For example, it would be possible to imagine constraints on the maximum displacement increment of actuators or constraints on the position of the end effector by defining zones where it is not allowed to go. Another extension would be to consider perturbations in the trajectory. This would enable the development of tasks where a force is applied to the robot at a given point in the trajectory. Finally, we are using a line search approach to update the variables, but other methods, such as trust region approaches, could be considered.

Fig. 3 .

 3 Fig. 3. Representation of the robots used in this work, simulated using SOFA. 1) The Diamond robot. 2) The Stiff-Flop robot. 3) The Liandford robot.

Fig. 4 .

 4 Fig. 4. Results obtained for Diamond robot control. End effector position along x (A), y (B), and z (C) axes. End effector speed along x (D), y (E), and z (F) axes. Green dot: the objective. Blue cross: results obtained by considering the derivative of the non-linear part of the condensed dynamics. Orange dot: results obtained by considering only the linear part. Three robot positions along the trajectory (1, 2, 3) are shown.

Fig. 5 .

 5 Fig. 5. Results obtained for the Stiff-Flop robot. First row: Effector at the middle of the robot. Second row: Effector at the end of the robot. Example of actuation (A-D); position of the effectors along one axis (B-E); velocity of the effectors along the same axis (C-F); three configurations of the robots (1, 2, 3). Green: the objective. Orange: position and velocity of the effector. Pink: actuation bounds.

Fig. 6 .

 6 Fig. 6. Results obtained for the Liandford robot control for the first case: position on the x-z plane for the spiral trajectory. Green: the objective. Orange: position of the effector on the considering plane.

Fig. 7 .

 7 Fig. 7. Results obtained for the Liandford robot control for the second and third cases. First row: actuation (A), position (B), velocity (C), and three configurations of the robot for the pick and place task. Green: the objective. Orange: position and velocity of the effector. Pink: actuation bounds. Second row: actuation (D), position (E), velocity (F), and three configurations of the robot for the orientation task. Green and Blue: the objective. Orange and Red: position and velocity of the effector. Pink: actuation bounds.

ACKNOWLEDGMENTS

This work was supported in part by the French government under the management of Agence Nationale de la Recherche through the project NIMBLE (ANR-22-CE33-0008) and as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute), and the Louis Vuitton ENS Chair on Artificial Intelligence. It was also supported by a grant overseen by the French National Research Agency (ANR) and France 2030 as part of the Organic Robotics Program (PEPR O2R). Finally, this work was supported by the TIRREX project, grant ANR-21-ESRE-0015.