Myriam Winance
email: myriam.winance@cnrs.fr

' From disruption to intimacy in relations with one's wheelchair. An analysis of relational modalities between persons and objects

Keywords: Prosthesis, embodiment, relationships, wheelchair, disability, materiality, attachment

'Don't touch/push me!' From disruption to intimacy in relations with one's wheelchair. An analysis of relational modalities between persons and objects

Myriam Winance

Introduction

The concept of prosthesis has been used by several authors to describe relationships between humans and technology [START_REF] Jain | The prosthetic imagination: Enabling and disabling the prosthesis trope[END_REF][START_REF] Smith | The prosthetic impulse: From a posthuman present to a biocultural future[END_REF][START_REF] Van Der Ploeg | Prosthetic bodies: the construction of the fetus and the couple as patients in reproductive technologies[END_REF]. In particular, it has been used to call attention to a change in the status of technical objects that we all use on a daily basis. Increasingly ubiquitous, these are considered an extension of the human body that augments the human subject's capabilities. For example, drawing both on publications -such as [START_REF] Sobchack | Living a 'Phantom Limb': On the Phenomenology of Bobily Integrity[END_REF]Nancy, 2010, etc.)-and her own research on heart transplants, Shildrick (2008[START_REF] Shildrick | Re-imagining Embodiment: Prostheses, Supplements and Boundaries[END_REF] shows that life with a prosthesis -internal or external, mechanical or organic -constitutes a process of bodily and subjective transformation of the person, which she describes as a process of hybridization. She uses the concept of assemblage for designating the 'incorporated self'. This is always the temporary and conditional result of a set of connections and relationships among different elements (technical, human, animal, material or cultural, etc.). Whether or not we are disabled, from the outset we are all made up of multiple relationships, of heterogeneous entities, technical objects, animals, data, etc. In other words, 'We are all always already prosthetic'. At stake in this expression is its emphasis on heterogeneous assemblages that define each person. It is used to emphasize the incompleteness of every human body and the necessary recourse to various other entities that, by transforming it, enable it to act or to acquire certain qualities. It is also used to emphasize that the body, far from being a distinct entity -circumscribed in particular by skin -is considered to be a process, unfinished from the start and always in a state of becoming.

Shildrick's work reflects the present use of a broad notion of prosthesis as designating any entity joined to the body and transforming its dis/abilities. However, as [START_REF] Dalibert | Exploring the Intimate Relations between Humans and Technologies[END_REF] has noted, this broad use of the concept of prosthesis prevents an understanding of the concrete realities of relationships that exist and take shape between humans and non-humans.

It focuses analysis on the relationship between the individual, their body and the object, with the presupposition that this relationship always transforms the embodied self. It thus avoids questioning the status of the non-human entity attached to the body, and more precisely, questioning the different statuses that this entity can acquire in the relationship: as a part of me, as not a part of me, as a part of the world. In other words, by using the term prosthesis, researchers fail to analyze the way in which the object and the person are relationally defined -that is, are defined by how they attach and detach from each other, what Latimer refers to with the key concept of 'relational extension' 1 [START_REF] Latimer | Being Alongside: Rethinking Relations amongst Different Kinds[END_REF][START_REF] Latimer | Keeping & dwelling: relational extension, the idea of home, and otherness[END_REF].

In this article, I re-examine the question of the correlative status of the entity and the person, defined by and in their relationships, which are at the same time relationships of attachment and detachment. The manners of attachment and detachment define relational modalities. In the first place, they define the nature of the entities, human or non-human, that make up each assemblage. They define their status: as part of me, as not part of me or as part of the world. In this sense, these relationships produce differences between the entities of an assemblage. Secondly, these relationships define the nature of assemblages and differentiate among them. I use the term 'entanglement' to refer to these relational modalities. The qualifier 'intimacy' allows a rethinking of the question of the incorporation of the object -in ways other than that of 'material incorporation' -into a whole that is the embodied self. It enables thinking about the different ways -material, emotional intellectual, etc. -in which an object transforms -or does not transform -who a person is. 'Intimacy', understood as an effect of relationships (Latimer and Lopez, 2019, p. 3) thus denotes the way in which a person lets themselves be affected -or not -in and by their relationship to the wheelchair. In this analysis of relational modalities, I draw on an ethnographic study of the use of the wheelchair.

From the outset, this is an ambivalent object, since it is in contact with the body while remaining detachable. This initial ambivalence renders visible the variation in relational modalities between the person and their chair. Finally, this analysis is based on a theoretical approach, drawn from Science and Technology Studies, which I develop in the following section.

From adjustment to entanglements

In order to explore the nature of relationships that are formed between persons and objects, and which define their status, I draw on a body of work from the field of Science and Technology Studies. This work has re-examined theories of action and, in relation to these theories, the correlative nature of objects and subjects [START_REF] Callon | Agency and the hybrid collectif[END_REF][START_REF] Latour | Technology is society made durable[END_REF][START_REF] Law | Networks, relations, cyborgs: on the social study of technology[END_REF]. As concerns action, it has been shown that it emerges through heterogeneous networks composed of humans and non-humans, hence the notion of Actor-Network-Theory, as well as that of attachment [START_REF] Gomart | A sociology of attachment: music amateurs, drug users[END_REF][START_REF] Hennion | Attachments, you say? How a concept collectively emerges in one research group[END_REF][START_REF] Latour | Factures/fractures. From the concept of network to the concept of attachment[END_REF]. The concept of attachment refers to the notion that agency materializes through an interweaving of relationships and interactions that all contribute to agency, which all 'make it happen' without knowledge of who does what, who acts or who is acted upon. Far from being defined beforehand by any essential nature, the objects and subjects, their status, but also their qualities, are produced by these attachments, which are constantly being performed. Some authors [START_REF] Akrich | Embodiment and Disembodiment in Childbirth Narratives[END_REF][START_REF] Berg | Introduction -Bodies on Trial: Performances and Politics in Medicine and Biology[END_REF][START_REF] Mol | The body multiple[END_REF][START_REF] Mol | Embodied Action, Enacted Bodies: the Example of Hypoglycaemia[END_REF][START_REF] Moreira | Self, agency and the surgical collective: detachment[END_REF], following this approach, have re-examined the notion of body.

Drawing on ethnographic studies, they suspend any a priori definition of the subject (as an autonomous or dependent subject, as a capable/incapable subject) and of the body (as an organism, a set of perceptions, etc.). They study how different embodied selves are constituted in practice, through specific and heterogeneous arrangements made up of actions, knowledge, techniques, places and experiences. In this work, embodiment is considered as a series of trials enabling (or not) the emergence of an embodied self that brings together different bodies. The concept of embodiment (and similar concepts such as those of performance or enactment) 'can be seen to point at the ongoing nature of the process: these are entities that have to be continually performed and maintained for them to persist' (Berg & Akrich, 2004, p.4). This work thus highlights the processes of learning, of being affected [START_REF] Latour | How to talk about the body? The normative dimension of science studies[END_REF], of adjustment [START_REF] Winance | Trying out wheelchair. The mutual shaping of people and devices through adjustment[END_REF] or even of bricolage [START_REF] Mol | Care in Practice. On tinkering in clinics, homes and farms[END_REF]) that underlie relationships between humans and non-humans, between things and bodies, and that define what they are. What the embodied self is -its qualities, its in/abilities [START_REF] Law | Managing, Subjectivities and Desires[END_REF][START_REF] Moser | Good passages, bad passages[END_REF][START_REF] Moser | A body that matters? The role of embodiment in the recomposition of life after a road traffic accident[END_REF], its in/ter/dependency and autonomy [START_REF] López | Embodying Autonomy in a Home Telecare Service[END_REF][START_REF] López | Little arrangements that matter. Rethinking autonomy-enabling innovations for later life[END_REF] -is defined, performed or enacted through this continuous process.

While I concur with this body of work 2 , in this article I shift my attention to examine not just the processes of adjustment that underlie relationships, but to look as well at those relationships as they take form at a certain moment and through which the status and qualities of the involved entities are defined. This will lead me to analyze the tension between two dimensions of the embodied self: simultaneously becoming and defined, unfinished and constituted, incomplete and complete. In other words, while the challenge is to pay attention to the temporal dynamic, emphasis is no longer on 'the becoming dimension' of this dynamic, but rather on 'what is defined through it'. I will then analyze the nature of the relationships formed between humans and non-humans entities that generate differences that are not only specificities, but which create distinctions between entities and between embodied selves. These distinctions are discernable in the ways people describe their wheelchair and describe themselves. After presenting my methodology, I focus on the various relational modalities that underlie the use of a wheelchair. In conclusion, I discuss the way this study transforms the analysis of the status of technical objects as prostheses.

Methodology

This article is based on an ethnographic study carried out between 2007 and 2009 among manual wheelchair users aged between 6 and 96 years old who had used a wheelchair at different points in their life and for various reasons (disease, accident, disability at birth, the aging process, etc.). The study consisted of observations of people trying out wheelchairs and of interviews with wheelchair users. The observations (15) were made at a wheelchair testing centre. The centre did not sell wheelchairs, but informed people about those available for sale, let them try out several wheelchairs and counselled them on their choice. The interviews (23 females and 32 males) were carried out with people met through the testing centre, through mutual acquaintances, and through retirement homes for elderly persons 3 . Among people we met, some had used a wheelchair for only a short time (less than 1 year), others for a very long time (up to 45 years). Place of residence was varied (city or country, private home or institution).

Content analysis was carried out using the body of observations and interviews in order to identify significant differences in the way wheelchairs were used and related to, but also differences in ways of talking about and perceiving the wheelchair. Above all, it was the contrast between two types of data -observations versus interviews -that made it possible to shift from an approach in terms of adjustment to an approach in terms of relational modalities.

These two types of data offered two contrasting entry points into relationships between the person and the wheelchair: their temporality and their spatial dimension. While the testing provided a 'snapshot', the interview described 'a life with'. Indeed, testing of wheelchairs appeared to be a critical moment. The wheelchair became the focal point bringing together the person using it, a professional who directed the trial, and at times a relative of the person or a professional who cared for them on a daily basis. During the trial, these actors examined the relationships between the person and the wheelchair and how these relationships defined the body and the environment of the person.

They attempted to redefine these relationships. This collective work was based both on perceptions of the person (exploration of the way they perceived themselves and their 'bodyin-this-wheelchair') as well as on a process of bricolage (of mutual transformation of the person, the wheelchair and their environment). This process, during which the two dimensions of embodiment were articulated -'having a body' and 'being one's body'allowed the person to feel at home in their wheelchair and to act in and with their wheelchair.

Testing of the wheelchair thus rendered highly visible the process of adjustment between the person, their body, the wheelchair and their environment -a process that defines each one of them.

While the observed testing appeared to be a critical moment for questioning the relationship between the person and their wheelchair, the interview with wheelchair users appeared to be a moment of reflection during which people recounted their experiences (on the role of interviews in making attachments perceptible, see Callen and López, this volume).

They explained how in daily life they managed to do this or that. These interviews were no longer about the process of adjustment between the person and their wheelchair, but more broadly about a process of organization through which the person sets up a mode of action by marshalling a series of resources that are of different types -objects, persons, knowledge, etc.

-and by arranging them in a way that makes action possible. This arrangement still assumes there to be a process of complementary adaptation of the person and their surroundings. But primarily, the interviews brought to light another dimension of attachment, not the indeterminate process but rather the nature and intensity of attachment that is created over time. This is because, while the trial is an instant, a moment in the life of a person, the interview reveals a long temporality, a duration. Consequently, the continual process of adjustment through which entanglements are defined no longer appears, but rather the entanglements themselves. Differences between users make it possible to identify modalities of 'being entangled', which may or may not lead to a form of intimacy, and which define different subjective configurations-some of whose characteristics are fixed -as well as different worlds. I have identified 5 relational modalities, which I describe below. For clarity of discussion, I chose to describe each modality from a typical example, even if several cases shared similarities.

The variable status of both the wheelchair and the person

An intruder that makes me feel 'non-human'

Madame Berger is quadriplegic following a road accident. She lives alone in her house, which was retrofitted after her accident. Her bedroom is on the ground floor in the former dining room. To get around in the house she uses an electric wheelchair that she controls with a highly sensitive joystick. For trips outside she is always accompanied and uses a manual wheelchair pushed by the accompanying person. This is how she described to me her experience of the wheelchair:

What strikes me most is the global and technological aspect … that is that it is very … well, the more technology advances, the more it increases the disability because I have the impression of being in a … you know, in a quad, in an immense thing … while I would have preferred a little manual, if it wasn't necessary to propel oneself … a little thing … when I see that, it's absolutely … it's not human … (interview, January 2008, Madame Berger, aged 50 years, road accident in 2005).

In this quotation, Madame Berger evoked the massive aspect of her electric wheelchair. One could interpret this remark as being related to the aesthetic dimension of the wheelchair.

However, the end of the quotation suggests a different analysis. Indeed, the way Madame Berger described her wheelchair and her situation -'it's not human' -is paradoxical, since her wheelchair is an object, a technology, and thus is indeed 'non-human', as defined by STS.

But what Madame Berger pointed out is the way in which an object, in certain situations and at particular moments, is felt to be non-human and transforms the person, what they are, their qualities. The wheelchair is no longer an object but something non-human. It is viewed as an intruder, making one feel a stranger to oneself. This feeling of intrusion caused by the relationship with another entity -and which remains another entity -is explained by Nancy when he describes his heart transplant: 'Once the [intruder/stranger] is there, if it remains a stranger, as long as it stays that way instead of simply "naturalizing itself", its visit never ceases. It continues to draw closer and it never stops being an intrusion in a way; that is, without rights and unfamiliar, without becoming intimate, and on the contrary being an inconvenience, a disruption of private life' (our translation) (Nancy, 2010, pp.11-12). This description of the wheelchair as 'non-human' is not an a priori definition, but results from experience and a temporal process. Madame Berger uses an electric wheelchair that she hastily chose during her stay in rehabilitation soon after the accident. This is her first wheelchair. But after using it, she realized it did not suit her. She finds it massive, and above all, she no longer feels safe since she has slipped out of the seat several times without noticing it until 'ending up with my legs on the floor' (interview, January 2008). In her case, there was no adjustment process that might have led to 'an intimacy, a familiarity'. On the contrary, a reverse process of distancing is taking place that, again quoting Nancy, 'disrupts privacy' and makes one feel and define oneself as disabled ('it increases the disability'). Paradoxically, the proximity of the wheelchair, which is in contact with the person, becomes a distance that denotes the difference in nature between the person and the object and defines the status of the person as 'disabled'. This process can also be observed among some family members of wheelchair users. For example, in reference to specially made pushchairs and manual wheelchairs suggested for her 8 year old boy with cerebral palsy, a mother told me 'these were things…really ugly…if they had put a sign over it: "warning, disabilities"…it would have been the same…I find that…I don't know why they make such ugly things' (observation, January 2008). The status of the wheelchair as 'non-human' affects the human status of the person, who becomes 'disabled'.

A tool that makes places accessible

Madame Debra has always had difficulty walking. She was born with undiagnosed hip dysplasia and not immediately treated, which gave rise to significant sequelae. Madame Debra walks over short distances using crutches. For the past 10 years, she has participated in monthly touristic outings organized by an association of disabled persons. She rapidly found these trips to be tiring and acquired a wheelchair, which she only uses during these outings.

During an interview, she told how she had chosen her wheelchair:

When I entered the pharmacy and said I wanted a wheelchair, the pharmacist told me 'ok, let's go into the other room'. It turned out that that wheelchair, which was for demonstration purposes, it suited me, as if by magic. So I decided not to look any further … and besides, I think it was the simplest one since it was not expensive; I didn't want a sophisticated wheelchair. I wanted one with solid tires and a reclining seat back. I didn't try to find out what colour it was; for me, what was important was that it had solid tires and a reclining seat back. And of course I wasn't going to take a wheelchair with a head rest or … it's just to help me out in case I'm tired when we go out.

[…] Because that chair there, it's called a wheelchair but it's like putting even just a chair for me, as long as I have something under my bottom, it doesn't matter what it's called. Except I'm not going to take my wicker chair, because I couldn't do anything because it doesn't have wheels. It's clear that I need a wheelchair with wheels.

(interview, September 2008, Madame Debra, 57 years old, hip dysplasia not treated at birth).

In Madame Debra's case, the relationship with the wheelchair is based on instrumental logic. The wheelchair is an instrument that allows her to participate in touristic outings without tiring since it facilitates her moving about thanks to 'its wheels'. The wheelchair is not 'a prosthesis attached to the person', but a tool for accessibility to the extent that it gives Madame Debra access to places that are not normally accessible to her way of getting about (her way of walking). In other words, use of the wheelchair depends on the place she visits and the manner of the visit (group visit, which assumes moving at the same pace as the others).

In line with this logic, the choice of the wheelchair was not problematic for Madame Debra. Moreover, there was not really a choice made since she bought the wheelchair that was on demonstration at the pharmacy, 'which suited her, as if by magic'. This logic was also seen in some of the wheelchair trials, often very short, during which the person wished to acquire the least expensive wheelchair possible, just to move about occasionally. For example, Madame Thin, aged 65 years, has polyarthritis. Like Madame Debra, she still walks and is looking for a wheelchair to facilitate outings with her husband. The trial led to the choice of a basic wheelchair with a reclining seat back, the only criterion enabling the occupational therapist overseeing the trial to select that wheelchair rather than another. Furthermore, after the trial, the occupational therapist explained to me that she could have suggested a whole series of standard wheelchairs to the lady without there being any single criterion to distinguish them.

When the wheelchair is described as a tool for accessibility, the adjustment process is minimal, almost non existent. Whether it was at the pharmacy for Madame Debra or during the trial for Madame Thin, there was little or no collective work needed to adapt the person to the wheelchair, and vice versa. On the other hand, the interview with Madame Debra revealed a process of adjustment [START_REF] Winance | From repair to enhancement: the use of technical aids in the field of disability[END_REF]. Indeed, Madame Debra decided to remove the foot rests, which she found to be too high, leading to an uncomfortable position for her legs. Without the foot rests, either she places her feet on the ground and uses them to push the wheelchair, or she holds her legs straight out in front when she is being pushed. In this way, she minimally adapted the wheelchair for her particular use. This process of minimal adjustment and of adaptation does not lead to a feeling of ownership for the wheelchair. The wheelchair is defined as a tool that remains outside and different from the person. When not using it, Madame Debra leaves her wheelchair in the garage situated in the courtyard of her building. The wheelchair never enters her apartment.

No intimate entanglement is created. The wheelchair is kept at a distance, as a separate entity and different from the person, unable to transform his or her qualities. This is what Madame Debra expressed by using the term in French 'situation de handicap' (literally 'disabling Thus, in this case, the wheelchair is a tool that the person uses. It facilitates some activities and extends their area of mobility. It becomes familiar to a certain extent, but in the end it is conceived of as an object that is part of their world, useful and separate, easily replaceable and replaced. The person does not let themselves be changed or affected by it.

The means for a functional transformation

Madame Chini, 60 years old, is hemiplegic following the rupture of an aneurysm. When I met her, she was in a period of recovery and had regular rehabilitation sessions. She could walk short distances on flat surfaces using a tripod cane, or by leaning on someone. Inside, she used an old manual wheelchair, especially convenient for sitting at the table. For outside trips, she used a manual wheelchair. When I asked her what the wheelchair meant to her, she answered: I'd say it was useful. A useful tool. It's indispensable for my progress to the extent that I can go out where I can see more things, so I think it is somehow necessary for recovery. I don't take it as a restriction, but on the contrary as a springboard. It's a springboard for my more…To go further. But having as a goal one day to do without it.[…] The wheelchair allows me to have a window on the world. To see other things. That's how I take it. (interview April 2008, Madame Chini, 60 years old, hemiplegic following the rupture of an aneurysm at age 60).

Following functional logic, the wheelchair is also considered as a tool, but its role is no longer solely one of providing access to a space, it is one of modifying bodily functions, the person's functional inabilities and abilities. As noted by Madame Chini, the wheelchair is an instrument useful for her progress, for gaining abilities. In other words, it is assimilated into a person's present and future transformation, in body and in identity. This logic manifests itself in wheelchair testing. During Madame Chini's session, the occupational therapist's attention was focused more on Madame Chini's mobility than on her comfort. The therapist focused on the way the wheelchair would or would not make Madame Chini mobile, thanks to its features (a system of double handrims on the left, her nonparalyzed side) and on its settings (especially lowering of the seat so she could touch the ground and thus use foot propulsion in addition to manual propulsion). A process of adjustment, exploration and bricolage takes place, which leads to a different body and brings about a transformation in the person. Dealing with a wheelchair thus makes it possible for them to think about another potential body since it brings to light a new dynamic articulating 'being one's body' and 'having a body'.

In this dynamic, the wheelchair remains external to the person, an object of the world, while being a means for regaining ownership of one's evolving body. By using their wheelchair, the person gains a new awareness of their different body and discovers its new dis/abilities. The wheelchair thus brings about a double transformation for the person: on the one hand, in the way they perceive their body and use it to act, and on the other, the way they perceive the world through their body. Thus, Madame Chini described how the sitting position has changed what she sees when, for example, she does her shopping (she doesn't see the same objects on the shelves) or has changed her relationships with children who are now on the same level as she. As she says: 'use of the wheelchair turns out to be a window on the world, a possibility to go towards…the outside'. In this functional logic the wheelchair, as external object, is what helps the person reconstruct themselves, re-establish ownership of their changed body, and redefine its qualities and abilities. It's the mediator of a bodily transformation.

In Madame Chini's case, this functional dynamic is part of a dynamic of progress but the former can also be found in a dynamic of regression. This is the case with some elderly For her, the wheelchair is part of a dynamic of loss of ability, a functional regression. This dynamic produces a 'loss of ownership' and 'alienation' in relation to her body. The person dissociates themselves from their body, which becomes a manipulated and passive body, and is experienced as such by the person.

Internalized as a way of being and doing

Monsieur Dos, 47 years old, has paraplegia following a motorbike accident when he was 17.

He has been using a wheelchair for 30 years. He participates in lots of disabled sports, especially tennis. He belonged to the national team when he was younger, and is still very involved in a club of adapted tennis. He owns 5 manual wheelchairs: a light town wheelchair that he uses when he travels outside and that stays in his car when he is home; 2 older manual wheelchairs that he uses at home and that he leaves outside in front of his home to facilitate transferring from wheelchair to car; 1 older wheelchair converted into a handbike; 1 manual sports wheelchair. In addition, he uses an electric wheelchair for verticalization and to tinker or work in his garden. When I asked him what the wheelchair represented for him, he answered:

It's my legs. We can say that…it's legs and then, it's like you, when you walk you take a good pair of shoes. Well me, to feel good I take a good wheelchair, it's the same thing. You have to sit in it all your life. You have to sit comfortably, to avoid having painful hips, or to hurt anywhere else. (interview July 2008, Monsieur Dos, 47 years old, paraplegia following a motorbike accident at age 17).

In this excerpt, we can see ambivalence concerning the status of the wheelchair, described by Monsieur Dos as both 'legs' and 'shoes'; on the one hand part of the body, on the other, an article of clothing, that is, an external object, different from the body. This ambivalence points to a process of adoption of the wheelchair; over time, it was assimilated by the person through a double process of adjustment and arrangement. Not only was there a process of correlative adaptation between Monsieur Dos, his body and the wheelchair, but a process of practical organization as well. That is, Monsieur Dos has set up ways of acting in a wheelchair by drawing on networks of heterogeneous entities. His in/abilities are made up of and emerge from these practical arrangements. But in Monsieur Dos' case, what is adopted is not so much the wheelchair, the object as such, as it is a certain savoir-faire and knowledge resulting from the handling of his wheelchair.

By operating his wheelchair, by adjusting to it, Monsieur Dos has acquired savoirfaire. Monsieur Dos uses several wheelchairs, depending on his different activities. Each wheelchair is adapted to Monsieur Dos, and especially to each one of his activities. This is the practical dimension of the wheelchair. This practical dimension reflects the ability of the person to act on a daily basis and to manage in various places, their ability to adapt when they are not in their usual environment. For example, Monsieur Dos often travels in groups, with other people in wheelchairs. Yet hotels rarely have more than two or three accessible rooms, so he is obliged to accept a standard room.

Monsieur Dos: We get by with nothing.

Sociologist: a room that is not necessarily accessible? Monsieur Dos: That's it. If we need to climb onto a chair to get to the bathroom, we put a chair in it, we put our wheelchair inside, we climb on a chair. These are things we do for a week, when we are on vacation. (July 2008, Monsieur Dos).

Thanks to this savoir-faire, Monsieur Dos can manage and act by marshalling the resources available in the environment. He has also acquired knowledge, that is, an understanding of the wheelchair, or more precisely, the wheelchair that suits him in terms of characteristics, settings, etc. Over the course of changing wheelchairs -but also at sporting events in which he has participated and which bring together large numbers of wheelchair users who compare their wheelchairs -he has formed an idea of what he needs. When he needs to change wheelchairs, he knows in advance exactly what wheelchair will be suitable for him: 'A salesman can never know how you are sitting. Everything depends as well on the size of your body' (interview, July 2008). While Monsieur Dos does not go to a testing centre when he changes wheelchairs, some users who fit this profile continue to do so. The trial is then very technical and essentially revolves around an examination of technical characteristics that distinguish wheelchairs (for example, the method of folding or nuances in the settings) and identify the one best adapted to the person and to which the person will best adapt.

This logic brings about a bodily transformation, but above all, an adoption of the wheelchair, an 'incorporation'. At the same time, what is incorporated is not the object itself, but the wheelchair as a form or mode of moving about and a way of acting. Monsieur Dos' case reveals a paradox. While there exists a strong attachment to the wheelchair as a way of being and acting, there's no attachment to a specific wheelchair, to a particular object. On the contrary, the savoir-faire and knowledge he has acquired enable him to easily change wheelchairs, to easily detach himself from any specific wheelchair. Indeed, these changes are very frequent in his case (every 2 years) as he makes rough use of his wheelchair and wears it out until it breaks down. The ambivalence in the status of the wheelchair, noted at the beginning of this section, is a result of this tension; the wheelchair is internalized and incorporated as a mode of being and of action, while remaining an object external to the in an armchair like that.' (interview March 2008, Monsieur Jon, 55 years old, left leg amputated following a motorbike accident at age 35).

In Monsieur Jon's case, there is not the ambivalence noted in quotes from Monsieur Dos. For Monsieur Jon, 'the wheelchair is part of him'. A process of adoption has taken place concerning not only the acquisition of savoir-faire and knowledge, but especially the wheelchair itself, that is, that particular wheelchair. Thus, Monsieur Jon doesn't often change wheelchairs (he has had 3 since his accident); he tries to make them last and takes care of them.

From time to time, once a month, I take it apart and clean it. [...] I take off the wheels and put a little oil or grease on so that it stays in good shape. We take care of it. It's vital for us. (March 2008, Monsieur Jon).

Following this logic, the adjustment process produces an incorporation of the wheelchair that is no longer an external object, but becomes a part of the person. It produces a material attachment to the wheelchair, incorporated as such. For this type of user, this often leads to the difficulty or even the impossibility of changing a wheelchair when it's worn out, and even more so when it's worn out since this wear is the result of a very advanced process of adjustment. In a sense, they have been bound together through this process. This may lead as well to a feeling of mutilation when the person, for one reason or another, is deprived of their wheelchair. For example, Mademoiselle Babet, who has paraplegia because of spina bifida, The wheelchair -as that particular wheelchair-becomes a constituent part of a person's bodily integrity, to the point that being deprived of it is experienced as an attack on bodily wholeness.

Conclusion

In this article, I have re-examined the question of the relationships between persons and technical objects. To this end, I have attempted to articulate an analysis in terms of adjustment with an analysis in terms of entanglement. Beyond the diversity of the hybrid assemblages that define every person, I identified relational modalities -ways of connecting and disconnecting oneself to the wheelchair -that correlatively define the status of the wheelchair and of the person. Describing these forms of entanglement, some of which define and redefine the 'intimacy' of the person, enables an understanding of the process of embodiment by emphasizing not only its processual dimension -as some STS approaches have done -but also what is produced through this, as described by the people involved. These approaches have deconstructed the autonomous subject and shown that their qualities and skills, their autonomy, are the result of the relationships of which they are constituted; they are emergent qualities of a hybrid assemblage. The concept of hybrid assemblage enables emphasizing the relational and heterogeneous nature of each embodied self. However, these approaches do not analyze how the nature of the entities is defined in these assemblages, producing differences not only between entities, but also between assemblages. As I have suggested in this article, the analysis of relational modalities between the person and their wheelchair allows us to do this and shows that the process of embodiment is not a process of hybridization that is infinitely adaptable. It is also a process of definition.

Certainly, we are all abled and disabled, and 'prosthetic', but what varies, or what distinguishes assemblages is the ways entities attach to and detach from each other. This dynamic defines relational modalities. I have described 5, perhaps there are others. These relational modalities do not correspond to types of users. Over time, a person can move from one to the other without following a predetermined order. These modalities are not either distributed over a continuum that goes from intrusion to being bound. But they do characterize 'ways of being together' that define what the wheelchair and the person are, and what 'the person in/with/far away from/etc. their chair' is 4 . They correspond to identifications/differentiations, more or less stable depending on people (on their history, on their other characteristics as age, type of impairments, etc.). They make it possible to account not only for the specificity of each 'hybrid assemblage', but also, and especially, for their differences, which are erased by the affirmation 'we are all dis/abled and prosthetic'. Indeed, the relational modalities that I described differ in several aspects. Their intensity varies and produces a greater or lesser distance between the person and their wheelchair as object. While the proximity between the person and the wheelchair can produce a form of intimacy from day to day, it can also, conversely, be perceived as an intrusion, distancing the person not only from the object that is the wheelchair, but also from themselves. These forms of entanglement offer various possibilities and impossibilities for action, and produce different forms of knowledge, both perceptions and reflective knowledge.

They also provide different ways and opportunities for becoming attached and of disengaging oneself. Above all, these relational modalities enable defining the status of the wheelchair on the one hand and the particular qualities of the person on the other. Firstly, while the wheelchair always transforms the dis/abilities of the person, this does not initially make it a prosthesis. Its status varies: it can be seen as part of me, as not part of me, or as part of the world. These statuses are not always exclusive of each other. Moreover, the separation between what is me, what is not me and what is part of the world does not correspond to a clear boundary between the entities, but rather to a set of attachments and detachments. The notion of intimate entanglement enables assigning these relational modalities to an entity, which in one form becomes me, while remaining both distinct and of a different nature. It enables modifying the process of incorporation of an object, which can connote an embodiment of the modes of acting and being or an incorporation of the material object as such.

Secondly, these forms of entanglement are linked to certain qualities. Relational modalities don't all have the same normalizing or 'ab/normalizing' potential. On this point, the contrast between Madame Berger and Monsieur Dos is of interest because they are, in a way, at the two extremes of a continuum. In the interview, Madame Berger said she 'feels disabled', while Monsieur Dos says 'he is athletic and self-sufficient'. Taking these selfperceptions and self-portraits seriously enables linking the analysis in terms of process with the analysis in terms of entanglement, and understanding how relative qualities can become what defines the person at a particular moment in their history and over a certain time.

Finally, this study opens interesting avenues of research for re-examining the question of time. This subject was approached by the STS in terms of dynamics, becoming, and process, focusing on the body as embodiment. However, time is not just a continuous flow, a sequence that produces a past, a present and a future. These dimensions are reworked by people in their narratives to define, to redefine, their bodily and social subjectivity. The relational modalities I have identified suggest that the way in which abilities and disabilities are fixed or modified results from a dynamic incarnation that associates a body as it has become with im/possibilities of transformation. This analysis invites us to reinstate the question of the narrative construction of the self as a constituent element of the reality of the person; in other words, that of the articulation between practice and discourse, between what the person does and what they say they are, between forms of entanglement and the definition of an identity.

Notes

1. Drawing on the work of Strathern, Latimer proposes this notion of 'relational extension as another way to think through being with non-humans that does not reduce human/nonhuman animal assemblages to hybrids.' (Latimer, 2013, p.98). It allows thinking about how, in the relationship between diverse entities, connection and unity are produced that preserve some form of division and alterity. Latimer also describes these relationships with the term 'being alongside', which enriches the term 'being with'. 2. My work on the political and scientific history of disability has however led me to underline the limits of these approaches and to suggest new directions of analysis [START_REF] Winance | Rethinking disability: Lessons from the past, questions for the future. Contributions and limits of the social model, the sociology of science and technology, and the ethics of care[END_REF], which I wanted to study further by again examining past research on the use of the wheelchair. 3. I wish to thank M. Pierret for conducting 16 interviews with elderly persons. I conducted the other ones. 4. At this stage in my thinking, the nature of these relational modalities partly remains a question. I thank one of the referees for having brought to my attention both the uncertainty of my text on this point and on possible options, as well as their risks. However, my work on Disability Studies has convinced me that one should not be satisfied with an idea that dilutes 'disability' in 'specificities', 'being' in 'continual becoming'. It's necessary to understand what assumes a pattern and defines any one person over time, without falling back into an essentialist and deterministic position. In other words, not all relationships between diverse entities are equivalent, but they produce 'relative differences which are essential'.

  situation') and by identifying herself as a 'falsely disabled person'. When I propel myself, I raise my feet a little and I put my legs in front; that surely makes people who see me laugh. They must wonder what…especially sometimes when they see me stand up. It even embarrasses me in relation to them because they must be thinking 'is that a falsely disabled person, or what?' In a way, I am a falsely disabled person, I'm a person 'in a disabling situation', but I'm not going to cup my hands and shout to them 'watch out, I'm going to stand up' (September 2008, Madame Debra).

  persons. Madame Moulin, 86 years old, talked about her wheelchair as follows: Sociologist: What does this wheelchair represent for you? Madame Moulin: What does it represent? I'll tell you that it represents what? My life now. No, that's not it…. I don't know. I can't tell you what it represents. I'm OK in it, now that I've gotten used to it. But to tell you what it represents, I'll tell you, it represents the end of my life. I can't do anything anymore, since I'm taken out of the wheelchair to go to the toilet; it's taken over, it's put in front and the wheels are blocked then someone grabs me under the arms. I stand and someone lowers my panties and then puts me on the toilet. And in the evening, then I come here, they take you up with the elevator, I come here, I take off my top and put on my nightgown. And then I go there and brush my teeth and wash my face. And then I wait. […] [A]nd then they put me to bed. There, that's my day! (interview, August 2009, Madame Moulin, 86 years old, in a retirement home).

  example, I can't move about and I don't have a wheelchair, if it's broken, I'm disgusting. I have a very aggressive attitude. If I can't get about and I have to stay sitting on a chair, even for an hour, I am not easy to get along with. Half my body has been taken away. (interview January 2008, Mademoiselle Babet, 44 years old, paraplegia following spina bifida).

Acknowledgements

I warmly thank Jon Cook for the translation and the editors, Joanna Latimer and Daniel Lopez, as well as the anonymous reviewers, for their challenging comments which still make me think.

Funding

This research was part of an inter-disciplinary project: 'Approches de la Biomécanique, de l'Intelligence Artificielle Distribuée et de la Sociologie pour la réalisation d'un Système d'Aide au Choix et au Réglages des fauteuils Roulants Manuels', co-ordinated by F.-X. Lepoutre and financed by the ANR (TecSan2006) from 2006 until 2010.

person. This is the origin of an intimate entanglement: the wheelchair, in its form, constitutes the bodily and subjective wholeness of the person, even though the object as such remains detachable. This is found in what Monsieur Dos said. When he talked about himself, he talked about 'his disability' as well as about 'his wheelchair' and asserted the fact of 'being autonomous'. His disability and his wheelchair are two of his characteristics that define his way of being, that define his intimacy, so much so that he can't stand for his wheelchair to be washed or to be pushed by someone.

Even my wife sometimes takes up the cleaning cloth. I tell her to leave the dirt on it.

You can't clean a wheelchair. I hate that. […] Sociologist: Does someone sometimes push you? Monsieur Dos: I hate that. Oh my! Sometimes my grandchildren come to the house, for fun, because they love to push me. But I hate that. For them it's a game to push me in my wheelchair, but I hate that. The day I can no longer get by alone, I'll stay at home. I could never stand that. The day when I arrived in a wheelchair I wanted to be independent; even though my mother and my relatives, they were there but I told them 'I don't need you, I'll manage, if I can't manage my life will soon be over'. Since I was

Bound to my wheelchair, like to a part of my body

Monsieur Jon, 55 years old, had a leg amputated following a motorbike accident when he was 35. He has a leg prosthesis. He can walk, either with his prosthesis, or with crutches. When he was working he moved about primarily using his prosthesis, but since his retirement, he mainly uses his wheelchair. Indeed, he has difficulty putting up with his prosthesis, which ended up giving him a stump wound when he wore it all day. Here is the way he talked about his wheelchair: 'You get used to it, the wheelchair is part of you, I couldn't get by without it anymore'.

And likewise, he explained to me he has become incapable of sitting in another chair: 'I already tried to sit in it [a living room armchair]…. Before my accident it was always my armchair for watching television. Now, I can't stand it anymore. I can't stay in it, I find it uncomfortable. I prefer to stay in this [in his wheelchair] when I watch TV with my wife; I sit to the side, and I prefer staying in it, I am a lot more comfortable in it than

Author biography

Myriam Winance is a sociologist at INSERM (National Institute of Health and Medical

Research). She is based at CERMES3 (Research Center, Medicine, Sciences, Health, Mental Health, Society). In her research, she is interested in how the concept of disability is defined in our society through institutional policies and systems on the one hand, and through the practices and experience of people on the other hand. Her work is built at the intersection of a political socio-history of disability, the sociology of health and the sociology of science and technology. It questions notions of care, of the person, of the body, of disability. In previous work, she became interested in the interaction between the person and the wheelchair. Her attention subsequently focused on the evolution of disability policies, particularly in France, and conceptual models of disability. Currently, her research focuses on the evolution of care practices and how these practices are organized for people with rare disabilities or multiple disabilities. She explores the notion of resistance in relation to that of fragility. She is one of three co-editors-in-chief of the journal Alter, European Journal of Disability Research.


A Basic definitions, results and facts from measure theory B Measurable functions and integration C Image of a measure

Notations

Mathematics

Proofs are ended by the symbol. Sets

• N: set of natural numbers.

• Z: set of integers.

• R: set of real numbers.

• C: set of complex numbers.

• N * = {n ∈ N s.t. n 1}.

• c

A B := A\B = {x ∈ A s.t. x / ∈ B}: complement of B in A. The prescript A may be omitted if the context makes it clear and obvious.

• ∂A: boundary of the topological space A.

• cl (A) = A ∪ ∂A: closure of the topological space A.

• B (x, R) = {y ∈ R n s.t. x -y 2 R} (n ∈ N, x ∈ R n , R > 0). • supp(f ) = cl ({x ∈ A s.t. f (x) = 0}): support of f : A → R.
• supp(µ): support of the measure µ.

Functions and Measures

• id: identity function i.e. id : R n → R n , x → x.

• 1 B : indicator function of B.

• |.|: absolute value.

• x 2 = x ∈ R n → n i=1 x 2
i : Euclidean norm.

• µ 1 ⊗ µ 2 : product measure of the measures µ 1 and µ 2 .

Derivatives and vector analysis

Let T ⊆ R, n ∈ N, m ∈ N, f : T × R n → R be a scalar field, and g : T × R n → R n be a vector field.

• ∂ x i f = ∂f ∂x i : partial derivative of f with respect to the variable

x i (i = 1, • • • , n). • ġ = ∂ t g = (∂ t g 1 , ∂ t g 2 , .
. . , ∂ t g m ): element-wise partial time derivative of g.

• ∇f = (∂ t f, ∂ x 1 f, ∂ x 2 f, . . . ) T : gradient operator.

• div(g) = ∂ x 1 g 1 + ∂ x 2 g 2 + • • • + ∂ xn g n : divergence operator.

• J(g) = (∂ x 1 g, ∂ x 2 g, . . . ∂ xn g): Jacobian operator.

• d t f = df dt = ∂ t f + (∇f )
• g: total derivative of f under the vector field g.

Linear Algebra

• M m,n (K): matrices with m ∈ N rows and n ∈ N columns, and which components are in the field K (R or C).

• I n : identity square matrix of dimension n.

• Let (A, B) ∈ (M n,n (K)) 2 (n ∈ N). The notation "A 0" means A is positive semidefinite. "A B" means "A -B 0".

• u • v: scalar product between the vectors u and v.

• A, B F = trace A T B = i,j

A ij B ij : Frobenius inner product of two matrices (A, B) ∈ (M n,m (K)) 2 (n, m ∈ N).

• 0 n,m is a matrix of zeros of dimensions n × m.

Probabilities and Statistics

• P (E): probability of the event E.

• x ∼ N (µ, Σ): x is a n-dimensional (n ∈ N) random vector following a multivariate Gaussian distribution with mean vector µ ∈ R n and with covariance matrix Σ ∈ M n,n (R) (Σ 0).

Orbital Model

Those notations describe what the adopted letters usually denote in this report, unless otherwise stated.

• r ⋆ = (r ⋆1 , r ⋆2 , r ⋆3 ) T : position vector of object ⋆ in some reference frame, (primary: ⋆ = p, secondary: ⋆ = s, relative: ⋆ = r).

• v ⋆ = (v ⋆1 , v ⋆2 , v ⋆3 ) T : velocity vector of object ⋆.

• x ⋆ = (r T ⋆ , v T ⋆ ) T : state vector of object ⋆. • X ⋆♭ = (x T ⋆ , x T ♭ ) T : state vector of combined object composed of ⋆ and ♭. • f (., .): system dynamics.

• X .|X 0 : trajectory, solution of the equations of motion starting from the initial condition X 0 .

Abbreviations

• CSM: Conjunction Summary Report.

• CAR: Conjunction Assessment Report.

• TLE: Two-Line Elements.

• TCA: Time of Closest Approach.

State of the Art of Collision Risk Assessment

Since the collision between the Russian satellite COSMOS 1934 and one debris of COSMOS 926 in December 1991, no less than eight orbital collisions have been reported between operational satellites (e.g. IRIDIUM [START_REF] Serra | A new method to compute the probability of collision for short-term space encounters[END_REF] and COSMOS 2251 collision in February 2009, the 10th), or between satellites and debris (e.g. the French satellite CERISE hit by a debris in July 1996, the 24th or the collision between THOR BURNER and a debris of Long March in January 2005, the 17th). Collision risk is particularly high in low orbits and the different space agencies (CNES, ESA, NASA) and the operators of the field (Airbus Defense and Space (ADS), GMV) have established alert procedures to assess the risks of collision for controlled satellites, and to authorize avoidance maneuvers if the predicted risk exceeds some tolerance threshold. These procedures have undergone many changes in recent years and the field of collision avoidance techniques is currently in full development. For the different evolutions of these procedures, according to the agencies and the operators, we refer to [START_REF] Flohrer | Tailoring the observation scenarios and data processing techniques for supporting conjunction event assessments[END_REF], [START_REF] Fletcher | Conjunction evolutions: The process of adapting and evolving operational collision warning software from server to service oriented architecture[END_REF] for ESA, [START_REF] Laporte | Operational Management of Collision Risks for LEO Satellites at CNES[END_REF][START_REF] Laporte | CAESAR: An initiative of public service for collision risks mitigation[END_REF] for CNES and [START_REF] Bonaventure | Collision risk management in astrium satellites[END_REF] for ADS. At the origin of any procedure of collision avoidance between two objects controlled or not in orbit, is the information of conjunction between the two objects. ), the miss distance between the two objects, statistical and geometrical information on the position and the velocities of each object. These messages are sent only three days before the date of the encounter. To obtain more accurate information on the possible encounter, it is necessary to subscribe to a service which will in return provide a Conjunction Summary Report (CSM) from which is extracted the information needed to calculate the risk of collision between both objects. The most general methods to accurately compute the global collision probability, without any additional assumption, are based on Monte-Carlo simulations, see e.g. [START_REF] Alfano | Satellite conjunction Monte Carlo analysis[END_REF][START_REF] Dolado-Perez | Satellite Collision Probability Computation for Long Term Encounters[END_REF] in the context of a simple encounter or [START_REF] Garmier | Collision Risk Assessment for Multiple Encounters[END_REF] in the context of a multiple encounter. These methods use a random sampling of N vectors in the space of initial conditions. For each of them, the corresponding trajectories are propagated according to the dynamical model adopted on the discretized time interval [0, T ]. We count 1 if there is a collision, 0 otherwise. At the end, the collision probability is given by the formula:

P c ([0, T ]) = 1 N N i=1 δ i .
The number of trials to be made depends on the requested precision as well as the value of the probability: a low value requires a lot of samples to be correctly estimated, and simulations can be dramatically time-consuming. This is one of the major disadvantages of Monte Carlo methods which makes them unsuitable for detecting low probability events in high dimension such as multiple events [START_REF] Garmier | Collision Risk Assessment for Multiple Encounters[END_REF]. Therefore alternative approaches had to be explored to assess the risk of collision between two or more objects. In the particular context of encounters between two objects, encounters are usually classified into two families: the short-term encounters [START_REF] Akella | Probability of collision between space objects[END_REF][START_REF] Chan | Spacecraft Collision Probability[END_REF][START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF][START_REF] Foster | A Parametric Analysis of orbital Debris Collision Probability and Maneuver Rate for Space Vehicles[END_REF] and the long-term encounters [START_REF] Chan | Spacecraft Collision Probability[END_REF][START_REF] Dolado-Perez | Satellite Collision Probability Computation for Long Term Encounters[END_REF] for which many simplifying assumptions, enabling the computation of the collision probability can be made. In the context of short-term encounters, conjunctions are assumed to be short and rare. The relative velocity between the two objects is assumed to be very high (higher than the km/s) and the relative motion is assumed rectilinear on the time interval of the encounter. Finally, it is also assumed that the cross-correlations between the estimated states of the two objects are very small and therefore negligible. Such encounters typically occur in low orbits where the orbital velocities are high. Long-term encounters are characterized by relative velocities of the order of m/s, and correspond to situations where both objects spend significant time in proximity to each other. The motion equations of both objects are linearized around the reference orbit. This type of encounter is more common in the context of formation flying or proximity operations. In the specific context of short-term encounters, several techniques for calculating the probability of collision have been developed. In historical order, the main methods are identified as those of [START_REF] Foster | A Parametric Analysis of orbital Debris Collision Probability and Maneuver Rate for Space Vehicles[END_REF] [START_REF] Foster | A Parametric Analysis of orbital Debris Collision Probability and Maneuver Rate for Space Vehicles[END_REF], [START_REF] Patera | General Method for Calculating Satellite Collision Probability[END_REF] [START_REF] Patera | General Method for Calculating Satellite Collision Probability[END_REF] and [START_REF] Alfano | Aerospace Support to Space Situation Awareness[END_REF] [START_REF] Alfano | Aerospace Support to Space Situation Awareness[END_REF], based on numerical integration schemes and the method of [START_REF] Chan | Collision Probability Analysis for Earth-orbiting Satellites[END_REF] [START_REF] Chan | Collision Probability Analysis for Earth-orbiting Satellites[END_REF][START_REF] Chan | Spacecraft Collision Probability[END_REF] based on an analytic formula of the probability of collision in the form of a convergent series with positive terms. All these methods have already been the subject of several comparative studies [START_REF] Alfano | Review of conjunction probability methods for short-term encounters (aas 07-148)[END_REF][START_REF] Chan | Spacecraft Collision Probability[END_REF]. A new method for calculating the collision probability in short-term encounter and the instantaneous probability of collision under Gaussian uncertainty, has been developed during the PhD thesis of R. Serra [START_REF] Serra | Opérations de proximité en orbite : évaluation du risque de collision et calcul de manoeuvres optimales pour l'évitement et le rendez-vous[END_REF] and published in [START_REF] Serra | A new method to compute the probability of collision for short-term space encounters[END_REF][START_REF] Serra | Probabilistic collision avoidance for long-term space encounters via risk selection[END_REF][START_REF] Serra | Fast and accurate computation of orbital collision probability for short-term encounters[END_REF]. The first gives a conservative estimate of the risk for a large part of actual encounters, those falling within the shortterm encounter framework, while the second provides, in a more general context, an instantaneous information which then makes it possible to obtain lower bounds for the risk of collision over time intervals. This new method has the advantage of being based on an exact analytical formula, in the form of a convergent series with positive terms, and is the exact version of the approximate one by Chan [START_REF] Chan | Collision Probability Analysis for Earth-orbiting Satellites[END_REF][START_REF] Chan | Spacecraft Collision Probability[END_REF]. Numerical examples are given and demonstrate the efficiency of the proposed formulas compared to existing methods. In addition, analytical bounds are obtained for the probability of collision: they represent a significant tool for mission analysis since their efficient evaluation, allows in many real cases to rule on the importance of the risk. In the general context of satellite flying formation or proximity operations, the hypothesis of shortterm encounters can no longer be considered valid for the calculation of the overall risk of collision [START_REF] Garmier | Collision Risk Assessment for Multiple Encounters[END_REF][START_REF] Carpenter | Conservative analytical collision probability for design of orbital formations[END_REF]. The assumption of short-time spent in the encounter area is no longer verified when considering a cluster of satellites and, even in the case of only two space objects, the encounter could sometimes occur several times per orbit, for several consecutive orbits [START_REF] Garmier | Collision Risk Assessment for Multiple Encounters[END_REF]. For such multiple encounters, the cross-correlations between the estimated states are not negligible anymore, and assessing the risk separately for each individual encounter as if they were independent events, is not sufficient to characterize the real risk of such a conjunction. The extension of the probability calculation in the case of long-term encounters, also called nonlinear framework, represents a first generalization attempt for specific cases of configurations [START_REF] Patera | Satellite collision probability for nonlinear relative motion[END_REF][START_REF] Mckinley | Development of a nonlinear probability of collision tool for the Earth observing system[END_REF][START_REF] Chan | Spacecraft Collision Probability[END_REF][START_REF] Dolado-Perez | Satellite Collision Probability Computation for Long Term Encounters[END_REF]. Unfortunately, these approaches are relatively limited because of their characterization for particular relative trajectories and are only imperfectly generalized to other cases [START_REF] Phillips | On-board estimation of collision probability for cluster flight[END_REF]. In this last reference or more recently in [START_REF] Demars | Information-Theoretic Approaches to Space Object Collision[END_REF], different metrics are proposed: distribution sampling, the Mahalanobis minimum distance and its upper bound on the collision probability, the Maximum Instantaneous probability and its lower bound on the collision probability, a hybrid probability combining the last two metrics for which the calculations can be shared or the symmetric Kullback-Leibler (KL) divergence. Roughly, the Mahalanobis distance or the symmetric KL divergence can be seen as a measure of similarity between two pdfs. The idea is to reduce the computational complexity of the distribution sampling approach by improving the determination of an interval over which two objects are in close proximity while accounting for the uncertainties of these objects. A similar approach based on computational tools of increasing complexity and precision (approximate ellipsoids for the probability density, probability density level curves, etc.) is also proposed in [START_REF] Campbell | Collision monitoring within satellite clusters[END_REF]. In [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF], V. Coppola proposes a mathematical formalization and generalization of the short-term encounter formula: the two objects are modeled with uncertainty in both position and velocity, and their trajectories are not assumed to be rectilinear anymore. More recently, the Coppola's formula has been revisited and generalized to non-spherical objects [START_REF] Krier | Satellite Collision Probability for Long-term Encounters and Arbitrary Primary Satellite Shape[END_REF]: instead of directly computing the collision probability, G. Krier first computes the collision probability per time unit, called the hazard function. The total probability of collision is the time integral of this hazard function, and this integration is performed numerically. This method has been successfully validated by comparing the obtained results with Monte Carlo simulations. The objective of this report is to give an exact and rigorous mathematical modelling of the problem of the computation of the probability of collision as derived by V. Coppola in the reference [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF].

Preliminaries

Encounter Modeling and Problem Statement

Consider an operational spacecraft (called the primary) in orbit around the Earth and a space debris (called the secondary). The state of each orbiting object is described by their position and velocity in a reference frame R, and gathered in a global state vector X ps :

X ps = (r p , v p , r s , v s ) T ∈ R n ,
where n = 12. Let [0, T ] be the time interval of the encounter. Consider now the dynamics of the 2 objects:

Ẋps (t) = f (t, X ps (t)), t ∈ [0, T ], X ps (0) = X 0 ps . ( 1
)
where f is a real vector field, which is supposed to be at least Lipschitz continuous, and T is a given terminal time defining the time interval of the encounter. The initial conditions X 0 ps ∈ R n (e.g. position and velocity) are usually subject to uncertainties, and so, they are supposed to be distributed according to a given probability measure µ I with its density function ρ I = dµ I dλ (cf. the notation defined in Appendix B). These equations include the Newtonian gravitational central field and possible orbital perturbations (non spherical Earth, atmospheric drag, e.g.). Whatever model is adopted, it is assumed that, for given initial conditions, the solutions of the system (1) exist and are unique. For each fixed initial condition X 0 ps ∈ R n , a trajectory, or sample path, is then defined as follows:

Definition 1 (Trajectory/Sample path). Given an initial condition X 0 ps ∈ R n , a trajectory, or sample path, starting from X 0 ps is the unique solution of:

X ps (t|X 0 ps ) = X 0 ps + t 0 f (τ, X ps (τ |X 0 ps ))dτ. (2)
Classically the objects are assumed to be spherical [START_REF] Akella | Probability of collision between space objects[END_REF][START_REF] Patera | Satellite collision probability for nonlinear relative motion[END_REF][START_REF] Alfano | Satellite collision probability enhancements[END_REF][START_REF] Chan | Spacecraft Collision Probability[END_REF][START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF]: this assumption enables one to ignore the orientation of the objects, and to model conservatively the secondary object whose geometry is often poorly known (see Figure 1).

Assumption 1.

The objects involved in the conjunction have a spherical shape. Let us now define the notion of a collision. Suppose that for any initial condition X 0 ps ∈ R n the corresponding trajectory exists on the time interval [0, T ], then roughly speaking, a collision occurs when a trajectory enters a certain forbidden region X R . For instance, in practice, this describes the fact that the relative distance between two objects is less than a certain given radius threshold R > 0.

If the state x is given by the positions and velocities of 2 objects, then the forbidden region is expressed as

X R = {X ps = (r p , v p , r s , v s ) ∈ R 12 | r s -r p 2 -R 2 < 0}. (3)
Note that, in case of two objects, the forbidden region (which depends on R) X R is also called the combined spherical object and let its complement, so-called safe/admissible region, be c X R := R n \X R .

Definition 2 (Collision). Given an initial condition X 0 ps ∈ R n , a maximum time of interest T > 0, and a forbidden region X R , a collision occurs if there exists t ∈ [0, T ] such that X ps (t|X 0 ps ) ∈ X R .

Definition 3 (Collision domain). The domain of collision X 0 T over the time interval [0, T ] is the set of initial conditions leading to a collision between any pair of objects during [0, T ], namely:

X 0 T = {X 0 ps ∈ R n | ∃t ∈ [0, T ], X ps (t|X 0 ps ) ∈ X R }.
Remark 1. In Equation ( 3) and Definition 3, the sets X R and X 0 T have been defined based on the combined object state vector X ps . Equivalent definitions could be given by using any combined state vector such as X rp for instance, as it is the case in the rest of this report.

Historically, the first criterion for collision risk assessment between two space objects was geometric: it relies on checking whether or not the nominal trajectory enters a no-go zone [START_REF] Alfriend | Probability of collision error analysis[END_REF][START_REF] Coppola | Determination of close approaches based on ellipsoidal threat volumes[END_REF] (including the forbidden region) whose size is either arbitrarily set or calculated with respect to uncertainties' standard deviations. A more natural formulation of the problem of collision risk assessment consists in computing the probability that a collision occurs: Problem 1. Let the dynamics in (1), a maximum time of interest T > 0 and a safe region c X R . Provided that the initial conditions X 0 ps ∈ R n are distributed according to a given probability measure µ I with its density function ρ I , the probability that a collision occurs is computed simply by:

P c ([0, T ]) = P c = P(X 0 ps ∈ X 0 T ) = µ I (X 0 T ) = X 0 T dµ I . (4)
The analytical calculation of the collision probability as defined in ( 4) is a very difficult problem: the first issue is to determine the domain of integration, which strongly depends on the chosen model for the dynamics when propagating the distribution of probability of the initial state. In addition, the integration of the density of probability on this set may be very complex, even for a Gaussian distribution [START_REF] Chan | Spacecraft Collision Probability[END_REF].

Measures for the collision probability

As in [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF], we define the set X 00 T of initial states for which a collision occurs at t = 0 and the set X 0tc T := X 0 T \X 00 T of the remaining initial states for which a collision occurs later. The collision probability P c := µ I (X 0 T ) is thus, P c = µ I (X 00 T ) + µ I (X 0tc T ). Often, in practice, the probability that a collision occurs at t = 0 is very small. However, the general computation of an instantaneous collision probability for a fixed t = t 0 (when the distribution of states at t 0 is Gaussian) is of interest and may be analyzed independently.

We now focus on computing µ I (X 0tc T ). More precisely, we are given the distribution (measure) of all initial states µ I , the dynamics, as well as the forbidden region X R . The computation of µ I (X 0tc T ) amounts in finding (in some sense) an unknown initial measure µ 0 which can be seen as the restriction of µ I to initial points leading to a collision on the time interval (0, T ], denoted by 1 X 0tc T µ I . Assume that trajectories X ps (•|X 0 ps ) starting at X 0 ps ∈ c X R are continuous functions and let ( c X R ) • be the interior of c X R . Then, over the fixed period [0, T ], these trajectories either are in ( c X R ) • or will touch its topological boundary

∂ c X R := c X R \( c X R ) • at a so-called first hitting time τ (X 0 ps ) ∈ (0, T ], τ (X 0 ps ) := min{T, inf(t 0 s.t. X ps (t|X 0 ps ) ∈ ∂ c X R )}.
Finally, one defines the final measure µ F ∈ M ([0, T ] × R n ) + which captures the distribution of the first hitting times τ (X 0 ps ) and the corresponding state X ps (t|X 0 ps ) after it has been propagated by the dynamics starting at t = 0 from the initial measure µ 0 :

µ F (A × B) := [0,T ]×R n 1 A×B (τ (X 0 ps ), X ps (τ (X 0 ps )|X 0 ps ))dµ 0 X 0 ps , (5)
for all Borel measurable sets A × B ⊆ (0, T ] × R n . The final measure µ F is the pushforward measure h τ ⋆ µ 0 of µ 0 , via the mapping:

h τ : R n → [0, T ] × R n , X 0 ps → (τ (X 0 ps ), X ps (τ (X 0 ps )|X 0 ps )), (6)
that is:

µ F (A × B) = µ 0 (h -1 τ (A × B)), (7)
for any Borel measurable sets A × B ⊆ (0, T ] × R n (cf. Figure 2).

X R µ F µ (1) 0 µ (2) 0 τ 2 τ 1 τ 3
Given initial Gaussian measure µ I 0 T

X ps

Figure 2: Yellow points: X 0 ps ∈ X 00 T , red points: X 0 ps ∈ X 0τ i T , blue points: X 0 ps ∈ c X 0 T .

Coppola's formula revisited

The objective of this section is to derive the formula for the computation of the probability of collision given in [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF] and obtained in a general context. Our intention is to give a thorough and rigorous treatment to the solution proposed in [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF].

We assume that the dynamical model under consideration (regardless of how the state vector X * * of the combined object is defined) Ẋ(t) = f (t, X(t)) gives rise to an invertible flow ϕ t 0 , behaving as a C 1 -diffeomorphism for t ∈ [0, T ]: 12 remains unchanged under time translation, we can set t 0 = 0 without loss of generality [START_REF] Haddad | Nonlinear Dynamical Systems and Control[END_REF]. The two parameter family of mappings satisfies:

ϕ t 0 : R 12 → R 12 X 0 → X(t|X 0 ),
where ϕ t 0 (•) is the map ϕ(t, t 0 , •) for a given (t, t 0 ) ∈ R × R. Assuming that the function ϕ(•, •, •) : R × R × R 12 → R
∂ ∂t ϕ t 0 (X 0 ) = f (t, ϕ t 0 (X 0 )).
Its inverse is given by:

ϕ t 0 -1 = ϕ 0 t : X(t|X 0 ) → X 0 .
When manipulating density functions and the change of variable formula in the calculations below, we will also need the local linearization of the flow with respect to the initial conditions, denoted by Dϕ t 0 (X 0 ) ∈ R 12×12 . Indeed, if X0 is close to X 0 , then:

ϕ t 0 ( X0 ) ≈ ϕ t 0 (X 0 ) + Dϕ t 0 (X 0 )( X0 -X 0 ).
Note that the local linearization of the inverse flow ϕ 0 t satisfies the relation:

Dϕ 0 t (ϕ t 0 (X 0 )) = Dϕ t 0 (X 0 ) -1 ,
since by differentiating with respect to X 0 the equation

ϕ 0 t • ϕ t 0 = Id R 12 , we get Dϕ 0 t (ϕ t 0 (X 0 )) • Dϕ t 0 (X 0 ) = Id R 12 .
Example 1.

Under the assumption that the flow is linear, Ẋ(t) = A(t)X(t), the solution of the dynamics equation via the state transition matrix Φ(t, 0), is such that

X(t|X 0 ) = ϕ t 0 (X 0 ) = Φ(t, 0)X 0 ,
that is, the initial conditions are propagated from time 0 to time t via the matrix Φ(t, 0). One has also X 0 = Φ(t, 0) -1 X(t) = Φ(0, t)X(t).

Also, suppose that there is a suitable surface parametrization of the whole or part of ∂ c X R given by P : S → R n , with S ⊆ R n-1 and with P (s) = (p 1 (s), . . . , p n (s)) ∈ ∂ c X R . (b) (Sphere) The coordinates (x, y, z) ∈ R 3 of a point on the sphere of radius R are parameterized on the sphere via the change of coordinates:

(x, y, z) T := P (θ, φ) = (R cos(θ) sin(φ), R sin(θ) sin(φ), R cos(φ)) T , with θ ∈ [0, 2π), φ ∈ [0, π] and S = [0, 2π) × [0, π]. (c) (Practical case in Coppola's article) For X rp = (r r1 , r r2 , r r3 , v r1 , v r2 , v r3 , r p1 , r p2 , r p3 , v p1 , v p2 , v p3 ) T ∈ R 12 the surface r r1 2 + r r2 2 + r r3 2 = R 2
, is parameterized on R 11 via the change of coordinates:

X rp = (r r1 , r r2 , r r3 , v r1 , v r2 , v r3 , r p1 , r p2 , r p3 , v p1 , v p2 , v p3 ) T := P (θ, φ, v r1 , . . . , v p3 ), (8)
= (R cos(θ) sin(φ), R sin(θ) sin(φ), R cos(φ), v r1 , v r2 , v r3 , r p1 , r p2 , r p3 , v p1 , v p2 , v p3 ) T , ( 9
)
with θ ∈ [0, 2π), φ ∈ [0, π] and S = [0, 2π) × [0, π] × R 9 .
Note that this parametrization is based on spherical coordinates used in Mathematics and which convention is different (θ and φ are swapped) from the one preferentially used in Physics and in [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF]. The a priori unknown and possibly complicated shape of X 0tc T makes it difficult to get the probability of collision by directly computing an integral over this set. The principle of Coppola's formula is to compute an equivalent integral over the set of collision states instead, via a change of variables given by the function ψ : [0, T ] × S → X 0tc T , ψ(t, s) = ϕ 0 t (P (s)) (note that this definition is a partial composition only, since ϕ 0 t depends on the first argument t of ψ). For this substitution to be licit, as stated by the following lemma, we need ψ to be 1-1 (more precisely, a C 1 -diffeomorphism). The following assumption is sufficient for the bijectivity of ψ.

Assumption 2.

For each trajectory leading to a collision, there is only one entering crossing in (or only one tangent to) the forbidden region X R for all t ∈ (0, T ]. Therefore, the collision domain X 0tc T and the surface S are respectively defined by:

X 0tc T = X 0 ∈ R n | ∃!t ∈ (0, T ], X(t|X 0 ) ∈ X R and v r (t) T r r (t) 0 , (10)
and

S = (θ, φ, v r , r p , v p ) ∈ [0, 2π) × [0, π] × R 9 : v r1 cos(θ) + v r2 sin(θ) sin(φ) + v r3 cos(φ) 0 . ( 11
)
In order to compute the part of the collision probability P c given by µ I (X 0tc T ) := µ 0 (R n ), the next derivations rely on the following lemma:

Lemma 1. (i) µ F = ((Id R × P ) • ψ -1 ) ⋆ µ 0 ; (ii) R n dµ 0 = X 0tc T ρ I (X 0 )dλ(X 0 ) = [0,T ]×S (ρ I • ψ)(t, s) |det(J(ψ)(t, s))| dλ(t, s), ( 12
)
where J(ψ) is the Jacobian matrix of the mapping ψ.

Proof.

(i) One notices that since ψ is 1-1, each point which achieves a collision,

X 0 ∈ X 0tc T ⊂ R n is transported via (Id R × P ) • ψ -1 on the surface ∂ c X R
at a certain time τ (X 0 ) ∈ (0, T ] and reciprocally since:

X 0tc T ⊂ R n ψ -1 --→ [0, T ] × S Id R ×P ----→ [0, T ] × R n , X 0 -→ (τ (X 0 ), s(X 0 )) -→ (τ (X 0 ), X(τ (X 0 )|X 0 )).
Therefore, we have that h τ = ((Id R × P ) • ψ -1 ) and by 2 in Appendix C,

[0,T ]×R n 1 A×B (t, X)dµ F = R n 1 A×B • (Id R × P ) • ψ -1 (X 0 )dµ 0 .
(ii) One uses the Jacobi's change of variables formula [START_REF] Schilling | Measures, Integrals and Martingales[END_REF]Chapter 15], since ψ is a C 1 -diffeomorphism. Note that this formula is obtained by applying (42) with

X = [0, T ] × S, Y = X 0t T , g(ψ) = ρ I (ψ) |det(J(ψ))|, µ = λ and noting that d(ψ ⋆ µ) = d(ψ ⋆ λ) = det(J(ψ -1 )) dλ [30, Remark 15.10]. Indeed, [0,T ]×S (ρ I • ψ)(t, s) |det(J(ψ)(t, s))| dλ(t, s) = X 0t T ρ I (X 0 ) det(J(ψ)(ψ -1 (X 0 ))) dψ ⋆ λ(X 0 ) = X 0t T ρ I (X 0 ) det(J(ψ)(ψ -1 (X 0 ))) det(J(ψ -1 )(X 0 )) dλ(X 0 ) = X 0t T ρ I (X 0 ) det(J(ψ)(ψ -1 (X 0 ))) det(J(ψ)(ψ -1 (X 0 ))) -1 dλ(X 0 ) = X 0t T ρ I (X 0 )dλ(X 0 ) = X 0t T dµ(X 0 ). Note that J(ψ)(ψ -1 (X 0 ))J(ψ -1 )(X 0 ) = I n . Example 3.
If the flow is linear, the function ψ is now defined as ψ : [0, T ] × S → X 0tc T , ψ(t, s) = Φ(t, 0) -1 P (s). For computing the Jacobian of ψ, one has:

∂ψ ∂t (t, s) = -Dϕ 0 t (P (s))f (t, P (s)) ∈ R 12 .
This indeed follows from differentiating with respect to t the equality P (s) = ϕ t 0 (ψ(t, s)) by applying the chain rule:

0 = ∂ ∂t ϕ t 0 (ψ(t, s)) = ∂ϕ t 0 ∂t (ψ(t, s)) + Dϕ t 0 (ψ(t, s)) ∂ψ ∂t (t, s) = f (t, ϕ t 0 (ψ(t, s))) + Dϕ t 0 (ψ(t, s)) ∂ψ ∂t (t, s)
= f (t, P (s)) + Dϕ 0 t (P (s))

-1 ∂ψ ∂t (t, s).

In addition, we have that:

∂ψ ∂s (t, s) = Dϕ 0 t (P (s)) ∂P (s) ∂s ∈ R 12×11 . Hence J(ψ)(t, s) = Dϕ 0 t (P (s)) -f (t, P (s)) ∂P (s) ∂s ∈ R 12×12 . ( 13
)
Example 4.

In the case of linear dynamics, the computations above become

∂ψ ∂t = -Φ -1 (t, 0) ∂Φ(t, 0) ∂t Φ -1 (t, 0)P (s) = -Φ -1 (t, 0)A(t)P (s),
since ∂Φ(t,0) ∂t = A(t)Φ(t, 0). Moreover, we have

∂ψ ∂s = Φ(t, 0) -1 ∂P (s) ∂s ,
and (13) becomes

J(ψ) = Φ -1 (t, 0) -A(t)P (s) ∂P (s) ∂s .
Practical computations for the Jacobian matrix in Coppola's case (Example 2(c)), are as follows:

∂P (s) ∂s =     -R sin (θ) sin (φ) R cos (θ) cos (φ) R cos (θ) sin (φ) R sin (θ) cos (φ) 0 -R sin (φ) 0 3,9 0 9,2 I 9     .
Due to 0 blocks in the above matrix, to compute the determinant of J(ψ), one only needs the first three coordinates of the f (t, P (s)) vector, and since f (t, •) is the dynamics, one has

f (t, P (s)) = (v r1 , v r2 , v r3 , vr1 , vr2 , vr3 , v p1 , v p2 , vp3 , vp1 , vp2 , vp3 ) T ∈ R 12 .
This gives:

det(J(ψ)(t, s)) = -v r1 -R sin (θ) sin (φ) R cos (θ) cos (φ) -v r2 R cos (θ) sin (φ) R sin (θ) cos (φ) -v r3 0 -R sin (φ) det Dϕ 0 t (P (s)) ,
which amounts to:

det(J(ψ)) = R 2 sin (φ) (cos (θ) sin (φ) v r1 + sin (φ) sin (θ) v r2 + cos (φ) v r3 ) det Dϕ 0 t (P (s)) = R 2 sin (φ) v r • n det Dϕ 0 t (P (s)) , (14)
where n = [cos (θ) sin (φ) sin (φ) sin (θ) cos (φ)] T is the unit vector normal to the sphere.

Example 5.

For linear dynamics, one may easily deduce that

det(J(ψ)) = -v r1 -R sin (θ) sin (φ) R cos (θ) cos (φ) -v r2 R cos (θ) sin (φ) R sin (θ) cos (φ) -v r3 0 -R sin (φ) det(Φ -1 (t, 0)),
and finally

det(J(ψ)) = R 2 sin (φ) (cos (θ) sin (φ) v r1 + sin (φ) sin (θ) v r2 + cos (φ) v r3 ) det(Φ -1 (t, 0)) = R 2 sin (φ) v r • n det(Φ -1 (t, 0)). (15)
Now, since ψ is 1 -1, one can apply Equation ( 12) and get:

R n dµ 0 = T 0 2π 0 π 0 ∞ -∞ ∞ -∞ ∞ -∞ ρ I (ψ(t, θ, φ, v r , r p , v p )) det Dϕ 0 t (P (t, θ, φ, v r , r p , v p )) 1 vr •n 0 (v r ) R 2 sin (φ) v r • n dv p dr p dv r dφdθdt (16) Example 6.
For linear dynamics, we have

R n dµ 0 = T t=0 2π 0 π 0 ∞ -∞ ∞ -∞ ∞ -∞ ρ I (ψ(t, θ, φ, v r , r p , v p )) det(Φ -1 (t, 0)) 1 vr•n 0 (v r ) R 2 sin (φ) v r • n dv p dr p dv r dφdθdt (17)
Lemma 2. Let a given random vector X with probability density ρ X and another random vector Y related to X by the equation y = g(x), and g bijective. The probability density ρ Y for Y is:

ρ Y (y) = ρ X (g -1 (y)) det(J(g -1 )(y)) . (18)
Proof. Let P(Y ∈ S) be the probability that Y takes a value in some particular subset S, so that we get:

P(Y ∈ S) = S ρ Y (y) dy.
Since Y takes a value in S whenever X takes a value in g -1 (S), one has:

P(Y ∈ S) = g -1 (S) ρ X (x) dx.
Now, changing from variable x to y gives:

P(Y ∈ S) = g -1 (S)
ρ X (x) dx = S ρ X (g -1 (y)) det(J(g -1 )(y)) dy.

In our case, let g be the flow ϕ t 0 , and ρ I be the probability density of the initial random state X 0 , one obtains the probability density ρ t (X) of the random state X at a given fixed time t from Equation (18):

ρ t (X) = ρ I (ϕ 0 t (X)) det Dϕ t 0 (X) . (19)
Example 7.

In the linear case, the flow is given by X 0 → Φ(t, 0)X 0 and then, we have that the probability density ρ t (X) of the random state X at a given fixed time t is given by

ρ t (X) = ρ I (Φ -1 (t, 0)X) det(Φ -1 (t, 0)) . (20)
From Equations ( 19) and ( 16), one has:

P I = R n dµ 0 = T t=0 2π θ=0 π φ=0 ∞ vr=-∞ ∞ rp=-∞ ∞ vp=-∞ ρ t (P (θ, φ, v r , r p , v p ))1 vr •n 0 (v r ) R 2 sin (φ) v r • n dv p dr p dv r dφdθdt. ( 21
)
Remark 3. Equation ( 21) is similar to formula (15) in Coppola's article, with two differences. Firstly, the parametrization in spherical coordinates is different, but this does not affect the result in any way. Secondly, there is a slight abuse of notation in formula (15) of Coppola's article, concerning the probability density function ρ t (X) which appears instead of ours ρ t (P (s)). More precisely, this means that ρ t is the probability density of the state random variable X at each fixed time t, but this function should be applied to P (s) (after the reparametrization).

There is also a slight abuse of notations in this document regarding the definition of a probability density function ρ t (P (s)) as shown in Lemma 2 since the random vector and the variable of the density function coincide.

We move on to the next assumption made in [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF]:

Assumption 3. Independence of the two random vectors x p and x s .

The dynamic model and probability distribution function for one object are independent from the dynamic model and probability distribution function of the other object.

Keeping in mind the slight abuse of notations mentioned in Remark 3 and if ρ tps (x p , x s ), ρ tp (x p ), ρ ts (x s ) denote respectively the joint density functions of the random vectors X T = x T p x T s , x p and x s then Assumption 3 means that:

ρ tps (x p , x s ) = ρ tp (x p )ρ ts (x s ). (22)
This implies that the density ρ t (P (s)) = ρ t (x r (θ, φ, v r ), x p ) satisfies:

ρ t (x r , x p ) = ρ tps (x p , x s ) = ρ tp (x p )ρ ts (x p + x r ), (23)
where x r = x px s is the relative state on the sphere of radius R. This may be proved by noting that X rp = x r x p = -I 6 I 6 I 6 0 6,6

x p x s and by applying Lemma 2. Therefore, Equation [START_REF] Demars | Information-Theoretic Approaches to Space Object Collision[END_REF] becomes:

P I = T 0 2π 0 π 0 |sin (φ)| ∞ -∞ R 2 |v r • n| 1 vr•n 0 (v r ) ∞ -∞ ∞ -∞
ρ tp (x p )ρ ts (x p + x r )dv p dr p dv r dφdθdt. [START_REF] Laporte | CAESAR: An initiative of public service for collision risks mitigation[END_REF] The next assumption from [START_REF] Coppola | Including Velocity Uncertainty in the Probability of Collision between Space Objects[END_REF] is related to the Gaussian nature of the two density functions of the random vectors x p and x s .

Assumption 4. Gaussian distributions.

The probability distribution functions ρ tp ∼ N (m p (t), P p (t)) and ρ ts ∼ N (m s (t), P s (t)) remain Gaussian at each time t.

It may be deduced that x r = x sx p ∼ N (m s (t)m p (t), P p (t) + P s (t)

Pr(t)
).

Let us define the following matrices:

G -1 = P -1 p + P -1 s , T = GP -1 s , ( 25
)
where G is a symmetric positive semidefinite matrix. Keeping in mind that:

ρ tp (x p ) = 1 (2π) 6 det(P p ) e - 1 2
then

ρ t (x r , x p ) = ρ tp (x p )ρ ts (x p + x r ) = ρ t (w, x r ) = ρ tw (w)ρ tr (x r ), (27)
is obtained after tedious algebraic manipulations for which the following identities are used:

T P s T T + (I 6 -T )P p (I 6 -T ) T = GT T + G(I 6 -T ) T = G, det(P p ) det(P s ) = det(P p P s ) = det(GP ) = det(G) det(P ),

and:

ρ tw ∼ N (m p (t) + T (m s (t) -m p (t)) mr(t)
, G(t)),

ρ tr ∼ N (m r (t), P r (t)). (29)
Equation [START_REF] Patera | General Method for Calculating Satellite Collision Probability[END_REF] shows that the random vectors w = x p + T x r and x r are independent. Therefore, we get that:

P I = T 0 2π 0 π 0 |sin (φ)| ∞ -∞ R 2 |v r • n| 1 vr •n 0 (v r )ρ tr (x r ) ∞ -∞ ρ tw (w)dw =1 dv r dφdθdt = T 0 2π 0 π 0 |sin (φ)| ∞ -∞ R 2 |v r • n| 1 vr •n 0 (v r )ρ tr (x r (θ, φ, v r ))dv r dφdθdt. ( 30
)
The Consequently, we get that:

- 1 2 (x r -m r ) T P -1 r (x r -m r ) = - 1 2 (r r -m rr ) T P -1 11 (r r -m rr ) • • • - 1 2 (•) T ∆ -1 ( v ′ v r -P T 12 P -1 11 r r - m ′ v (m vr -P T 12 P -1 11 m rr )) v ′ -m ′ v , = - 1 2 (r r -m rr ) T P -1 11 (r r -m rr ) - 1 2 (v ′ -m ′ v ) T ∆ -1 (v ′ -m ′ v ),
and ρ tr (x r ) = ρ tr (r r )ρ tv ′ (v ′ ) with:

ρ tr ∼ N (m rr (t), P 11 (t)),

ρ tv ′ ∼ N (m ′ v (t), ∆(t)). (31)
Moreover, |v r • n| = v ′ (t) • n + Rn T P -1 11 (t)P 12 (t)n = |v ′ (t) • n + ǫ 0 (n, t)| and we get:

P I = T 0 2π 0 π 0 R 2 |sin (φ)| ρ tr (r r )• ∞ -∞ v ′ (t) • n + ǫ 0 (n, t) 1 v ′ (t)•n+ǫ 0 (n,t) 0 (v ′ )ρ tv ′ (v ′ )dv ′ I(n,t) dφdθdt. (32)
The unit vector normal to the sphere n is defined by n = [cos (θ) sin (φ) sin (φ) sin (θ) cos (φ)] T and the orthogonal matrix T defined by:

T =   cos(θ) sin(φ) sin(φ) cos(θ) cos(φ) -sin(θ) cos(θ) 0 -cos(θ) cos(φ) -sin(θ) cos(φ) sin(φ)   , (33)
is such that i = 1 0 0 T = T n. With the notations:

v ′′ (t) = T v ′ (t) = ǫ(t) ζ(t) , σ 2 , ∆ 22 - ∆ T 12 ∆ 12 σ 2 .
Note that m ǫ (θ, φ, t) = nT (m vr -P T 12 (t)P -1 11 (t)m rr ) = i T m ′′ v (t). Therefore,

I(n, t) = ∞ -∞ v ′ (t) • n + ǫ 0 (n, t) 1 v ′ (t)•n+ǫ 0 (n,t) 0 (v ′ )ρ tv ′ (v ′ )dv ′ , = ∞ -∞ |ǫ(t) • n + ǫ 0 (n, t)| 1 ǫ+ǫ 0 (n,t) 0 (ǫ)ρ tv ′′ (v ′′ )dζdǫ, = ∞ -∞ |ǫ(t) • n + ǫ 0 (n, t)| 1 ǫ+ǫ 0 (n,t) 0 (ǫ)ρ tǫ (ǫ) ∞ -∞ ρ tζ (ζ)dζ 1 dǫ, = - 1 √ 2π σ -ǫ 0 (n,t) -∞ (ǫ(t) • n + ǫ 0 (n, t)) e - (ǫ -m ǫ ) 2 2σ 2 dǫ, (35)
since v ′ (t) • n = v ′′ (t) • T n = v ′′ (t) • i.
Finally, remembering that the error function is given by:

erf : R → R x → erf(x) = 1 √ π x 0 e -t 2 dt, (36)
the integral I(n, t) may be readily computed as:

I(n, t) = σ √ 2π e - (ǫ 0 (n, t) + m ǫ ) 2 2σ 2 - (m ǫ + ǫ 0 (n, t)) 2 1 -erf m ǫ + ǫ 0 (n, t) √ 2 σ , (37)
and the probability P I is finally obtained as:

P I = T 0 2π 0 π 0 R 2 |sin (φ)| ρ tr (r r (θ, φ))I(n, t)dφdθdt. ( 38
) 1. µ (∅) = 0, 2. for every sequence (E n ) n∈N of pairwise disjoint sets in A, µ n∈N E n = n∈N µ (E n ).
Example 9.

-Borel measure : A measure defined on the Borel σ-algebra B(R n ) is called a Borel measure.

-Dirac Measure at a : Let (A, A) be any measurable space and let a ∈ A. Then δ a : A → {0, 1}, defined for A ∈ A by:

δ a (A) = 0 if a ∈ A, 1 if a ∈ A,
is called the Dirac measure at the point a.

-Lebesgue measure: The set-function λ n on (R n , B(R n )) that assigns the value :

λ n ([a 1 , b 1 ) × • • • × [a n , b n )) = n j=1 (b j -a j ) to every half-open rectangle [a 1 , b 1 )ו • •×[a n , b n ) ∈ I, is called the n-dimensional Lebesgue measure. If A ⊂ R n then λ n (A) = inf A⊂ ∪ ∞ i=1 R i (b i 1 -a i 1 ) • • • • • (b i n -a i n ),
where the R i 's are half-open rectangles, i.e.

R i = [a i 1 , b i 1 ) × • • • × [a i n , b i n )
. The 1-dimensional-Lebesgue measure represents the conventional "length" of a segment:

∀ (a, b) ∈ R 2 s.t. a < b, λ ([a, b]) = b -a.
Remark that, for a ∈ R, λ ({a}) = 0. From the second point of Definition 9:

∀ (a, b) ∈ R 2 s.t. a < b, λ ([a, b]) = λ ({a}) + λ (]a, b[) + λ ({b}) = λ (]a, b[) .
Definition 10 (Measure Space). A measure space is a triplet (A, A, µ) where A is a set, A is a σ-algebra of subsets of A, and µ is a positive measure on (A, A). Definition 11 (σ-finite measure). A measure µ defined on (A, A) is said to be σ-finite and (A, A, µ) is called a σ-finite measure space if A contains an increasing sequence

(A j ) j∈N ⊂ A of sets A 1 ⊂ A 2 ⊂ • • • such that ∪ j∈N A j = A (exhausting sequence) verifying µ(A j ) < ∞ for all j ∈ N.
Definition 12 (Support of a Measure). Let (A, A, µ) be a measure space. The support of the positive measure µ is the smallest closed set defined as:

supp(µ) = {x ∈ A s.t. µ (U ) > 0 for every open neighborhood U of x} . Example 10.
-The support of the Dirac measure at 0 is the singleton {0}.

-Let (R, B(R)). The support of the Lebesgue measure λ is R. Note that every single point has Lebesgue measure zero and so has every countable union of points.

Definition 13 (Product Measure). Let (A 1 , A 1 , µ 1 ) and (A 2 , A 2 , µ 2 ) be two σ-finite measure spaces.

Then the set-function ρ :

A 1 × A 2 → [0, ∞], A 1 × A 2 → µ 1 (A 1 )µ 2 (A 2 )
is the unique measure called product measure, denoted by ρ = µ 1 ⊗ µ 2 and defined on the measurable space

(A 1 × A 2 , A 1 ⊗ A 2 ) where A 1 ⊗ A 2 = σ(A 1 × A 2 ) is a product σ-algebra. (A 1 × A 2 , A 1 ⊗ A 2 , µ 1 ⊗ µ 2
) is called the product measure space. The product measure ρ is a σ-finite measure on (A 1 × A 2 , A 1 ⊗ A 2 ) such that ρ(E) = 1 E (x, y)µ 1 (dx)µ 2 (dy) = 1 E (x, y)µ 2 (dy)µ 1 (dx), holds for all E ∈ A 1 ⊗ A 2 .

-For arbitrary f ,

f dµ = f + dµ -f -dµ,
where f + = max{f, 0} and f + = -min{f, 0}.

If (B, B) is measurable, the integral of f over B is defined by:

B f dµ = f 1 B dµ.
Theorem 1 (Fubini). Let (A 1 , A 1 , µ 1 ) and (A 2 , A 2 , µ 2 ) be σ-finite measure spaces and let f :

A 1 × A 2 → R be A 1 ⊗ A 2 -measurable.
If at least one of the following is finite For instance, if the measure µ I is Gaussian (µ I = µ g ), with mean m ∈ R n , and covariance matrix Σ, given by:

A 1 ×A 2 |f |d(µ 1 ⊗ µ 2 ),
µ I (A) := 1 (2πdet(Σ)) n A exp - (X -m) T Σ -1 (X -m) 2 dX, (39)
where A ∈ B(R n ), and B(R n ) denotes the completion of the Borel σ-algebra on R n and the integral in (39) is with respect to the standard n-dimensional Lebesgue measure. The density of the measure µ I is a function ρ I : R n → R defined by:

ρ I (X) = 1 (2πdet(Σ)) n exp - (X -m) T Σ -1 (X -m) 2 . ( 40
)
Using the notation of Definition 18, one has ρ I = dµ I dλ .

C Image of a measure

Let us denote by M (S) + the cone of all nonnegative Borel measures on the subset S of a measurable space. For completeness, let us recall the classical notions related to the pushforward measure, which is roughly speaking the image of a given measure under a given mapping:

Definition 19. [35, Theorem 1.44] (Pushforward measure) Let two measurable spaces (X, A) and (Y, B), a (A, B)-measurable mapping h : X → Y and a measure µ ∈ M (X) + . The pushforward measure (or image measure under the mapping h) ν = h ⋆ µ ∈ M (Y ) + defined on B is given by:

ν(B) = h ⋆ µ(B) = µ(h -1 (B)), (41)
for all Borel measurable sets B ⊆ Y .

For an arbitrary set B ⊆ Y , h -1 (B) is the preimage of B under the mapping h, i.e. (42)

h -1 (B) = {x ∈ X : h(x) ∈ B} . X Y 0 ν B h -1 (B) ν(B) = µ(h -1 (B)) R h µ

Figure 1 :

 1 Figure 1: Encounter between two spherical objects.

Example 2 .

 2 (a) (Circle) The coordinates (x, y) ∈ R 2 of a point on the circle of radius R are parameterized on the circle via the change of coordinates: (x, y) T := P (θ) = (R cos(θ), R sin(θ)) T , with θ ∈ S = [0, 2π].

Figure 3 :

 3 Figure 3: Spherical coordinates for the parametrization of a sphere.

Remark 2 .

 2 Assumption 2 is equivalent to the Assumptions (A1) and (A2) proposed in Coppola's article and reminded below. (A1) Only one crossing. (A2) Trajectories must cross.

( 2 (

 2 xp-mp) T P -1 p (xp-mp) , ρ ts (x s ) = 1 (2π) 6 det(P s ) e -1 xs-ms) T P -1 s (xs-ms),

Property 1 .

 1 Let (A, A, µ) be a measure space and B, C ∈ A. Then, C ⊂ B ⇒ µ (C) µ (B).

Figure 4 :

 4 Figure 4: Monotonicity of positive measures.

A 2 A 1 |fA 2 A 1 fA 1 A 2 f

 12112 (x, y)|µ 1 (dx)µ 2 (dy),A 1 A 2 |f (x, y)|µ 2 (dy)µ 1 (dx) then A 1 ×A 2 f d(µ 1 ⊗ µ 2 ) = (x, y)µ 1 (dx)µ 2 (dy) = (x, y)µ 2 (dy)µ 1 (dx).Definition 18 (Density). Let (A, A, µ) be a measure space and ρ a positive real A-measurable function, the set-functionν : A → A ρdµ = 1 A ρdµ = 1 A (x)ρ(x)µ(dx),is a measure on (A, A) called the measure with density function ρ with respect to µ and denoted ν = ρµ. Traditionally, the density is denoted ρ = dν dµ .

Figure 5 :

 5 Figure 5: Pushforward measure or image measure ν.

  next step consists in partitioning the mean vector m r (t) =

			m rr (t) m vr (t)	and the covariance
	matrix P r (t) as P r (t) =	P 11 (t) P 12 (t) P T 12 (t) P 22 (t)	leading to:
	P -1 r (t) = ∆(t) = P 22 (t) -P T P -1 11 (t) + P -1 11 (t)P 12 (t)∆ -1 (t)P T 12 (t)P -1 11 (t) -P -1 11 (t)P 12 (t)∆ -1 (t) -∆ -1 (t)P T 12 (t)P -1 ∆ -1 (t) 11 (t) 12 (t)P -1 11 (t)P 12 (t).	,

Appendix

This appendix quickly summarizes the main definitions from measure theory and integration that are needed for the developments presented in this report. The material here in is mainly borrowed from the references [START_REF] Bogachev | Measure theory, volume I and II[END_REF], [START_REF] Ambrosio | Calculus of variations and nonlinear partial differential equations[END_REF] and [START_REF] Schilling | Measures, Integrals and Martingales[END_REF].

A Basic definitions, results and facts from measure theory Definition 4 (Indicator Function). Let B ⊂ A. The indicator function 1 B of B is defined as:

Definition 5 (σ-algebra). Let A be a set. A σ-algebra of subsets of A (sometimes called a σ-field) is a family A of subsets of A such that:

Example 8.

-P(A) is a (the maximal) σ-algebra on A.

-{∅, A} is a (the minimal) σ-algebra on A.

-{∅, B, A\B, A} is a σ-algebra on A.

-If B = ∅ and B = A then {∅, B, A\B, A} is not a σ-algebra on A. Definition 6. For every system of sets G ∈ P(A), there exists a smallest σ-algebra containing G. This σ-algebra, denoted by σ(G) is the σ-algebra generated by G.

Definition 7 (Borel σ-algebra or topological σ-algebra). The σ-algebra σ(O n ), denoted B(R n ), generated by the system O n of all open sets of R n is called the Borel σ-algebra on R n and its members are the Borel sets or Borel measurable sets.

Note that the Borel σ-algebra B(R n ) may be generated by the family of half-open rectangles in R n , i.e. B(R n ) = I where

Definition 8 (Measurable Space). A measurable space is a pair (A, A) where A is a set and A is a σ-algebra of subsets of A.

Definition 9 (Measure). Let (A, A) be a measurable space (see Definition 8). A positive measure on A (or, more precisely, on (A, A)) is a mapping µ : A → [0, +∞] satisfying:

Example 11.

The n-dimensional Lebesgue measure λ n on (R n , B(R n )) may be defined as a product measure, i.e.

B Measurable functions and integration

Definition 14

Definition 15 (Measurable function). Let (A, A) be a measurable space. A function f :

Example 12.

Let (X, A) be a measurable space. The indicator function f

Definition 16 (Simple function). A simple function g : A → R on a measurable space (A, A) is a function of the form:

If y i 0 for all i, the function g is called a positive simple function.

Definition 17. Suppose µ is a positive measure on the measurable space (A, A) and f is measurable

y j 1 A j is a positive simple function then

-If f 0 then f dµ = sup gdg : g f, g is positive simple .