Valeria Andriano
email: valeria.andriano@liceogalfer.it

Roberto Capone

Carola Manolino

Ornella Robutti

Riccardo Minisola

Teachers' awareness on combinatorics during Lesson Study discussion

Keywords: Awareness, combinatorial thinking, lesson study, teacher professional development

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

Introduction and literature review

Collaboration among teachers is considered a key factor in improving teachers' education [START_REF] Bakker | Future themes of mathematics education research: An international survey before and during the pandemic[END_REF] and teaching itself [START_REF] Andriano | Teachers' awareness of classroom interactions in hybrid distance education through lesson study[END_REF]. In the frame of ICMI Study 25 (Borko & Potari, 2020), the academic community discussed how complex it can be to grasp and demonstrate the assertion. Lesson Study (LS) is one of the main sources for researchers to gain insight into teachers' attitudes towards working and learning, their mathematical knowledge for teaching, and teaching practices [START_REF] Huang | Improving teaching, developing teachers and teacher educators, and linking theory and practice through lesson study in mathematics: An international perspective[END_REF]. In particular, a number of studies focus on analysing teachers' discourses as a window on teacher learning (e.g., [START_REF] Clivaz | How do dialogic interactions contribute to the construction of teachers' mathematical problem-solving knowledge? Construction of a conceptual framework[END_REF].

LS is a collaborative model for teachers' professional development inside the institutions (schools). LS is usually modelled as a cycle. Here, it is structured in the following phases: 1) Goal setting: formulate a "research" theme of common interest to teachers around a topic to be taught; 2) Planning: collaboratively prepare a Lesson Plan to implement the research lesson; 3) Observing: one group member teaches the lesson while other members observe; 4) Discussing: members discuss the data collected during the observation and revise the lesson. Then the cycle could be repeated multiple times. This structure has many variants and has been adapted in various countries with different features according to diverse cultural aspects (e.g., [START_REF] Seleznyov | Lesson study: An exploration of its translation beyond Japan[END_REF].

In Italy the first experiences were in primary school [START_REF] Bartolini Bussi | Cultural transposition of Chinese lesson study to Italy: An exploratory study on fractions in a fourth-grade classroom[END_REF], nowadays LS is experienced at all school levels, from kindergarten to secondary school, and with prospective teachers [START_REF] Capone | Networking of theories for a multifaceted understanding on lesson study in the Italian context[END_REF][START_REF] Manolino | An elaboration of the Lotman's semiosphere theoretical construct for mathematics education: Analysis of the Chinese mathematics lesson study cultural transposition within the Italian context[END_REF]. Specifically, in the Italian experience, an essential emphasis is given to the cultural context in which the teaching-learning is situated. This is because LS was introduced as a teacher education proposal that, by leveraging cultural differences, stimulates teachers to reflect on themselves and their own habits (e.g., [START_REF] Arzarello | Networking hybridizations within the semiosphere: A research trajectory for the cultural transposition of the Chinese lesson study within a western context[END_REF].

LS is usually introduced by researchers to teachers, and emphasis is on the important role of the researcher as a cultural mediator [START_REF] Ponte | Fitting lesson study to the Portuguese context[END_REF]. In Italy, the researcher does not mediate cultural adaptation a priori, but uses it educationally as a stimulus for teachers' development (Ramploud et al., 2022). In this study, there is an additional peculiarity: LS was introduced to a group of mathematics teachers by one of their colleagues, in the role of facilitator. The facilitator is author of the paper and collaborates with the researchers. The focus of this study is teachers' awareness, in the sense of [START_REF] Mason | Learning about noticing, by, and through, noticing[END_REF], studied in the observation and discussion phases (3 and 4). Particularly we aim at analysing teachers' critical reflection during the discussion meetings, i.e., members' discussions after lessons observation. To conduct the analysis of a 6-cycles LS in an Italian secondary school focused on the introduction of combinatorics, in line with [START_REF] Andriano | Teachers' awareness of classroom interactions in hybrid distance education through lesson study[END_REF], we use the three-step theoretical lens based on the concept of awareness.

Theoretical framework

The theoretical background is based on the concept of awareness as introduced by [START_REF] Gattegno | The science of education part I: Theoretical considerations[END_REF]. Awareness triggers actions without our having to initiate them consciously, in the sense that they could lie under the surface of our consciousness. Using the term awareness in the sense of Gattegno, Mason (2021) proposed several valuable levels of awareness in educational contexts: 1) Awarenessin-action: the sensitivities to certain situations which enable action; 2) Awareness-in-discipline: awareness of awareness-in-action; 3) Awareness-in-counsel: awareness required in order to be sensitive to what others require to build their own awareness-in-action and in-discipline. As Mason sustains, being a teacher requires sensitivity to learners, and this is made possible because of awareness-in-discipline. According to Mason, awareness is educable and can be improved by the discipline of noticing "namely, imagining oneself acting appropriately in the future and using this to sharpen one's noticing in the moment, thus educating one's awareness." (Mason & Davis, 2013, p. 192). Conceptualization of noticing is varied: some authors define noticing solely as that to which teachers attend [START_REF] Sherin | Accessing mathematics teachers' in-themoment noticing[END_REF], others consider noticing to involve two processes, attending, and making sense, and making sense for some researchers is solely interpreting, whereas for others involves interpreting and deciding how to respond [START_REF] Jacobs | Professional noticing of children's mathematical thinking[END_REF]. In this paper we focus on a single component of noticing, that of attending. To improve one's awareness, habits need to come to the surface, in the sense that, in education, teachers should observe their own habits from a distance, and use them as objects of their own reflection. This process can be enhanced when teachers work together and are part of a community where specific practices are shared (Wenger, 1988). Implementing Lesson Study, teachers are part of the same community and in preparing their Lesson Plan they have to share common practices and to define where to direct their attention depending on the research problem [START_REF] Alston | Lesson study as a tool for developing teachers' close attention to students' mathematical thinking[END_REF]. During the observation of the lesson, teachers take distance from their habits, and pay attention to something new, they "notice" something. During the following discussion meeting, when teachers work together, these noticed things become objects of critical reflection. Critical reflection is a social process oriented to an action. "To notice is to make a distinction […] that distinction enables an appropriate action to be initiated. […] The coupling of a distinction and an action is what Gattegno refers to as an awareness." (Mason, 2021, p. 240). Based on the previous considerations, [START_REF] Andriano | Teachers' awareness of classroom interactions in hybrid distance education through lesson study[END_REF] proposed to frame the analysis of awareness in teachers' discussions and reflections in a qualitative study divided into three steps: noticing, objectifying, and acting. The terms are used with the following meaning: a) Noticing: teachers notice something, there is a shift of attention. It can take place below the level of consciousness; it is like stepping into a new point of view.

b) Objectifying: the noticing becomes an object of discussion. Teachers take distance from their habits and their actions become objects of collaborative reflection. It is a social process, the moment of critical reflection.

c) Acting: "The important question is whether discerning that distinction enables an appropriate action to be initiated" (Mason, 2021, p. 240). Critical reflection is oriented to action. It is a transformative process, i.e., a collaborative moment of teachers' development, where teachers agree on acting and change their own practice.

According to these theoretical elements, our research question is: How do teachers improve their awareness when involved in a professional practice inspired by Lesson Study?

Context and method

This study takes place in an Italian secondary scientific high school (grades 9-13) and it is part of a larger, three-year research project aimed at introducing LS in a group of mathematics teachers and studying the teachers' processes. These teachers volunteered to be introduced to LS by a colleague working in the same school (the first author of this paper), who acted as facilitator in disseminating LS main ideas and in organising the different phases of a cycle. Since the facilitator had the opportunity to participate as an observer in some LS research lessons and discussions in Shanghai, during a period in China, coming back to Italy she first shared her experience with the Mathematics Education researchers of the Department of Mathematics "G. Peano" of the University of Torino and Bari (the other authors), then introduced LS to her colleagues. A decision arising from the meetings with the researchers was to introduce LS to colleagues as an experience "in action".

The first year of the project was devoted to finding a suitable structure for implementing LS in the Italian institutional context, in line with the Italian background of Mathematics education [START_REF] Arzarello | Italian trends in research in mathematical education: A national case study from an international perspective[END_REF], and with the Cultural Transposition framework [START_REF] Ramploud | Chinese lesson study: Critical aspects of transfer from China to Italy[END_REF].

During the 2019/20 school year, only one LS cycle was implemented, due to the COVID-19 pandemic. In 2020/21, multiple cycles of LS were implemented. The data for this paper come from these cycles. The LS group consisted of 5 teachers with 10-20 years of teaching experience in secondary school. One of the five teachers, the first author, served as the LS facilitator. The group had taken part in one LS cycle in the previous year. Three teachers taught in grade 12, one teacher taught in grade 10 and one teacher taught both grade 10 and 12. The experiment involved four grade-12 classes and two grade-10 classes. The LS followed this structure:

-1 online "Goal Setting" meeting to choose the research theme and the topic (henceforth referred to as M1); -2 "Planning" meetings to design the research lesson and write the Lesson Plan (M2, M3). -6 research lessons: 4 lessons in grade 12 (L1, …, L4) taught by teacher T1, …, T4; 2 lessons in grade 10 (L5, L6) both taught by teacher T5 (there were 6 lessons because all teachers wished to teach the lesson and they were allowed to). -6 "Discussing" meetings, each following the respective research lesson in the same day (from here on, discussion meetings are referred to as DM1, …, DM6). -1 online meeting to summarise the project conclusion (M4).

Due to the pandemic, the classes involved in the project were only allowed to have 50% of students physically present in the classroom. The students were split into two groups: one attending class physically, and the other attending online. Therefore, in M1, teachers choose two research problems: 1) analyse students' combinatorial thinking; 2) compare the interactions among the students and between students and teachers according to the different modes of attendance: physical and online. In this research we focus on the first research problem. In M1, it was decided to plan one lesson for the introduction to combinatorics because it is a suitable topic both for grade 10 and 12 and furthermore the didactics of combinatorics is as challenging for students as it is for teachers [START_REF] Batanero | Statistics and probability in high school[END_REF]. In M2 and M3, the teachers designed the research lesson and wrote the Lesson Plan. The focus of the lesson was a group activity on a problem followed by whole class discussion. The lesson included three phases: 1) Introduction: the teacher introduces the topic starting from the concept of permutation.

2) Working group: students work in groups on a problem. 3) Institutionalisation: a whole class discussion.

The problem for the group activity was chosen in M2 and M3. Teachers were looking for a problem with a simple formulation for straightforward understanding, and at the same time engaging and challenging for all students. The teachers agreed upon the Alpinists' Problem reported on the textbook:

8 alpinists need to rope up in line to walk across a glacier. Two of them are beginners and cannot be placed neither in the first nor in the last position. How many ways are there to form the line?

During the lesson, students were given the problem and tasked to work in groups, without teacher's help. In-person students worked in groups of three. Online students worked in bigger groups in Zoom breakout rooms. Two teachers observed the physical groups, and two teachers observed the students in the Zoom breakout rooms. All LS meetings and lessons were video recorded. Teachers' observation notes, as well as student samples and interactive boards, were collected and saved during the lessons.

Method

The aim of this study is to analyse teachers' critical reflection during collaborative moments of discussion. The theoretical framework justifies the use of qualitative analysis. Therefore, the videos, the transcripts of teachers' words and teachers' observation notes are the basis of our data. To carry out the analysis we viewed many times the videos and we analysed transcripts of meeting discussions using the developed theoretical lens (noticing, objectifying, acting). First of all, we identified in the videos some moments of noticing, i.e., some moments in the discussion when a teacher reported to have noticed something, a shift of attention. Then we kept track of these noticings in the same and in the following meetings, considering these questions: do they evolve in a critical discussion, or are they left forever, or are they left and then return in following meetings? Is there a moment of critical reflection? Does this critical reflection result in teachers' action? In this way we collected some episodes where we can distinguish the three steps: noticing, objectifying, and acting. In this paper we focus on the analysis of one episode of noticing that evolves in a critical reflection and results in teacher action.

Results and data analysis

In this section, we analyse, through the developed theoretical lens (noticing, objectifying, acting), one episode where students' combinatorial thinking becomes objects of inquiry for teachers. The following transcripts are taken from the third "Discussing" meeting (DM3) that took place one hour after lesson 3 (L3) delivered by teacher T3. Teacher T3 introduced the meeting with some considerations about the lesson. Then every teacher added some comments.

Noticing

T5:

At a certain point, in the problem but also in the final solution, it was said that you can move the letter to a different position, or you can fix the position and choose a letter to put in. Because it is different… (DM3, 11:45) T5 notices something, makes a distinction, and that distinction is expressed in words. T5 notices a different way to see the permutations. The written calculation is the same, but it is seen in a different way. T5 notices that the same occurs in the whole class discussion. Even though the calculation is the same, 6x5x6!, students' reasoning in solving the problem was different from the one that the teacher used to summarise. The object of noticing becomes the object of critical reflection. Teachers start discussing combinatorial reasoning.

Objectifying

During the discussion, teachers became aware of different ways of solving the problem. Moreover, they became aware that summarising students' strategies could change the original idea. The discussion goes on comparing different solutions of the alpinists' problem proposed by the students.

T2: Yes, you can see it in two ways. T5:

Actually, also in the final solution. You had summed up the reasoning in a different way compared to what the student said. Writing 6x5, the student didn't say: 'for the first and the last place I have 6 and 5 possibilities'. He said: 'beginners can stay in 6 places. I take the first, he can stay in one of the six central places. T1:

Yes, right! T5:

Then one takes the second beginner and can put him in 5 places. So, he [the student] took the places and put in them the alpinists. T3:

Right, true! T5:

The student said this, then you summed up in a different way that was the way we solved the problem [when preparing the LP]. … (DM3, 12:10) T3:

Often, when a person has an idea, s/he doesn't listen carefully to the solution, and doesn't have time to understand deeply. I didn't get this solution, you [T5] are telling me now. I saw the right calculation; the student was explaining, and I summed up to the class in the way I understood it. This thing is interesting. (DM3, 13:14)

Combinatorial thinking becomes the object of critical reflection. During the Lesson Plan design, teachers' solution of the problem followed this reasoning: you have 8 places for the 8 alpinists. In the first place you can choose among 6 experts, in the last place you can choose among 5 experts, and then you must fill the 6 central places with the remaining 6 alpinists. In formula: D6,2•P6=6•5•6!.

During the lessons, many other strategies were found by the students. For example, in the transcript, T5 observed that the solution proposed by a group had the same calculation but a different reasoning: in the first solution (the teachers' one) the objects are the alpinists, in the second the objects are the positions in the rope. The noticing of T5 becomes an object of discussion. During the moment of critical reflection, T3, who delivered the lesson, became aware of the different strategies proposed by the student during the lesson and of the fact that his way of summarising had modified students' reasoning: "I didn't get this solution, you are telling me now. I saw the right calculation, the student was explaining, and I summed up for the class in the way I understood it". Other groups of students proposed strategies by subtraction, i.e., 8! -2 During the discussion T4 reports the fact that, observing the lesson delivered by another teacher, realised that in the same situation they would have acted differently: "I wouldn't have insisted on this kind of solution. I would have insisted on the other way. I would have dropped the solution by subtraction, pretending not to see it". The critical reflection made it possible for T4 to envision other ways of acting: "probably this is wrong, because we don't all have the same ways of reasoning".

Acting

The critical reflection is oriented to an action. The critical reflection made it possible for T4 to have other action coming to mind in a similar situation. Two days after this discussion, the next lesson (L4), was delivered by T4. In the group activity, 3 groups found the right solution with different strategies. Two of them by subtraction. T4 didn't "drop" it. On the contrary, during the whole class discussion, T4 asked students to explain their strategies.

T5:

It was really interesting because there was more than one method by subtraction. … One group, instead of taking 8! and then subtracting all the non-acceptable possibilities, and in this way, you have first to subtract then to add, and it is difficult.

They have considered the position, not the place. They said: 'I must subtract: when the first and the last are beginners, so 6!; then when the last and the first [are beginners]; then when the first is a beginner, and the other is in one of the central positions: 6 times 6!.' … In this way they have only subtracted 2•13•6!. … T1:

Yes! 12 and 12 and 2, 26•6!, (All teachers nodding) T4:

Yes! And they were also able to explain. Their idea was clear. Then a student said that she didn't understand, and I asked the group to explain, and they explained clearly.

(DM4, 20:04)

Conducting the lesson, T4 showed a different awareness of combinatorial thinking. Two days ahead, during the discussion, T4 said "I would have dropped the solution by subtraction, pretending not to see it". On the contrary, conducting the lesson T4 had recognized combinatorial reasoning by subtraction and had invited the group to explain the reasoning to the whole class. The new level of awareness is recognizable in two ways. On the one hand, the teacher was able to recognize different ways of solving the problem, different combinatorial thinking, and therefore was able to lead the whole class discussion asking students to explain the different solutions. On the other hand, the teacher was aware of the fact that summarising students' reasoning could lead to misinterpretations, therefore T4 asked students to explain their own reasoning. Moreover, the discussion among teachers during this discussion meeting (DM4) shows a different level of awareness of combinatorial thinking of the whole group of teachers. They all took part in the discussion, showing that they were able to recognize the different combinatorial thinking.

Discussion

The focus of this paper was to study how teachers' discussion in Lesson Study can promote changes in teachers' awareness and in their practice. To analyse collaborative moments of critical reflection in teachers' meetings, and to capture how awareness can promote changes in teachers' practice, we proposed a theoretical lens structured in three steps: a) Noticing: teachers notice something, they make a distinction and express it in words. b) Objectifying: the object of noticing becomes an object of critical reflection. c) Acting: critical reflection is oriented to action. There is a change in teachers' practice.

This framework allowed us to examine and interpret teachers' discussion. During the LS, teachers improve their awareness on combinatorial thinking. They become aware of the wide variety of strategies that students use in solving the problem, and this enables them to effectively lead the whole class discussion on the problem. The improvement in teachers' awareness regards two different aspects. The first one concerns combinatorial thinking: teachers become able to recognize different strategies and argumentations supporting the same written calculation. This awareness lets teachers choose the appropriate task, direct students' attention and effectively lead the whole class discussion enhancing students' thinking and contributions. The second improvement in teachers' awareness regards the communicative strategies used during a whole class discussion. One of them is summarising students' strategies and during the discussion they became aware that summarising students' strategies could change the original idea. These improvements in awareness have been reached through teachers noticing and moments of collaborative critical reflection, and it results in actions or changes in teachers' practice. This was possible because of the participation of teachers both in the observation of the lessons and in group discussions, which is typical of LS. In this paper only one episode is analysed. We need to expand our research analysing further episodes and identifying different aspects of teachers' awareness, especially focusing on the moments when noticing leads to objectifying and acting, and when it does not. Moreover, further research is needed in order to analyse which are the characteristics of the applied design of the LS which emphasise this learning of teachers.

  have been able to do it. And probably, for my idea, I wouldn't have insisted on this kind of solution, I would have insisted on the other way. I would have dropped the solution by subtraction, pretending not to see it. And probably this is wrong, because we don't all have the same ways of reasoning.

		•7! -2•7! + 2•6! or 8! -26•6!. These, and other solving
	strategies observed by teachers in the group work, were discussed.
	T4:	I really appreciate your explanation about the solution by subtraction, 8! minus….
		I would not It is not
		respectful of students' ways of working.	(DM3, 18:01)


Permeable viewpoints and parasitic influence in BBC and Netflix's Dracula

Can Dracula still surprise us? It hardly seemed possible when the TV series Dracula directed by Steven Moffat and Mark Gatiss was released in January 2020 on the BBC and on Netflix. And yet, although it marks a return to the original text, Bram Stoker's novel, it catches the spectators off guard with all its unexpected twists and turns. One of the first aspects commented on by the reviewers was the elaborate plotting, typical of the creators of the BBC series Sherlock (2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017). While Screenrant criticised it because the promise of "lavish climaxes" was not successfully fulfilled (Donaldson), The Guardian praised it because the "dramatic flourishes" that the series is strewn with "give you the feeling that this is not just a treat but a tribute" (Mangan). The TV show presents a series of mysteries and riddles within the riddles and they rely heavily on the use of viewpoints. Benjamin Poore has qualified Sherlock as "a trickster narrative" where "virtuoso sequences … which initially appear to be 'genuine' flashbacks of the show's 'real' storyline, … are then revealed as unconvincing theories or fantasies" (96). The truth is often disguised because "like the trickster straddling two worlds, the perspective of Sherlock veers unpredictably between the physical world and the world of the mind" (Poore 97). This view certainly applies to Dracula as well, where most of the riddles are deeply embedded in the representation of the characters' subjectivity. What is troubling however is the interaction between the various subjective perspectives offered to our view. The aim of this article is to analyse this permeability of the viewpoints of the various characters not just as part of the ploys of detective fiction but also as a clue indicating parasitic influence. But while the parasite is used in general as a metaphor to express the fear of the foreign, a shift occurs in BBC and Netflix's series with the focus on death: contagion is explained by the parasite's refusal to die with its host, a view that humanises it.

Viewpoint and parasitic detectives

As Mark Gatiss said about the creation of Dracula, it all started as a joke when the silhouette of Benedict Cumberbatch shot from the back in his cloak with his collar up suddenly reminded him of Dracula (Gilbert). The connection between the vampire and the detective runs however much deeper than that. Investigation is a key theme in Bram Stoker's Dracula, as several critics have pointed out. 1 Gatiss and Moffat's Dracula presents three main detectives in as many episodes: Sister Agatha Van Helsing, Dracula himself, and then Zoe Van Helsing. 2 This detective work implies a strong interaction between the subjectivities of the characters. Episodes one and two ("The Rules of the Beast" and "Blood Vessel") 1 Rosemary Jann in particular has brought attention to the link between Sherlock and Van Helsing and has demonstrated that the vampire hunters back their claims with "observation, deduction and hypothesis testing" (280). The principle of "abduction" is used when the characters arrive at the correct hypothesis by eliminating the false ones (281) and the characters "order facts into chains of deduction" (282). 2 There is an interesting echo of the vampire hunters' chase of Dracula when Zoe finds Dracula's home, using her knowledge to narrow down the possible buildings in episode three. A blend of Stoker's Van Helsing and Mina (although Mina does appear, though rather briefly, as a character too, in episode 1), she uses a "codified system" (Jann 283): she guesses Dracula's choices based on his breeding and ego ("The Dark Compass") and thus implicitly labels him as an aristocratic egotist, which makes him deterministically unable to hide, just as the "deterministic Lombrosoan model" (Jann 283) used by Van Helsing and Mina in Stoker's novel helps them infer his decisions based on the selfishness of his "child-brain" (296). Zoe's hypothesis is then verified thanks to his phone number, which is listed.

foreground the role of Sister Agatha3 in the discovery of the truth through her reading of the other characters' minds. This mind reading implies both a deep immersion in and a critical distance towards the viewpoints of the others. In episode one, she adopts the critical attitude expected from a detective dealing with a testimony, whose subjectivity might be misleading. Jonathan's perspective when he recounts his adventures in Dracula's castle to Sister Agatha is conveyed with what George Wilson calls "subjectively saturated" shots, that is to say shots where "the character's visual perspective is completely determined by his or her present state of dreaming, hallucination, or inner visualization" (85). However, those shots are constantly interrupted and the superimposition of Sister Agatha's voice repeatedly breaks the spell and leads the spectator to take some distance towards the narrative with those recurrent reminders that what is seen on screen is not "reality" but Jonathan's version of it.

Such interventions also re-orient the investigation: the riddles that Jonathan attempts to solve are akin to red herrings. In his story, he persists in presenting himself as the knight in shining armour and tries to find out who carved a call for help on his window pane and where the baby brought to the castle has gone while the woman he is trying to help is a predator, and he is the one in dire straits. Jonathan appears to be constantly on a wild goose chase, lost, as symbolised by his inability to find his way in the labyrinthine castle. Those riddles, along with the mystery of the hidden map or the lack of servants in the castle,4 obscure the real question, the one Sister Agatha is trying to answer: what caused Jonathan's present illness and what is the true nature of Dracula? The importance of the ordering of facts and the focus on details in testimonies was stressed in Stoker's Dracula5 and like Abraham Van Helsing, Gatiss and Moffat's Sister Agatha draws the spectator's attention to significant details that failed to be interpreted by Jonathan when he told his story. When she asks why the woman wrote her call for help in English and not in Rumanian, it is supposed to help Jonathan realize that the woman was a vampire: it was through his blood that she had imbibed his knowledge and language. Sister Agatha's probing lays bare the inherent unreliability of Jonathan as a narrator while guiding him towards the truth. She helps him gradually unearth his repressed memories and discover his true nature: he has become a vampire.

Her reading of Jonathan's mind leads however to a rather troubling appropriation. Her influence over Jonathan's narrative is first suggested by the use of her voice as a voice over commenting on the subjectively saturated shot which depicts Mina in Jonathan's dream ("The Rules of the Beast"). Her rising interference is then conveyed by the projection of her image inside Jonathan's memory at the moment when he is trying to understand how he escaped from the roof of the castle ("The Rules of the Beast"). The following shot, which pictures Jonathan framed by a window with Sister Agatha behind him, symbolically underlines her overbearing influence over his viewpoint. Her appropriation of his narrative and even his memories is complete once his account is over. When she needs further information to understand better what made Dracula scream before Jonathan fell from the battlements of the castle, his memory is simply retraced, as suggested by the rewinding of the shot showing his fall. This rewinding starts immediately after a close shot on Sister Agatha's lit up face which suggests that the truth is dawning on her. Consequently, the status of the rewound shot becomes very ambiguous: is it still Jonathan's memory or is it Sister Agatha's version of it as she conveniently manipulates it to find her answer? Jonathan says he cannot remember and she is the one who realizes that he was facing the sun. She knows that Dracula screamed because he was blinded by the golden crucifix on Jonathan's neck even before Jonathan remembers that detail. The epiphany it brings is more Sister Agatha's than Jonathan's and she concludes: "God is real!" The permeability of the two viewpoints unveils the vampiric quality of Sister Agatha's interpretative reading.

In the TV series as a whole, a parallel is drawn between the reading of stories and the reading of blood itself. As Dracula tells Zoe, "Everything is in the blood, if you know how to read it" ("The Dark Compass") because "blood is lives. Blood is testimony." The slight shift from the original "Blood is life" in Stoker's novel to "Blood is lives"6 utterly changes its meaning. It turns the vampire into a detective of sorts, whose thirst for knowledge sends him on a quest for testimonies, which he then carefully interprets. Dracula's motto is accurately explained by Sister Agatha: "stories flow in our veins, if you know how to read them" ("The Rules of the Beast"). The metaphor of reading as vampiric appropriation also appears implicitly at the beginning of Episode two. The attention paid by Dracula to the quality of the books he reads recalls his careful choice of the preys he feeds on. In Gatiss and Moffat's Dracula, stories are indeed food for thought and Sister Agatha's use of Jonathan's account evokes Mina Harker's appropriation of the other characters' narratives to produce meaning in Stoker's novel. As Valerie Pedlar has highlighted, Mina's "performance as collator and editor-in-chief of the volume we are reading" can be compared to the way she reprocesses Dracula's blood and makes it her own: she consumes the characters' written material when she puts it in order and types it out (224-225). 7 This vampiric assimilation is portrayed in the TV show through Sister Agatha's influence over Jonathan's account. Her role as a parasite is further emphasised in episode three: she has been carried in Dracula's veins for over a hundred years and is also in Zoe's veins once the latter has drunk Dracula's blood ("The Dark Compass"). She causes hallucinations and absorbs Zoe's knowledge and modern language ("my DNA syncs with yours").

It is however in episode two that the deepening of the bond between Sister Agatha and Dracula is explained. Dracula tells the story of the voyage of the Demeter to England, peppered with a series of murders. As was the case with Jonathan Harker's narrative, Dracula's account is repeatedly interrupted by her comments, which sometimes appear as voice over when the images are suddenly frozen ("Blood Vessel"). She is again challenging Dracula's narration and pointing to its unreliability. This however turns to a battle: mind reading becomes a mind game, as symbolised by the game of chess, a recurrent image in the many adaptations of Doyle's works to depict the battle between Sherlock Holmes's intelligence and that of the criminal. 8 The metaphor was used in Stoker's novel to depict the vampire hunters' fight against Dracula's cunning. 9 In BBC's Dracula, the game of chess during which the story of the mysterious deaths on the Demeter is told materialises the battle of Sister Agatha's and Dracula's minds. The characters' winning of losing moves on the chessboard reflect their proficiency in their verbal match.10 Sister Agatha's immersion into the mind of Dracula evokes the projection into the mind of the criminal that typically helps the detective overcome his foe. Like both Stoker's Mina and Van Helsing, she must form a psychic union with the criminal before she can expose the truth and fight him. Several critics have compared Mina's connection to the Count and Abraham Van Helsing's use of it in his investigation to Sherlock Holmes's approach. Catherine Wynne states that "like Holmes's corresponding relationship to the criminal, they must become Dracula in order to destroy him" (56) and Betsy Van Schlun has shown that Mina's brain, with her "power of insight," her special bond that "enables her to relate to the criminal" and her extraordinary memory of data, reminds the reader of Sherlock Holmes's great intellect (303). In the TV series, the projection into the mind of Dracula does provide a better understanding of the vampire's experience, notably with the "subjectively inflected" point of view shots "supposed to represent subjective enhancements and distortions of the character's field of vision at the time" or "internal properties of a character's perceptual state" (Wilson 85). The flashes, closeups and extreme close-ups on the wound of the sailor with the broken leg on the Demeter represent the flow of images overwhelming Dracula when he sees blood ("Blood Vessel"). This subjective representation is correctly interpreted by Sister Agatha as revealing his addiction to blood, a view that is not just confirmed by Dracula himself but also by the following close ups on the rare meat being cut, on the jugular in the throats of Adisa and Dr Sharma and on the vein inside Lord Ruthven's wrist, especially with the subjectively emphasised pulse of the veins ("Blood Vessel"). Such an interpretation also ties in, as Richard J Walker has demonstrated, with a possible reading of Stoker's Dracula. 11However, Sister Agatha's immersion into the mind of Dracula in episode two goes far beyond mere detective work, it goes deeper than she realizes. The extent of the fusion between the two characters utterly destabilises the narrative. While the voyage of the Demeter was presented as fiction, it turns out to be 'reality'. The game of chess on the other hand is unreal. This is a masterly instance of what George Wilson calls "twists" since "the epistemic structure of this segment is not specified straight away when it occurs" (89): while the spectator thought he was discovering the perspective of the vampire, he was also in fact in Sister Agatha's mind and what was presented as "real" was a shared hallucination. Sister Agatha was not projecting herself into Dracula's mind. His thoughts and memories were already within her, planted in her brain. Ironically, the little boat that Dracula places on the chess table reminds the spectator of the staggering twist in Bryan Singer's The Usual Suspects (1995) when the whole story turns out to have been invented by the protagonist telling it, as revealed by the close ups on the little objects and names that were the source of his inspiration. Here, the parallel between the boat in the bottle and the story of the Demeter seems to suggest that the events on the ship are all fabricated, while in fact they are real.

One is then led to reassess Dracula's little parody of a detective story. The very fact that he masquerades as a detective reveals his interaction with his victim: he has acquired the attitude of a sleuth through his consumption of Sister Agatha's blood. Dracula's ridiculous role-playing once again underlines the parasitic nature of the detective. It is because he is a vampire that he can so easily project himself into the mind of the others and unravel all the little mysteries so easily. If he has no trouble answering the Grand Duchess's little riddle (what was her birthday gift?), it is because she is a former prey and their minds are merged. When he answers the question, the angle reverse angle on their two faces makes the interspersed flashes ambiguous: one can no longer know whether they represent Valeria's memories or Dracula's ("Blood Vessel"). The depth of the bond between the two characters is conveyed through the gradual zooming in on their two faces. A parallel is here drawn with storytelling: the Grand Duchess is a wonderful storyteller, building up suspense, and Dracula appropriates her tale and finishes it. In his narrative, Dracula's appropriation of the viewpoints of the various characters is associated with his role as an omniscient narrator. If the whole detective plot around the deaths on board the Demeter is told from Dracula's perspective, the narration should then be limited to his viewpoint. However, as in a novel told by an omniscient narrator, the spectator also witnesses the events happening in other parts of the ship and has access to the other characters' memories and even their dreams. This would not be surprising if the whole story was fabricated but since the events turn out to be "real," it becomes problematic. The story starts with a horrific scene portraying the Captain of the Demeter in his cabin, waking up to be attacked by a decomposed sailor, later revealed to be the cook. This was however just a nightmare and there is here yet another twist: we were in fact immersed in the mind of the Captain. But how could we be if Dracula is the one telling the story? This immersion is possible because he is a vampire. Dracula's omniscience as a storyteller is explained by his ability as a vampire to imbibe the thoughts and knowledge of all the characters. This sway over his potential victims was represented in Stoker's novel through his power as a hypnotist. Many critics have studied Stoker's Dracula as a comment on contemporary fears and practices of mesmerism and hypnotism.12 Most interestingly, the anxiety around Dracula's invasion of his victims' minds has been interpreted as a projection of the contemporary fear of criminal mesmerism as a parasitic and contagious influence. As underlined by Lara Karpenko, John Bennett compared mesmerism to a contagion in his The Mesmeric mania of 1851 (Karpenko 151). In BBC's Dracula, the vampire's spell over the characters and the invasion of their minds are presented in a way that clearly evokes parasitic contagion.

Viewpoint and parasitic infection

Dracula's contaminating influence can be felt in Jonathan's narrative when he cuts his finger on one of the shards of the mirror that the vampire has broken in Jonathan's room ("The Rules of the Beast"): the extreme close up on the falling drops of blood, combined with the slow motion, represents Dracula's subjective perception rather than Jonathan's. This is confirmed by the following close up on Dracula's face: his blood-shot eyes and his monstrous mouth, with the fangs out, reveal his blood lust. This temporary disclosure of Dracula's true nature seems however to have gone unnoticed by Jonathan at the time. The vampire's ability to commune with the others' minds also appears when he intimates the deaf and mute daughter of Dr Sharma onboard the Demeter not to tell what she witnessed during the night, when she saw him attack Lord Ruthven's wife, Dorabella ("Blood Vessel"). The girl's memory is altered: contrary to what happened at the time, Dracula now notices her presence as she watches and he makes a shushing sign with his finger on his lips. This contamination of the characters' viewpoint is also recurrently conveyed with the superimposition of his face on the hallucinated image of the victim's beloved when Dracula is feeding on his preys. The sex scene between the injured sailor and Dorabella is revealed to be a hallucination when Dracula's hand, and then his face, replace the woman's ("Blood Vessel"). Similarly, a flash of his face interrupts the dreamed love scene between Jonathan and Mina ("The Rules of the Beast"). The vampire infects the others' viewpoint, as symbolised in the opening credits with the extreme close up on an eye looking upwards, its iris and its rim oozing blood.

This contamination is also represented metaphorically through the breaking of boundaries and the permeability between the inside and the outside. The window is used as an image to represent a potential way in. Episode one starts with a zooming in on a convent and its window, which is immediately followed by a zooming out on the window from the inside ("The Rules of the Beast"). The flowing movement that makes the spectator enter the convent although the window is closed suggests a certain permeability. When Dracula later arrives at that convent, the impending contagion is foreshadowed by the movement of the bats breaking the window and entering the room to attack Mina. The choice of the window as a point of entry highlights once again the infection of the characters' viewpoint since looking out of the window has become a common trope to refer to a character's perspective. The initial zooming out on the window features a fly on the windowpane. The following extreme close ups on Jonathan's eye and on the fly's eye lead the spectator to connect the two. The irruption of that unusual shot on the fly is so sudden that it feels invasive. When the insect finally lands on Jonathan's iris and goes inside it, it becomes finally clear that his view is infected. The contamination of the mind is then associated with the contamination of the body. It is metaphorically conveyed with Dracula's repeated attempts to break boundaries, as when he tries to enter the convent or the protective circles that Sister Agatha, like Stoker's Van Helsing, creates. When Dracula finally enters the convent and then the protective circle around Sister Agatha and Mina, he can also enter Sister Agatha's head, as his presence in her mind palace in episode two suggests.

Such situations seem to picture the breach of a cell by a virus. A parallel can be drawn with Stoker's novel. When Abraham Van Helsing tries to protect Lucy, he is bent on hermetically sealing the window and carefully rubs garlic "all over the sashes" (121). Garlic is then used both as an antiseptic13 and as a medicinal remedy boosting her immune system: the garlic changes the nature of all the foreign elements coming inside the room since the Doctor ensures that "every whiff of air that might get in would be laden with the garlic smell" (131). His first attempt fails because of the intervention of Lucy's mother, who opens the window and lets the virus in. The second time, a wolf summoned by Dracula breaks in, crashing the window, which seems to symbolise the breach of a cell membrane by the virus. The "whole myriad of little specks … blowing in through the broken window" (131) represents the viral microbes infecting Lucy's body. The obvious resemblance between the symptoms of Dracula's victims and the symptoms of syphilis has led several critics to study the representation of vampirism as a metaphor for disease in general.14 Martin Willis has shown that Dracula stages the conflict between two contemporary medical approaches, contagionism, the belief that infectious diseases were passed through close contact or touch (Willis 305), and miasmatism, the theory asserting that "diseases were the product of environmental factors" (Willis 305), to finally turn to the nascent germ theory, as the descriptions of the metamorphosis of the infectious vampires into little specks suggests. This particular feature has also been commented on by Jens Lofhert Jørgensen. He draws the attention to the "bacteria-like features" of Dracula when he adopts the shape of "infinitesimal dust particles … capable of entering even hermetically sealed room" (41) and shows that the vampire hunters' sterilisation of the crates of earth disseminated by Dracula evokes the containment of an "epidemic threat" (41). Hub Zwart on the other hand has paid closer attention to viral zoonosis, focusing on the role of the bats in the transmission of diseases (28).

Gatiss and Moffat also present vampirism as a viral infection: it is associated with physical symptoms conveying both illness and the loss of integrity of the self with the diseased fingers, the nails coming off, the loss of hair and the skin lesions. Dracula's attack on Zoe in episode three makes the comparison with a disease quite clear. When he drinks her blood, he is projected into her mind and then rejected because Zoe's body is already inhabited, it already harbours a parasite, a creature of death and decay ("The dark Compass"). This creature is the embodiment of cancer, the disease gnawing her from the inside. Zoe seems at first trapped with Dracula in an imaginary place reminiscent of Caspar Friedrich's The Abbey in the Forest (1809), like Sister Agatha in her mind palace. However, Dracula is the one who is contaminated and trapped with Zoe's parasite, death itself. What appeared to be a representation of Zoe's mind then becomes a representation of Dracula's mind and one is led to draw a parallel between the parasitic presence of the vampire in the characters' bodies and thoughts and the parasitic presence of the figure of death in Zoe.

The showrunners' interpretation seems even more literal in the title sequence, which showcases images of red cells put under a microscope. The very first image is an orange-red virus-like shape and its tentacles are fanning out, picturing the spread of the disease. In some shots the red cells look putrid and their membrane is broken by flies. As Stacey Abbott has demonstrated, "the emphasis on the close-up of the blood cell", and more generally the "preoccupation with analysing the minutiae of life, such as bacteria, microbes, viruses, genes and atoms," "has its place across a long history of visual representation of the vampire in film and television" (45). The vampires can be associated with viral microbes because they are in between the living and the non-living and like vampires, "are unable to reproduce until they are within a living host" (49). Once inside the host, they can get inside a cell and even genetically modify it to facilitate their own reproduction (49). This gradual modification is very strikingly emphasised in BBC's Dracula. Dracula is invading Jonathan's body as well as his mind. He is becoming Jonathan, up to the point where he actually wears his body like a suit and rips it open to shed it like a skin ("The Rules of the Beast"). This horrifying parasitic appropriation of the body is foreshadowed when Dracula claws his way out of the body of a wolf instead of shapeshifting back into his human form ("The Rules of the Beast"). Organic parasitism is here pushed to its extreme.

Why flies?

Emilie Taylor Brown and Ross Forman's interpretation of this kind of parasitism in Stoker's novel might help shed light on the choice of the association of flies and parasitic infection in the TV show. The method of transmission (biting), the symptoms (anaemia, fatigue and fever) and even the efficacy of garlic and the danger of leaving the window open at night are the same in malarial infection and vampirism (Taylor 18). More specifically, Dracula's fangs "resemble the proboscis of the mosquito, long thought to transmit tropical diseases" (Taylor 17). The metaphor of malaria certainly applies to Gatiss and Moffat's Dracula as well. When Dracula appropriates Jonathan's body in episode one, his aim is to use it as a disguise to enter the circle of wafer that protects Sister Agatha and Mina. His shapeshifting and his use of Jonathan's body as a trojan horse to cross the protective barrier call to mind the strategy deployed by the parasite that triggers malaria. Indeed the malaria parasite, the Plasmodium, is able to switch "between as many as 150 genes" to outsmart the immune system and avoid detection (Nowak 755) and uses host cells in the liver to develop buds of parasites. It also acts on the signals sent by those dead host cells to the immune system in order to ensure their safe passage into the bloodstream (Sturm et al.). However, mosquitoes are not featured in the TV-series. The flies seem to be playing their role. They do not bite but they can invade the body, as underlined earlier, and they are systematically used to mark the characters' illness.

The choice of flies in the TV show was certainly dictated by their association with decayin the title sequence, they both herald putrefaction and follow in its wake. It is also a hint at Renfield's obsessionand indeed, Renfield, a lawyer in the TV series, eats flies in episode three. In Stoker's novel, the metaphor of the fly is used to convey Renfield's rising influence over Dr Seward, whose mind is gradually contaminated by the madman's views. When the doctor obsessively attempts to find the right label to define the lunatic, the answer is said to be "buzzing in his mind" (71) during the whole day, a clear echo of the buzz of the fly "bloated with carrion food" that Renfield ate not long before (69). The buzzing thought seems to have been planted in Dr Seward's mind, like a growing parasite: "the rudimentary idea in his mind is growing" (69). Van Helsing too has a half thought "buzzing" in his brain (295). Renfield and his flies evoke vampiric assimilation. The madman's recreation of the Darwinian food chain and his "zoological ranking by sheer numbers of lives each species feeds upon for its survival" (Vrettos 166) are based on the assumption that, as Huxley emphasised, "all living things are reducible to universal protoplasm which, created by plants and consumed by animals, is plastic enough to be variously incarnated" (Blinderman 417). Renfield's fly eating is then a way to draw a connection between various lives, like Dracula when he is drinking blood. That particular aspect becomes a key element in the TV show. Blood is lives indeed, as graphically pictured in the opening credits: one shot features blood drops flowing from right to left. Each drop encapsulates the faces of various characters, which constantly shift and merge. Blood is more than testimonies: it is the protoplasm linking all lives.

This view sheds light on Dracula's words to Sister Agatha before drinking her blood at the convent: "You'll be part of me. You'll travel to the new world in my veins" ("The Dark Compass"). Through his blood, he makes her travel through time and establishes a bond with her great grandniece. The allusion to travel reminds the spectator that Dracula is a foreigner. As Stephen Arata has successfully demonstrated, with Dracula's arrival on English soil and his terrifying contamination of English victims, Bram stoker was playing on his contemporaries' fear of reverse colonisation: "the fear is that what has been represented as the 'civilised' world is on the point of being colonised by 'primitive" forces" (120). This is what Moffat and Gatiss seem to be hinting at with Dracula's allusion to his impending travel from Rumania, portrayed as primitive, to "the new world," that is to say England … until Dracula finally swims to the English shore to discover that this world is literally new: it is now the 21 st century. Similarly, the image of the parasite has traditionally been associated with foreigners, and more particularly the Jews. 15 The fear of the parasite is also a fear of the invasion of the foreign. But the shift from the metaphor to a more literal representation of the parasite leads to a more universal reflection on the fear of death. The end of episode three unravels what is presented as the greatest mystery of all, the mystery around Dracula's origins, his limitations and his allergies. As it turns out, Dracula has imbibed all the superstitious beliefs concerning vampires and has been limited by his fear that they might be true. Garlic, sunlight, crucifixes: all those weapons work on him because he is unwilling to take the risk of being hurt. His main preoccupation has always been to avoid death, which terrifies him. A reviewer called this revelation "a big thematic and emotional let-down" (Donaldson) but it ties in perfectly with the literal reinterpretation of Stoker's novel in the series as a whole: it is the parasite's survival instinct, its refusal to die with its host that leads to further contamination. Although Dracula is depicted as a monstrous villain, this conclusion tends to humanise him but at the same time it also brings us uncomfortably closer to the bacteria and microbes that inhabit our bodies. Common to all life-forms is the necessity to face death and it is what leads ultimately to the harmonious fusion between Zoe / Sister Agatha and Dracula. In death, they finally put an end to the fight and become one in the cell-like circle of fire ("The Dark Compass").

Works cited

She is a blend of three of Stoker's characters: Abraham Van Helsing, Mina and Sister Agatha, the secondary character that nurses Jonathan back to health.

This mystery is also explored in Stoker's book.

As Rosemary Jann pointed out, the very "methodical ordering" (280) of the testimonies plays in itself an important role in the investigation since it is through the arrangement of all the evidence gathered in chronological order "that the main characters suddenly see the pattern that points to Dracula" (280).

Moffat and Gatiss playfully underline the hint with Sister Agatha's question: "are you sure Dracula did not say 'blood is life?'" ("The Rules of the Beast").

Jennifer Wicke has also underlined that speech in Stoker's Dracula is vampirized by mass mediation and the "insinuation" of "invisible, or translated, stenography" (473). The text absorbs extraneous testimonies "just as Dracula assimilates the life-blood of his victims" (474). Rebecca Pope has more specifically called such appropriation parasitic: different discourses and languages are vamped together and the novel feeds off other discourses (199). Garrett Stewart presents Mina's transcription and retyping as a parody of vampirism (9) and Erik Butler explains that Mina's vampirism stems from the contaminating influence of the texts providing information on vampires. Because the texts are typed and thus mass-produced and mediated, they are no longer personal and Mina's immersion in them opens her body to infection (14).

Chess is however barely mentioned in Doyle's stories with the famous sleuth. One of the very rare mentions is when Amberley's proficiency as a chess-player is seen by Sherlock Holmes as the mark of a scheming mind in Doyle's "The Adventure of the Retired Colourman"(1926).

Van Helsing exclaims "Check to the King"(Stoker 135) when Lucy's health is temporarily preserved and he presents the little victories of the "Crew of light" as an "opportunity to cry 'check' in some ways in this chess game, which we play for the sake of human souls"(Stoker 223).

When Sister Agatha insults Dracula, calling him a pig because of his lack of restraint, and compares him to "a fox in a hen coop", he changes the derogatory simile into the meliorative "like a connoisseur in a wine cellar" ("Blood Vessel") and turns the insult into a compliment, associating the pig with the valued truffles it is able to find. This verbal battle goes on and on, as when Sister Agatha's description of Dracula ("a sophisticated gentleman nothing more than a veneer") is parred with "the sophistication of a gentleman is always a veneer" ("Blood Vessel").

According to him, the connection between vampirism and addiction is almost too obvious to make (257). Judith Halberstam describes vampirism in Dracula as "a psychological disorder, an addictive activity" (186).

According to Daniel Pick, at the end of the nineteenth century "the 'hypnotic menace' becomes a matter of forensic investigation and grave public concern" (77) and Dracula embodies that fear. For a further study of this fear of hypnotists and its relation to Stoker's novel, see alsoKelly Hurley.

In Martin Willis's terms, "surrounded by the garlic flowers, a metonymy for antiseptic that combats the metaphor of vampirism as a microbe, Lucy is well protected from the microbial infection that vampirism represents (313).

Pascale Krumm has shown that vampirism mimicks the pathology of syphilis with its three stages: hysteria, dementia and then death (7).

Monica Tomaszweska has related Stoker's depiction of the vampire as an infectious parasite, an invasive degenerate, to "the general apprehensions about degeneration of the imperial race" but also more specifically to the "fear of Judaisation" (6). The link between Dracula's monstrous parasitism and anti-Semitic discourse has also been established by Judith Halberstam. Jews crystallised many fears: they were linked to the spread of syphilis, to degeneration, to criminality and psychological disturbances (183).


Maxime Brunet

Florent Marine Pétriaux

Di Meglio

Nicolas Petit

Enabling safe walking rehabilitation on the exoskeleton Atalante: experimental results

This paper exposes a control architecture enabling rehabilitation of walking impaired patients with the lower-limb exoskeleton Atalante. Atalante's control system is modified to allow the patient to contribute to the walking motion through their efforts. Only the swing leg degree of freedom along the nominal path is relaxed. An online trajectory optimization checks that the muscle forces do not jeopardize stability. The optimization generates reference trajectories that satisfy several key constraints from the current point to the end of the step. One of the constraints requires that the center or pressure remains inside the support polygon, which ensures that the support leg subsystem successfully tracks the reference trajectory. As a result of the presented works, the robot provides a non-zero force in the direction of motion only when required, helping the patient go fast enough to maintain balance (or preventing him from going too fast). Experimental results are reported. They illustrate that variations of ±50% of the duration of the step can be achieved in response to the patient's efforts and that many steps are achieved without falling.

I. INTRODUCTION

Patients suffering from walking impairments are unable to produce the efforts required to achieve regular walk patterns. High-dosage walking rehabilitation has many benefits but is laborious for physiotherapists, particularly for the most impaired patients who struggle maintaining their balance. From these observations, the concept of robotic-assisted gait training has emerged. This concept involves motorized devices (in this paper a robotic exoskeleton) and aims at teaching patients how to produce the appropriate efforts to walk, following either a nominal gait pattern, ideally, or custom gait patterns tailored to their specific disabilities. All robotic-rehabilitation control laws [START_REF] Artemiadis | A Review of Robot-Assisted Lower-Limb Stroke Therapy: Unexplored Paths and Future Directions in Gait Rehabilitation[END_REF]- [START_REF] Martínez | A Velocity-Field-Based Controller for Assisting Leg Movement During Walking With a Bilateral Hip and Knee Lower Limb Exoskeleton[END_REF] introduce a certain level of freedom given to the patients. For exoskeletons, this constitutes a challenge because the patient muscle forces may jeopardize walk stability in unpredictable ways.

In this paper, we present control system updates enabling the use of the self-balanced lower-limb exoskeleton Atalante for rehabilitation. We choose to let the patient physically contribute to the motion of the swing leg and, consequently, we allow a modulation of the velocity at which a predefined gait is traveled. Because this directly impacts stability, a new reference trajectory has to be computed for the lowlevel controllers of the whole system. This is done online by an optimization-based trajectory planning algorithm. This methodology is presented in the article and tested experimentally. The two legs of Atalante are the subjects of very M. Brunet and M. Pétriaux are with Wandercraft, 88 Rue de Rivoli, 75004 Paris, France maxime.brunet@wandercraft.eu On the swing leg we proceed as follows. The nominal gait for the two-legged system, generated as in [START_REF] Harib | Feedback Control of an Exoskeleton for Paraplegics: Toward Robustly Stable, Hands-Free Dynamic Walking[END_REF], serves to define a geometric path for the swing leg. The exoskeleton efforts in the longitudinal direction of the path are nullified and left for the patient to produce, while the robot motion is strictly controlled in the hyperplane orthogonal to the path. To this end, we rely on the Virtual Guides (VG) methodology [START_REF] Joly | Imposing motion constraints to a force reflecting telerobot through real-time simulation of a virtual mechanism[END_REF], [START_REF] Restrepo | Intuitive, iterative and assisted virtual guides programming for human-robot comanipulation[END_REF]. Resembling [START_REF] Shu | Modulation of Prosthetic Ankle Plantarflexion Through Direct Myoelectric Control of a Subject-Optimized Neuromuscular Model[END_REF], our VG approach maps the high-dimensional user efforts to a one-dimensional quantity: the velocity at which the swing leg's geometric path is followed. Implicitly, this defines a new schedule for the path, the patient schedule, which we aim to follow as long as safety is not threatened.

On the support leg the control structure is also changed. The Atalante control system uses admittance [START_REF] Caron | Stair Climbing Stabilization of the HRP-4 Humanoid Robot using Whole-body Admittance Control[END_REF] to generate the contact forces that ensure the stabilization of the Centerof-Mass (CoM) dynamics around a reference trajectory. The reference trajectory tracked must satisfy the unilateral contact constraints (as a consequence, the Center of Pressure (CoP) must remain in the support polygon). Because the swing leg degrees of freedom (DoF) are used for the rehabilitation task, only the support leg can be used for the admittance controller.

To define this reference trajectory consistently with the patient schedule, a simple time rescaling of the nominal gait is possible but may violate of the aforementioned constraints. Therefore, combining VG and admittance control entails defining a more careful approach. We chose to adopt an online planning strategy. This approach is similar in spirit to [START_REF] Tucker | Preference-Based Learning for Exoskeleton Gait Optimization[END_REF], as we optimize the trajectory to best satisfy the patient's input, with the difference that we adapt the trajectory at a higher frequency, to maximize responsiveness to the patient efforts

The online planning strategy we implement solves an optimal control problem (OCP) over an unspecified horizon for a Linear Inverted Pendulum (LIP) model, which represents the overall balance dynamics of the system. The patient schedule is treated as a penalty on the final time. The unilateral contact constraints is a constraint on the CoP. To ensure the feasibility of the next step, it is sufficient (see [START_REF] Wieber | Modeling and Control of Legged Robots[END_REF]) to require that the trajectory endpoint satisfies some geometric constraints. These are included in the OCP. Similar online trajectory generation can be found in many related works for quadrupeds [START_REF] Bellicoso | Dynamic Locomotion Through Online Nonlinear Motion Optimization for Quadrupedal Robots[END_REF]- [START_REF] Mastalli | A feasibility-driven approach to control-limited DDP[END_REF], humanoids [START_REF] Dantec | Whole Body Model Predictive Control with a Memory of Motion: Experiments on a Torque-Controlled Talos[END_REF], [START_REF] Romualdi | Online Non-linear Centroidal MPC for Humanoid Robot Locomotion with Step Adjustment[END_REF], or manipulators [START_REF] Kleff | High-Frequency Nonlinear Model Predictive Control of a Manipulator[END_REF]. It is worth noting that of all these schemes, only [START_REF] Kleff | High-Frequency Nonlinear Model Predictive Control of a Manipulator[END_REF], [START_REF] Mastalli | Crocoddyl: An Efficient and Versatile Framework for Multi-Contact Optimal Control[END_REF] solves a non-linear MPC problem at 1kHz as we do (with a CPU twice as fast as Atalante onboard computer). The simplicity of the LIP model allows us to use a fast resolution method for this nonlinear OCP, relying on the theoretical study of [START_REF] Brunet | Fast replanning of a lower-limb exoskeleton trajectories for rehabilitation[END_REF].

The methodology has been experimentally tested. During the experiments, the optimization algorithm finds a CoM trajectory. As long as the duration of this trajectory matches the patient schedule, the user is allowed to drive the swing leg velocity. Otherwise, the patient schedule is considered non-feasible (i.e. it jeopardizes the balance of the system). It is overridden and the swing leg velocity is modified. Consequently, the swing leg actuators generate non-zero forces in the direction of motion, helping the patient go fast enough to maintain balance (or preventing him from going too fast). In all cases, the support leg controls the robot balance. As a result of the methodology, the robot assists the swing leg motion only when required.

The main contribution of the article is the detailed exposition of this control methodology, along with its experimental validation. In particular, we report an experiment consisting of a succession of 10 steps with a user-driven velocity variation of more than 50% of the baseline velocity.

The paper is organized as follows. In Section II we describe the dynamical models used for control design, the swing leg Virtual Guides controller, and the support leg admittance controller. In Section III, we describe the online planning strategy generating the controllers' reference trajectories and compare it with a naive approach. In Section IV we perform experiments to illustrate quantitatively the safety increase provided by the online planning strategy, and the performance of the overall approach.

II. FEEDBACK CONTROLLERS: A SPLIT-LEG DESIGN

In this section, we describe the Virtual Guide controller used to regulate the swing leg DoF and the admittance controller regulating the contact forces through the support leg actuators. First, we recall the equations of motion on which we rely throughout the article.

A. Dynamics of the patient-exoskeleton system

Consider the exoskeleton depicted in Fig. 1. The fastening system of the exoskeleton completely assigns the positions of the lower limbs of the patient with respect to the robot. Their torso is less firmly attached, but we consider it rigidly fixed to the exoskeleton. These positions define the major part of the patient's weight distribution, hence we neglect all the other DoF of the patient. However, the mass distribution of patients cannot be easily measured. As a surrogate, we assume they follow a normal distribution, which can be found in [START_REF] Winter | Biomechanics and Motor Control of Human Movement[END_REF]. Under these assumptions, the patient-exoskeleton system can be modeled as an articulated rigid-body system of total mass m with 12 actuated DoF (the joints of the exoskeleton) and 6 unactuated DoF (the position and orientation of the exoskeleton pelvis in the world frame). In details, the Lagrangian dynamics of the system write [START_REF] Wieber | Modeling and Control of Legged Robots[END_REF] with q = q sw , q sp , q un ⊤ ∈ R 18 the generalized coordinate vector, composed of the actuated positions vector q sw of the swing leg, q sp the actuated positions of the support leg, and the unactuated degrees of freedom q un , M (q) the generalized inertia matrix of the system, C(q, q) the combined gravity and inertia effects vector, (τ sw , τ sp ) the vectors of exoskeleton swing and support joint torques to be chosen by the controllers, (τ sw u , τ sp u ) the vector of swing and support joint torques created by the patient, f i ∈ R 3 the external forces and J i (q) the associated Jacobian matrices at each contact point p i .

M (q)q + C(q, q) =   τ sw + τ sw u τ sp + τ sp u 0   + i J i (q) ⊤ f i (1)
The dynamics of the system, in an inertial reference frame, taken as a whole, give the Newton and Euler equations [START_REF] Wieber | Modeling and Control of Legged Robots[END_REF]

m(c + g) = i f i , L = i (p i -c) × f i (2)
with c = c x,y , c z ∈ R 3 the CoM of the system and L the angular momentum of the system with respect to its CoM.

B. Virtual guides controller on the swing leg

The methodology of Virtual Guides [START_REF] Joly | Imposing motion constraints to a force reflecting telerobot through real-time simulation of a virtual mechanism[END_REF] allows a parametric curve P to be followed at a velocity prescribed by the patient's efforts on the robot. We propose to use it to estimate the user intent.

1) Constructing the parametric curve: The parametric curve used by the controller is built from a nominal gait trajectory 12 , readily computed as in [START_REF] Gurriet | Towards Restoring Locomotion for Paraplegics: Realizing Dynamically Stable Walking on Exoskeletons[END_REF], from which we extract the swing leg trajectory T sw : t → T sw (t) ∈ R 6 . The latter can be reparametrized with respect to its curvilinear abscissa s : τ

T : t ∈ [0, T f ] → T (t) ∈ R
→ τ 0 || Ṫsw (t)|| 2 dt ∈ [0, L max ],
with L max the total length (in the articular space) of the swing leg trajectory. By this formula, the curvilinear abscissa s is monotonous, therefore, assuming further that the Euclidean norm || Ṫsw (t)|| 2 is non-zero for all t, s can be inverted. Then, we can define the parametric curve as

P ≜ T sw • s -1
2) Virtual guides low-level controller: In the following

σ : t → σ(t) ∈ [0, L max ]
is a freely chosen control variable which defines the current set-point P (σ(t)) for the swing leg.

The Virtual Guides methodology minimizes the interacting force between the robot and the user while constraining the robot to the parametric path P . Defining the parametric path in joint-space, by constrast with Cartesian space, enforces inter-joint coordination. At the current point P (σ(t)), the Frenet-Serret unit tangent vector to the curve P , pointing in the direction of motion is

T (σ) ≜ dP dσ (σ)
To provide contraction property in the direction orthogonal to T , the joint torques of the swing leg are computed using a high-gain proportional-derivative controller

τ sw (σ, σ) = K sw p (P (σ) -q sw ) + K sw d (T (σ) σ -qsw ) (3)
with K sw p , K sw d ≻ 0 constant gain matrices. 3) Estimation of the target velocity: Following the Virtual Guides approach, we define the estimate of the user desired velocity σt such that the projection of the efforts τ sw (σ, σt ) along the path P is nullified, which reads

T (σ) ⊤ τ sw (σ, σt ) = 0 (4)
This yields [START_REF] Restrepo | Intuitive, iterative and assisted virtual guides programming for human-robot comanipulation[END_REF] σt ≜ T (σ) ⊤ K sw p (q sw -P (σ)

) + K sw d qsw T (σ) ⊤ K sw d T (σ) (5)
For the rest of the article, we consider that satisfying the user desire by imposing σ = σt is the rehabilitation objective. We now call σt the target velocity. The design of a safe σ such that σ is as close to σt as possible is the subject in Section III.

C. Admittance controller on the support leg

The only terms yet to be defined in Eq. ( 1) are the torques of the support leg τ sp , this is addressed below.

The Newton-Euler equations ( 2) can be simplified into the Linear Inverted Pendulum (LIP) dynamics to ease the stability analysis and the design of controllers. Indeed, along gait patterns of moderate velocity, the angular momentum variations are small and can be neglected. Assuming the robot walks on horizontal ground and the CoM remains at a constant height c z , Eq. (2) simplifies to the following LIP dynamics [START_REF] Wieber | Modeling and Control of Legged Robots[END_REF] cx,y = ω 2 (c x,y -u)

where ω ≜ Eq. ( 6) reproduces the unstable nature of the system. For stabilization, a State-of-the-art admittance controller such as the one detailed in [START_REF] Caron | Stair Climbing Stabilization of the HRP-4 Humanoid Robot using Whole-body Admittance Control[END_REF] is used

(q sw , qsw ) σ * (x * , u * ) τ sw τ sp (x, u) (q sw , qsw ) σt
u d = u * -(1+ k p ω )(ξ * -ξ)- k i ω (ξ * -ξ)+k d ( ξ * -ξ) (7)
with ξ ≜ c + ċ ω ∈ R 2 the Divergent Component of Motion (DCM), readily computed from the LIP state x, the DCM reference trajectory ξ * , computed from the state reference trajectory x * , and the associated CoP reference trajectory u * , and k p , k d , k i ≻ 0 three diagonal matrices. The choice of the reference and feedforward trajectories (x * , u * ) is the topic of Section III.

For implementation, u d ∈ R 2 is converted into articular targets (q sw * , q sp * ) ∈ R 6 × R 6 using inverse kinematics and admittance tasks inspired from [START_REF] Caron | Stair Climbing Stabilization of the HRP-4 Humanoid Robot using Whole-body Admittance Control[END_REF]. Finally, the support leg joint torques τ sp are computed using a high-gain proportional-derivative controller

τ sp = K sp p (q sp * -q sp ) + K sp d ( qsp * -qsp ) (8)
with K sw p , K sw d ≻ 0 constant gain matrices. More details on this admittance scheme can be found in Appendix A.

D. Summary of low-level controller updates

The updates of the control law that we propose are schematically depicted on Fig. 2. They consist in separated calculations of τ sw and τ sp , during single support phases. Classically, the admittance methodology is used on both legs during double support phases (see Appendix A for more details). On the one hand, a new calculation of τ sw is proposed according to the VG law Eq. ( 3). On the other hand, τ sp is calculated according to the admittance control law Eq. ( 8). These controllers require a reference trajectory (x * , u * ), satisfying unilateral contact constraints (the CoP should remain in the support polygon U), and a schedule σ * , taking into account the patient input σt . Their design is the topic of the next section.

III. REFERENCE TRAJECTORY DESIGN

In this section, we first expose a Time Rescaling (TR) strategy for the choice of the reference variables (σ * , x * , u * ). As will appear, it is not sufficient as it does not enforce the unilateral contact constraint. Then, we expose an Online Planning (OP) strategy which explicitly takes the constraints into account. We illustrate its benefits on an example. A more thorough experimental investigation of the stability benefits is provided in Section IV-A.

A. Time Rescaling (TR) strategy: a naive approach

A natural way to define the reference control variable σ * is to simply integrate the target velocity

σ * (t) = t 0 σt (τ )dτ ( 9
)
where σt is given by Eq. ( 5). Then, from the articular nominal gait T , and Eq. ( 9), the state reference trajectory x * can be computed using the time-rescaled nominal gait trajectory and Forward Kinematics (FK), see [START_REF] Carpentier | The Pinocchio C++ library : A fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives[END_REF] x

* = F K • T • s -1 • σ * (10)
Finally, the corresponding input u * can be readily computed from x * using Eq. ( 6). However, this TR strategy does not take the unilateral contact constraints into account. As a result, the feedforward trajectory u * is not confined to the support polygon U (and the state reference trajectory x * does not respect the inputconstrained LIP dynamics).

This shortcoming is illustrated by a simulation in Fig. 3. The reference CoP, in green, is not contained in the support polygon when modulating the trajectory at 60% of the nominal velocity. The final state (the endpoint of the red line) obtained by forward integration of the full-state dynamics [START_REF] Artemiadis | A Review of Robot-Assisted Lower-Limb Stroke Therapy: Unexplored Paths and Future Directions in Gait Rehabilitation[END_REF], using the open-source simulator Jiminy [START_REF] Duburcq | Jiminy: A fast and portable python/c++ simulator of poly-articulated systems for reinforcement learning[END_REF], is different from the nominal final state x f ≜ F K • T (T f ) (the endpoint of the blue line). The eight-centimeter resulting error is sufficient to make the robot fall at the end of the step.

B. Online Planning (OP) strategy

Instead of the previous naive approach, we consider a joint optimization of the variables σ * and (x * , u * ) taking into account the constraints U and a next-step LIP-feasibility [START_REF] Wieber | Modeling and Control of Legged Robots[END_REF] constraint x f .

The target velocity σt (saturated to be strictly positive) is converted into a target time T t to the end of the current step This equation does not exploit any behavioral description of the patient [START_REF] Ivanenko | Two-thirds power law in human locomotion: Role of ground contact forces[END_REF], but solely assumes that the patient's desire is to keep the velocity constant until the end of the step. A bi-level trajectory optimization problem for the inputconstrained LIP dynamics is formulated from the current state x 0 as follows Problem 1: Given (x 0 , x f ) and T t find u opt and T opt as

T t = L max -σ σt (
T opt = arg min T ∈T(x0,x f ) |T -T t | s.t. u opt = arg min u∈Ω(x0,x f ,T ) T 0 u 2 dt
where T(x 0 , x f ) ⊂ R is the set of times for which Ω is not empty, Ω is the set of feasible commands respecting the boundary conditions (x 0 , x f )

Ω(x 0 , x f , T ) ≜ {u ∈ U ad (T ), x u (0) = x 0 , x u (T ) = x f }
with x u the solution of Eq. ( 6) from x 0 and U ad (T ) ≜ {u s.t. ∀t ∈ [0, T ], u(t) ∈ U} is the set of admissible controls. Problem 1 is in fact a minimal time problem for an input-constrained linear dynamics of dimension 4. Its phase plane analysis, decoupling the X and Y directions, has been performed in [START_REF] Brunet | Fast replanning of a lower-limb exoskeleton trajectories for rehabilitation[END_REF], covering all possible cases of initial and final conditions. The main finding of [START_REF] Brunet | Fast replanning of a lower-limb exoskeleton trajectories for rehabilitation[END_REF] is that T is the union of at most two intervals, such that the solutions T opt and u opt can be easily determined using a bisection method, granting high-numerical efficiency. Computation times are reported in table Fig. 4. They allow a 1 kHz update of the reference trajectory. This is consistent with our aim to maximize responsiveness to the patient efforts.

Then, the reference control variable σ * is computed from the optimal time T opt as follows

σ * = t 0 σopt ≜ L max -σ(τ ) T opt (τ ) dτ
The reference trajectories (x * , u * ) are obtained from the optimal command u opt as follows

x * = x u opt , u * = u opt
The effect of this OP strategy is illustrated on Fig. 5, where, as opposed to Fig. 3, the CoP reference trajectory (green) is entirely contained in the support polygon (black horizontal lines) and offset towards the y = 0.0 line. As a result, the endpoint of the forward integration of the fullstate dynamics (1) (the endpoint of the red line) is close to the nominal final state (the blue cross): the state is successfully driven to the final state using the stabilization controlled Section II-C. This recursively ensures the success of the walk. To further illustrate the merits of this approach, we report experimental results in the next section.

IV. EXPERIMENTAL RESULTS

Two types of experiments are conducted. First, we compare the stability properties of the two strategies and conclude on the vast superiority of the OP strategy. Then, we explore the performance of the OP strategy in terms of compliance to the patient schedule.

A. Stability comparison

We perform several stability comparisons replacing the patient with a dummy. To simulate the behavior of the patient, we consider pre-recorded1 piecewise constant velocity signals σt (t) consisting of a square wave whose duration and magnitude are varied. An experiment consists of a 10 steps walk in straight line. A practitioner keeps hold of the two lateral exoskeleton handles and is allowed to create an effort with one finger on each hand only. This creates a very low upper-bound on the external forces.

The reported results on Fig. 6 show a great safety improvement offered by the OP strategy in the low-velocity range, below 90% of nominal velocity, compared to the TR strategy. They stress that, using the OP strategy, the proposed controller is completely preserving the balance of the system for velocities about as low as 70% of the nominal velocity, and velocities as low as 50% of the nominal velocity provided the change duration is lower or equal to 300 ms. These results are in complete alignment with the previous stability results obtained in simulation and reported in [START_REF] Brunet | Fast replanning of a lower-limb exoskeleton trajectories for rehabilitation[END_REF].

B. Rehabilitation: experiment with an able-bodied user

We report below the results of a 10-steps walking experiment with an able-bodied user using the proposed controller with the OP strategy. We first report a single step velocity (on Fig. 7) and CoM trajectory (on Fig. 8). Figure 7 reveals how the OP strategy accounts for the unilateral contact constraints. In detail, during the first 450ms of the single Fig. 6: Comparison of experimental stability for velocity variations having various durations and magnitudes2 . Green: stable without external help. Orange: stable with practitioner help. Red: unstable. Left: TR strategy. Right: OP strategy. Fig. 7: Effective velocity σopt and target velocity σt over a step with an able-bodied user (step 6 of Fig. 9). Black curves: lower and upper limits of the feasible velocities set. support phase, the OP strategy leaves the target velocity unchanged because the solution of Problem 1 is T opt = T t ( σopt in green completely overlaps σt in blue). Hence, the patient's schedule is fully respected. During the remaining 500 ms, the OP strategy starts filtering the target velocity to preserve the balance of the system, σ opt ̸ = σt . Gradually, σ opt is constrained around 71% normalized velocity.

The OP strategy also wisely chooses the CoM reference trajectory and satisfies the final state constraint. Figure 8 shows the final constraint is satisfied as the replanned (OP) and time-rescaled (TR) CoM trajectories' endpoints are identical. The CoM trajectory computed with OP strategy is very different from the one with TR strategy, and, in particular, exhibits a minimum on the Y axis 2.5 cm closer to the support foot (centered at 0.0 cm) than the nominal trajectory: the exoskeleton sways its hip toward the support foot to accommodate for the user's low-velocity desire. This Fig. 8: CoM from TR and OP strategies over a step with an able-bodied user (step 6 of Fig. 9). Grey areas: double support phases. White areas: single support phases. Left: X axis. Right: Y axis. Finally, Fig. 9 shows σopt over the whole experiment, with double support areas in grey. During this experiment, the user varies the level of efforts produced by their legs during the single support phases. Note that Eq. ( 5) is only used during these phases while the user's desire is ignored during double support phases 3 .

V. CONCLUSION

The control architecture proposed in this article extends the functionalities of Atalante and enables rehabilitation tasks for walking impaired patients. Since stability of the walk is handled by the self-balanced exoskeleton, the physiotherapist is relieved from this tedious but critical task.

The quantitative evaluation on physical health improvement remains to be done. In particular, the evaluation of the controller from a medical perspective will certainly be insightful to guide future developments.

In anticipation, several points in the methodology could be improved further. The new controller is not active during the double support phases of the walk. However, the CoM transfer during double support is a question of interest for walking rehabilitation and could be addressed with further developments of the presented method. It would also be interesting to consider adapting the step length to the patient efforts. This is a very natural extension to be addressed in the online planning strategy. For this, a library of predefined gaits could be used. Finally, the LIP model could be enhanced to address more dynamical gait patterns where the angular momentum variations can not be neglected. They are solved for the articular target acceleration qt that best satisfy their objectives. Numerical integration from the previous articular position and velocity targets yields the support leg target position and velocity (q sp * , qsp * ) tracked using [START_REF] Shu | Modulation of Prosthetic Ankle Plantarflexion Through Direct Myoelectric Control of a Subject-Optimized Neuromuscular Model[END_REF]. Set-points for Level 2 and 3 of both SoT 1 and SoT 2 are computed from the nominal gait T .

We validated this custom SoT 2 by performing a performance comparison with SoT 1 , both experimentally and in simulation, and found no noticeable impact on the overall stability of the walk (their transient responses differ in shape but not in error magnitude).

F

  . Di Meglio and N. Petit are with MINES Paris, Centre Automatique et Systèmes, PSL University, 60 bd. St Michel, 75272 Paris Cedex, France distinct changes.

Fig. 1 :

 1 Fig. 1: Atalante's kinematics (revolute joints in blue)-left. An able-bodied user performs a rehabilitation exercise-right.

g c z

  is the angular-frequency, and the CoP u ≜ p x,y i f z i f z i lies inside the support polygon (rectangle) U by definition. We note x the state of the LIP x(t) ≜ (c(t), ċ(t))

Fig. 2 :

 2 Fig. 2: Proposed control architecture for rehabilitation.

Fig. 3 :

 3 Fig. 3: Reference CoP u T R y , CoM c T R y and measured CoM c y positions along the Y axis of the inertial frame. Reference quantities computed using the TR strategy and a simulated target velocity as low as 60%. Black horizontal lines represent the support polygon limits.

Fig. 4 :

 4 Fig. 4: CPU time of the OP strategy (running on a i7-1185G7E at fixed 1.8 GHz frequency).

Fig. 5 :

 5 Fig. 5: Reference CoP u opt y , CoM c opt y and measured CoM c y positions along the Y axis of the inertial frame. Reference quantities computed using the OP strategy and a simulated target velocity as low as 60%. Black horizontal lines represent the support polygon limits.

Fig. 9 :

 9 Fig. 9: Effective velocity σopt over a 10-steps experiment with an able-bodied user. Grey areas: double support phases. White areas: single support phases.

  APPENDIX A. CoP control using admittance on the support leg The admittance scheme is adapted from [9] by reorganizing the so-called Stack-of-Tasks (SoT) [29]. The original SoT 1 reads, in decreasing order of priority as follows Level 0: support and swing foot position and velocity tracking; Level 1: CoM acceleration tracking; Level 2: pelvis roll and pitch tracking; Level 3: static standing articular configuration tracking. We use SoT 1 during double support phases, and the SoT 2 , described below, during single support phases in order to account for the swing leg effect on the CoM acceleration Level 0: support foot and swing leg articular position P (σ * ) and velocity σ * T (σ * ) tracking; Level 1: CoM acceleration tracking; Level 2: pelvis roll and pitch tracking; Level 3: static standing articular configuration tracking. SoT 1 and SoT 2 are Hierarchical Quadratic Programs.

Authorized licensed use limited to: ECOLE DES MINES PARIS. Downloaded on January 22,2024 at 17:31:06 UTC from IEEE Xplore. Restrictions apply.

i.e. replacing Eq. (5) with a predefined function of time

The white spaces in this figure corresponds to unfeasible values of the parameters violating the constraint σ * ≤ Lmax.

More precisely, the reference trajectory used during double support is computed once, at the beginning of the step. For this, we use the OP strategy and the mean velocity of the previous step, for sake of continuity