Manon Lelandais

Gaëlle Ferré

Multimodal Analysis of Parentheticals in Conversational Speech

Keywords: Parenthesis, discourse structure, prosody, gesture

Based on a video recording of conversational British English, this paper aims at describing the relation between verbal and non-verbal signals in the production process of parentheticals within the framework of Multimodal Discourse Analysis. Parentheticals are described in linguistics as side sequences interrupting linear development. Although their syntactic, prosodic and discursive characteristics have been deeply analysed, few studies have focused on the articulation of the different communicative modes in their production process. Beyond

showing that gesture brings complementary information in regard to prosody, contributing to a composite collateral message, the results allow better delineation and understanding of skipconnecting phenomena as constructing coherence. Changes in the modal configuration throughout the parenthetical sequence suggest modes are dynamic and flexible resources for indexing parentheticals and their particular framing function.

INTRODUCTION

In linguistics, parentheticals have often been relegated to editing or processing errors, indexing the unplanned character of speech and contravening certain co-operating norms.

From the point of view of Discourse Analysis, parentheticals are short segments inserted into the current stream of discourse, momentarily suspending the progression of a sequence to construct another unit, bringing information at a second discursive level [START_REF] Payà | Prosody and Pragmatics in Parenthetical Insertions in Catalan[END_REF] such as in example (1) below 1 . When analysing how semiotic units form larger sequences of action in discourse and conversation, spontaneous speech presents disruptions where the parts no longer follow one another, some interrupting or abandoning an ongoing utterance, others *Corresponding authors: Manon Lelandais, Gaëlle Ferré, LLING, Université de Nantes, Chemin de la Censive du Tertre BP 81227, 44312 Nantes cedex 3, France. E-mails : manon.lelandais@etu.univ-nantes.fr; Gaelle.Ferre@univ-nantes.fr returning to previous statements. Conversely, some can project an upcoming utterance, such as this reflection announcing an upcoming French term:

(1) Rhianna # (h) # I don't wanna be a judge # I don't wanna be a prosecutor

Alex yeah

Rhianna so I'd kind of be more interested in uh well there's no English equivalent but uh commissaire de police (h)

These breaks in contiguity are extremely relevant to examine language production and comprehension in real time: a particular negotiation of meaning takes place between participants, presenting implications for discourse modelling. Often taken as momentary 'deviations', parentheticals show an interesting degree of flexibility in speech planning and production, in which segments and other objects of discourse are organised into another hierarchy than that of temporal succession. The study of parenthetical sequences also questions the construction of coherence, i.e. the logical relations created by the linkage between constitutive elements of a unit.

Parentheticals display particular interactional strategies, yet their analysis has often concentrated on speech, taking precedence over other communicative modes. In the past, substantial work on such sequences has focused on the relation of parenthetical segments to their host from the perspectives of syntax, relevance theory, or prosody alone, describing them as forms of side sequences which can often be deleted without affecting the syntax or semantics of their host. The development of analytical tools and schemes now facilitates an account of parenthetical sequences as multimodal phenomena.

In face-to-face conversation, participants negotiate meaning through multimodal contributions, in which the linguistic resources of speech interface with gesture. These modes do not work independently from one another, although a particular mode may weigh more than the others at some points [START_REF] Norris | Analyzing Multimodal Interaction: A Methodological Framework[END_REF]). Within speech itself, the vocal or the verbal mode may stand out as more prominent at particular points in time. The present qualitative analysis aims at establishing links between prosody and gesture in discourse organisation, shedding light on the production process of parentheticals as particular activities structuring information, constructed as resources for conversation.

Viewing language as integrating speech and gesture in an organised system makes it possible to investigate why parentheticals are traditionally seen as disruptive (Quirk et al. 1985: 911;[START_REF] Mazeland | Parenthetical Sequences[END_REF]Mazeland : 1856) ) on the one hand, but somehow related to mainline speech on the other (Mondada & Zay 1999: 401). If parentheticals create a break (Payà 2003: 223) and complicate the flow of discourse, we may wonder how they are recognised as 'side sequences' by participants. Regarding them as incremental audio-visual signals oversteps the opposition between competence and performance, which had constrained parentheticals to special formal categories.

From the hypothesis that parentheticals are signalled in an integrative way according to their specific function with a set of features that co-participants attend to, this paper investigates the participants' use of the array of communicative modes in their construction, with a grounding in Discourse Analysis. Through an empirical annotation scheme, we first enlarge the set of their formal and functional verbal characteristics, allowing the formulation of specific hypotheses for their vocal and visual cues. We then test whether both sets of cues index a break, bring background information, and demarcate the specific types of parentheticals. The results confirm that parentheticals are composite framing devices (i.e.

projecting a space in which a semiotic entity makes meaning, providing unity, relation, and coherence) showing a multimodal distribution of labour between discourse organisation and information status. Modal density [START_REF] Norris | Analyzing Multimodal Interaction: A Methodological Framework[END_REF]) is suggested as a dynamic resource for constructing coherence.

I-THEORETICAL BACKGROUND

1.

Parentheticals and discourse

The vast terminology concerning side sequences suggests little consensus over the description of insertion phenomena (Payà, op. cit.), given the various perspectives along which they are observed. The expression derives from 'parenthesis', expressing 'the external and marginal source of the inserted element ' (p.207). Recent studies indeed refer to 'parenthetical' sequences/clauses/structures or 'parentheses' [START_REF] Payà | Prosody and Pragmatics in Parenthetical Insertions in Catalan[END_REF][START_REF] Mazeland | Parenthetical Sequences[END_REF][START_REF] Banik | An Integrated Architecture for Generating Parenthetical Constructions[END_REF][START_REF] Blakemore | On the Relevance of Parentheticals[END_REF], not indicating any syntactic form or discourse function of the segment in question. 'Parenthetical' is used here as a generic term to cover the phenomenon of interpolating an element in discourse, with expressions of varying length and complexity [START_REF] Dehé | Clausal Parentheticals, Intonational Phrasing, and Prosodic Theory[END_REF]) halting an ongoing utterance or turn. These terms sometimes apply to fragments introducing direct speech (she said) or to expressions containing verbs of opinion (I think) or declaratives (I must say), which are excluded from the definition in this study. The term does not cover comment clauses (as you know) either, described in Quirk et al. (op. cit.) and in [START_REF] Brinton | The Comment Clause in English: Syntactic Origins and Pragmatic Development[END_REF]. The present paper rather links up to the 'incidental sequential practice' referred to in Mazeland (op. cit.: 1817).

Parentheticals are said to organise information in turns and shape the participation framework [START_REF] Duvallon | Parenthesis as a Resource in the Grammar of Conversation[END_REF][START_REF] Bolden | Implementing Delayed Actions[END_REF]), which reflects the way the speakers and hearers can relate to one another as well as their orientation towards utterances. They are subsidiary activities referred to as skip-connecting, i.e. relating an ongoing utterance to some early talk (Sacks 1995). Likewise, they are considered a useful resource for the handling of topics (Mondada & Zay, op. cit.).

They also allow interactional reconstruction [START_REF] Fox | Interactional Reconstruction in Real-Time Language Processing[END_REF], as a specific means of framing in discourse with cues directing interpretation [START_REF] Tannen | Framing in Discourse[END_REF]. Through the production of a parenthetical, the speaker indicates the co-speaker has to re-interpret a part of her utterance. 'Speaking while monitoring addressees' (Clark & Krych 2004: 62) shows the speaker's ability to reflect and modify her utterances in real time for the benefit of the cospeaker, giving background information (Duvallon & Routarinne, op. cit.).

In addition, Clark & Fox Tree (2002:78) regard parentheticals as collateral signals in a discourse model where speakers refer to the 'official business, or topics, of discourse with primary signals', and to the 'performance itself with collateral signals' [START_REF] Clark | Using Language[END_REF]. Such devices manage the on-going performance [START_REF] Goffman | Forms of Talk[END_REF], commenting upon the problems encountered. Collateral signals can either be integrated in a same mode or produced in another form of modality, although only referred to as 'concomitant' when produced otherwise than verbally. Recent studies extend this view in suggesting collateral messages can be integrated verbally, prosodically and/or gesturally [START_REF] Rodrigues | Verbal and Nonverbal Signals in Face-to-Face Interaction: A Theoretical Framework for a Holistic Micro-Analysis: The Example of a Parenthesis[END_REF].

Prosody of parentheticals

Parentheticals are marked off from their hosts by special intonation contours, as well as a by a variation of tempo and of loudness [START_REF] Crystal | Prosodic Systems and Intonation in English[END_REF][START_REF] Bolinger | Intonation and Its Part: Melody in Spoken English[END_REF]. They are uttered in a separate Intonational Phrase [START_REF] Selkirk | Phonology and Syntax: The Relation between Sound and Structure[END_REF]. A falling-rising tone on the preceding domain and potential pauses signal their boundaries [START_REF] Local | Continuing and Restarting[END_REF]Payà, op.cit.). [START_REF] Bolinger | Intonation and Its Part: Melody in Spoken English[END_REF] and [START_REF] Delomier | Caractéristiques Intonatives et Syntaxiques Des Incises[END_REF] agree on a true set of common features for parentheticals which are not representative of the phenomenon if singled out, although some of them might not be present in a same insertion sequence. Wichmann (2001: 181) calls these common features the 'canonical' or typical prosodic pattern for parentheticals.

Intonation has more specifically been confirmed as the main cue of such sequences, through a disjunction of levels [START_REF] Debaisieux | Les Parenthèses: Étude Macrosyntaxique et Prosodique Sur Corpus[END_REF]. The typical 'parenthetical intonation' is lower F0 (i.e. fundamental frequency = pitch) and no modulation [START_REF] Hirschberg | Intonational Features of Local and Global Discourse Structure[END_REF]. [START_REF] Local | Phonetic Detail and the Organisation of Talk-in-Interaction[END_REF] pinpoints that the pre-and post-insert talk match each other in pitch height and that the post-insert talk is higher. F0 generally rises on the final syllable of the inserted segment, indexing it as prefacing further speech, continuing the paragraph and the point being treated [START_REF] Morel | Grammaire de l'intonation[END_REF]. Editing out a parenthetical then leaves the remaining utterance sounding prosodically well-formed and coherent, speakers resuming the pitch contour at the point at which it was interrupted [START_REF] Wichmann | Intonation in Text and Discourse[END_REF].

However, if some canonical prosodic features have been observed, parentheticals often show more variety than described. Their prosody depends on a wide range of criteria such as length, syntactic weight and position (Bolinger, op. cit.). Certain short sequences are prosodically integrated into an adjacent segment [START_REF] Wells | English Intonation PB and Audio CD: An Introduction[END_REF][START_REF] Lee | On the Role of Intonation and Pauses in the Interpretation of English Parenthetical Adverbs[END_REF].

Gesture and side sequences

A considerable body of research regards gesture as part of a speaker's message together with language, sharing the same integration system [START_REF] Mcneill | Hand and Mind: What Gestures Reveal about Thought[END_REF]2005;[START_REF] Bernardis | Speech and Gesture Share the Same Communication System[END_REF][START_REF] Ferré | A Multimodal Approach to Markedness in Spoken French[END_REF]. If a communicative intention fuels the production of a signal, speech is regarded as gesture at a multimodal communication level, as a composite auditory-visual message. Since parentheticals essentially index changes in these communicative intentions, this theoretical stance presents interesting implications.

Additionally, recent research on the construction of coherence in discourse [START_REF] Laursen | Towards an Embodied Grammar: Gesture in Tying Practices Constructing Obvious Cohesion[END_REF][START_REF] Lascarides | Discourse Coherence and Gesture Interpretation[END_REF] focused on repeated gestures throughout an interaction by same or other participants in terms of connecting, relating to McNeill's (op. cit. 1992) theory of gesture catchments, considering repetition as correlated to underlying cognitive units in language production.

3.1.

Hand gestures Calbris (2011: 47) studies how the hand gestures related to verbal units render a visual hierarchic organisation of discourse through patterns. If maintaining a gestural shape corresponds to the 'maintaining of the idea it represents', concurrent change in gestures segments discourse. 'Change creates a gap in the continuum and performs a demarcative function, whereas the type of change provides information and performs a referential function ' (p.46). An earlier study (Calbris 2008: 40) refers to 'side issues' as presented by switching hands, the process of utterance production rendered as a 'path towards a goal to which the central line is the topic'. The initial hand movement is suspended for the duration of the parenthesis, and reused after its completion, the speaker continuing along the same path.

Furthermore, Rodrigues (op.cit.: 7) notes that 'discontinuity happens in movements with opposing direction', i.e. to the right and then to the left. Continuity is then established with movements with the same direction, or the progression of circular ones.

Besides, Deppermann (2013) observes pointing gestures as devices for establishing reference as early as possible. Preceding verbal reference, they can orientate recipients to relevant spatial domains and possible referential targets of the turn [START_REF] Schegloff | On Some Gestures' Relation to Talk[END_REF][START_REF] Mondada | Temporalité, Séquentialité et Multimodalité Au Fondement de L'organisation de L'interaction: Le Pointage Comme Pratique de Prise de Tour[END_REF]). Lascarides & Stone (op.cit.) approach deictic meaning in gesture as reusing salient information: 'when [a pointing gesture] is interpreted with respect to a previously established virtual space, they offer similar forms in similar spatial configurations to represent the same objects over time ' (p.19).

In this sense, gestures for the same referent have been observed as more schematic, while gestures for new information are sharper and clearer [START_REF] Kita | Movement Phases in Signs and Co-Speech Gestures, and Their Transcription by Human Coders[END_REF].

Head gestures

Correspondingly, the significant distinction for head gestures can be that of 'dynamic units', i.e. of change of direction as opposed to 'static units' (Rodrigues,op. cit.: 8).

Among the specific formal types, head nods can provide segmental information, singling out particular entities and participating to their salience, while side-to-side shakes correlate with expressions of intensification and inclusivity. Some other gestures such as beats are connected with discourse structure in their function (Kendon 1972: 7), having a 'visual rhythmic character that seems to mark out the rhythmic organization of the utterance'.

Gaze direction and eyebrow movement

A shared visual perspective, as part of common ground, is extremely useful in that speakers can choose to convey information visually so that the collaborative effort to establish mutual understanding is reduced to a minimum [START_REF] Goodwin | Conversational Organization: Interaction between Speakers and Hearers[END_REF].

Gaze and head traditionally move away from the co-speaker for discourse elaboration as soon as the speaking turn is taken and secured. The head comes back towards the cospeaker followed by gaze at the end of the turn. A gaze towards the co-speaker functions either as a call for attention or as potential turn-yielding.

Often used for punctuation functions and word search [START_REF] Ekman | The Repertoire of Nonverbal Behavior: Categories, Origins, Usage, and Coding[END_REF], eyebrow movement may provide suprasegmental information in participating to focalisation and increasing the detection of prominence.

II-CORPUS AND METHODOLOGY

Corpus recording

A spontaneous conversation between two native British young women of 20 and 23 was recorded and filmed for 30 minutes. The few criteria along which the two speakers were selected were the absence of any heavy regional accent and English as a mother tongue.

Alexandra was 23 by then and had just settled in France after finishing her degree at UCL in London. Alexandra is from Cambridge. Rhianna was 20 when she was recorded and has mostly lived in France, although she comes from Wimbledon. They had only met once before the recording.

Little information concerning the object of study was given to the speakers before the recording. Alexandra and Rhianna signed an information and consent form, which guaranteed the confidentiality of the data in any other context than academic studies. Disclosing the real purpose of the study would certainly have influenced their conversation as well as their attitude, especially their gestures. The object of study was only referred to as an analysis of spoken English. The consent form also stipulated the duration of the recording.

The speakers were asked to talk just as they do in everyday life, and some specific themes had been suggested, should their conversation be at a standstill (e.g. plans for the forthcoming summer holidays, unusual situations they had lived). Some were used during the recording, others were not.

Sitting in armchairs in a recording studio offering good lighting and soundproofing, they were filmed in a static, wide-angled shot. The two participants were visible from head to ankle, sitting three-quarters turned towards each other: if on the one hand they had been sitting next to each other, they would have instinctively turned their heads towards each other while talking, making it impossible to know whether looking away was deliberate or a means to rest their heads. If on the other hand they had been completely face-to-face, the camera would not have caught subtle face movements or facial expressions such as raising one's eyebrows. They were left alone during the session and knew they were not overheard by the authors.

Both had a lavalier microphone, providing two separate audio tracks. The native video recording was transformed into an MPEG-4 stereo file, with a rate of 25 frames per second.

Two audio files corresponding to each microphone were created in a WAV format, facilitating the analysis of overlapping speech. This material was decided upon as a compromise between the necessity of a quality recording, enabling detailed prosodic and gestural analyses, and that of naturally occurring data.

Corpus transcription

Prosodic coding

The corpus was first edited in Praat (Boersma & Weenink 2011) for a standard orthographic transcription segmented into tone-units, according to the British school of intonation [START_REF] Halliday | Intonation and Grammar in British English[END_REF][START_REF] Crystal | Prosodic Systems and Intonation in English[END_REF][START_REF] Wells | English Intonation PB and Audio CD: An Introduction[END_REF] found in greater number in our sequences, would indicate that there is no break in between the different elements of the sequence.

Gestural coding

The gestures were annotated in Elan [START_REF] Sloetjes | Annotation by Category: ELAN and ISO DCR[END_REF], based on Kendon's (2004) definition of a gesture as a visible action of any body part that is used as an utterance or part of an utterance. Hand gestures, head and eyebrow movement as well as gaze direction were manually given labels.

Descriptive annotation

The descriptive annotation was based on gesture units. Each gesture unit considered here starts at the onset of the gesture and ends at the return to rest position if there is one. In the case of two consecutive gestures, the first gesture phrase (adopting Kendon's terminology) ends at a significant change in shape and/or trajectory. A main dimension or function regarding the co-occurring speech corresponds to each gesture unit.

In a series of different tiers, head movement was labelled into nods, shakes, tilts, beats, or jerks. Gaze direction was annotated as either towards the co-participant or away, eyebrow movement distinguished between rise and frown, and hand gestures, which values are described below, were categorised into iconics, metaphorics, pointings, beats, emblems, butterworths, and adaptators.

Functional categories

The Adaptators, i.e. self-contact gestures used for comfort, were included given their important number and the fact that they give the experimenters information on the organisation of turns, being more frequent when the participant is listening. As hand gestures may have several dimensions (McNeill 2005), two values were noted in the same label and were both counted in such a case.

Treatment of parentheticals

Localisation

The parentheticals were localised and transcribed as Ps over the course of the whole interaction. The preceding clause was called A (antecedent), the subsequent one called R (resumption). The object of study was then defined as an APR sequence.

A total of 28 APR sequences were annotated in the thirty-minute conversation. The data were analysed separately for the two speakers because of an asymmetrical distribution of Ps: one speaker produces 22 parenthetical segments, making up a total of 1 min 15 s out of 13 min 31 s of speaking time (i.e. 1.65 P/min) while the other produces only 6, totalling 14 s out of 15 min 10 s of speaking time (i.e. 0.40 P/min)2 . The similarities and differences between the speakers were nonetheless studied.

Verbal annotation

Syntactic classification

Mainly (2) Rhianna (A) no [NUCL] it's a special program # uh for (h) people who speak good English (P) [SUF] so hopefully # hem # that wouldn't be a problem (h)

From the point of view of parataxis as a form of structuring beyond other realisation modes of complex constructions, they are propositions juxtaposed in a junction mode between subordination and syntactic independence. This 'third dimension' of phrase structuring, closer to discourse than to sentence, has further semantic and illocutionary implications. If Ps are in majority grammatically autonomous, independent enunciations (introduced with a nongoverning conjunctive morpheme such as 'so', or non-introduced), their semantic, modal, and illocutionary autonomy is relative. Their level of autonomy was ranked as shown in Figure 1 below: some Ps are speech acts and syntactically independent, whereas others have a syntactic fixed form and/or are semantically bound to their hosts. Their global autonomy can then be seen on a scale rather than with absolute values. If most of the syntactically independent Ps are speech acts as well, the Ps with no verbal complex are likely to be illocutionarily dependent on their hosts. Example ( 3) is a reference-tracking P where the pronoun 'that' specifies the identity of a head referent located in A.

(3) Alex (A) cause it was during freshers' week (P) oh you don't have that in France (R) anyway so it was during freshers' week

Speech acts

Each P was then indexed as either anaphoric or cataphoric, i.e. either commenting or orientating information. Example (4) is a cataphoric APR in a narrative, where the pronoun 'it' points forward to a concept that has not yet been introduced. Representational Ps should be characterised by high pitch, modulation, rising contours and slow speech rate. Gaze should be towards the co-participant, and the gestures should be addressee-centered.

III-RESULTS

Prosody

Speech rate and length

The results show that P is indeed faster than A and R; P is also the shortest segment whereas R is the longest. Table 3. Mean speech rate (in syll/sec) and length (in sec) for each type of parenthetical

The following discourse P working as an evaluation in an argumentation is indeed longer and slower:

(10) Alex (A) and then they've got their scratch card it's like 4.55 syll/sec (P) (h) we're on a plane for an hour you know wh-# I don't want a scratch card (laughs) like (laughs) seriously (laughs)

3.92 syll/sec (R) so always pointless things 6.29 syll/sec

Pauses

Filled and silent inter-unit pauses occur more between P and R than between A and P, marking Ps at their ends. As far as intra-unit gaps are concerned, R features the greatest number of pauses. Ps are more characterised with 'hem', signalling a local interruption, than with 'uh', signalling a strong boundary. Discourse Ps feature more pauses than the other types.

This metadiscursive P, produced in an answer to Rhianna's enquiry about Alex's master's degree, features a pause between P and R. Silent and filled intra-unit pauses create a prosodic break in R whereas there are no pauses in P:

(11) Alex (A) (h) mh to use the French word médiation culturelle

(P) cause I don't know what it is in English

(R) (h) hem # and then hopefully do something like kind of # work in culture

Intonation

Pitch is lower in P than in A and R, especially in the discourse type, as shown in Table 4 gathering a mean F0 height for each segment.

Restarts and resumptions

Frequent interruptions mark the end of A, either creating orphans, i.e. suspension of speech between words, or word fragments, i.e. suspension of speech within words. The majority of orphans over word fragments suggests a preferential preliminary syntactic commitment, in that syntax weighs more than phonology in the process planning of speech during APRs.

Orphans and near repeats can be seen in this discourse P working as a justification in a narrative about luggage restriction:

(14) Alex (A) because # they didn't # (P) cause you know they have the like cabin baggage restriction that says that all the stuff fits in the overhead # lockers (R) (h) and they obviously didn't have enough space

Gesture

Hand gestures

Crucially, speakers gesture more during P and R. If no hand gesture is produced in A, metaphoric gestures preferentially appear in P or R. As for adaptators, the same gesture unit is produced throughout the whole sequence or appears in R, indexing processing or retrieval.

The following P, working on discourse as an explanation in the context of a narrative about changing subjects at University, features emerging gestures co-occuring with different segments. In (b), she produces the emblematic gesture for TWO, in (c), she presents the emblematic gesture for MAYBE with an oscillating movement of her right hand (her wrist shifting from the vertical plane to the horizontal plane), and at last produces an adaptator in (c), touching her lips during the silent pause that follows the filler 'hem': If the higher proportion of metaphorics over iconics only marks a general trend, we also note that gestural tendencies are increased in P rather than contradicted: if a specific type of hand gesture is produced in A, more of this specific type is produced in P, such as in the following APR, increasing the common ground in Rhianna's description of her childhood in Montmartre. Figure 5 below shows a succession of abstract pointing gestures sharing similar forms and values regarding the co-occurring speech. What is interesting though is that each abstract pointing does not correspond to a precise referent (the town hall as opposed to the next district), but rather serves as a demarcation in between speech acts. Yet, the same hand shape used by the speaker maintains the gestural coherence of the passage:

VIDEO 4: TOWN HALL The prosodic cues fully confirm our first hypothesis according to which APRs create a break, in that they feature a difference in terms of pitch, speech rate and pauses. However, this break is not indexed where it was expected: it lies in the resumption part rather than in P itself, meaning that a break is not signalled between the antecedent and the parenthetical, but rather between the parenthetical and the resumption. The gestural features, conversely, fully confirm our second premise about APRs bringing background information in that the proportion of metaphorics and abstract pointing gestures are increased in P and R, subscribing to the previous typological tendencies of a sequence in producing more gestures of a same type.

Prosody and gesture, then, do not bring the same message regarding parentheticals.

If the different types can be distinguished, only discourse Ps feature both distinct prosodic and visual characteristics: they are slower and longer than the other P types, featuring more pauses. They are also accompanied with more head movement, eyebrow rises, and abstract gestures. Metadiscursive Ps are signalled gesturally with eyebrow movement (frowns) and adaptators, whereas representation Ps are modulated, not signalled with gesture at all. Some Ps show self-attentiveness (lower F0, faster speech rate, gaze away, adaptators) whereas others show other-attentiveness (change of F0, shorter, implicational tones, gaze moved or maintained towards the co-speaker). Other-attentiveness is however dominant in parentheticals, whatever their pragmatic type.

Gesture brings extra information in regard to prosody, contributing to a complex and composite message in which modes simultaneously bring information at local and global levels: APR sequences feature prosodic action at a linear level with segmental cues indexing an unexpected disruption (demarcative function), while gesture is used at a sequential level, defining the status of the information presented in the APR as background (referential function). This distribution of labour between modes can also be observed through several temporalities: gesture, relying on space as well as on time, 'ties' A, P, and R together, escaping the linearity of sound. Prosody is preferred for its local segmental functions, framing P or R alone. This does not mean, however, that both segmental and suprasegmental cues always index discontinuity: key and final contours can mark further incoming speech, organising information [START_REF] Coulthard | Exchange Structure. Discourse Analysis Monograph No. 5[END_REF][START_REF] Hirschberg | The Intonational Structuring of Discourse[END_REF], but there are more prosodic cues expressing demarcation than prosodic signs of continuum in APRs.

'Complicating locally to simplify globally' allows Ps to take local configuration and form, without becoming the central activity of the interaction; information is marked as different just as the sequence is indexed as not finished, the conversational agenda still pending and attended to. The speaker is identified as 'doing' the clarification: the opening of a specific sequence by the co-speaker is not wanted. Turn-holding, if not already clearly indicated, is secured.

Change in modal configuration

The highest modal density, i.e. the moment where all modes are the most active, drawn on and interwoven at the same time [START_REF] Norris | Analyzing Multimodal Interaction: A Methodological Framework[END_REF]2011) lies in R, where most of the 'tying up' to the main sequence is achieved and where P is also indexed as 'particular' speech retrospectively. The negotiation of the communicative importance between the different modes fluctuates not only as different action sequences take place, but also within a particular sequence.

Although one mode is generally dominant in a discourse segment, both gesture and prosody play a part in language integration: neither of these two modes can be taken into account independently from one another. Recent work goes towards a shared integration system between action observation and language processing: language and action-related information (i.e. gesture) contribute to a single yet composite, incremental speaker's message.

The multimodal configuration can be seen in Figure 9 below, in which the squares stand for each particular action performed through discourse. As APRs are conversational phenomena, the verbal discourse is each time at the core of this action. P stands for prosody while G stands for gesture, weighing differently in the distinct segments. When the visual,

Global coherence

This dialectic between speech and gesture can also be seen as semantic interaction. In terms of coherence relations, which 'structure communicative actions by showing how the speaker is grouping ideas together to highlight the meaningful relationships among them' [START_REF] Lascarides | Discourse Coherence and Gesture Interpretation[END_REF] 3), information is not presented in isolation but in a contiguity relationship. [START_REF] Laursen | Towards an Embodied Grammar: Gesture in Tying Practices Constructing Obvious Cohesion[END_REF] observes that when gesture projects an increment, indicating which element is followed by an increment and the element to which it is tied, the verbal mode specifies the Verbal discourse Verbal discourse Verbal discourse P P G G P G type of relation that is to be expected in the increment (i.e. that makes the incremented action relevant and coherent), while gesture makes the action visibly cohesive. Likewise, the function of some gestures constructing coherence and cohesion throughout the APR in establishing reference (e.g. small pointing gestures, repetition of metaphorics, continuation of a frozen gesture) can be compared to that of repeats and pronouns in the verbal mode. APRs are action sequences in which the referent is likely but not certain to be recognisable for interlocutors, in the way [START_REF] Enfield | Primary and Secondary Pragmatic Functions of Pointing Gestures[END_REF]: 1733) describe 'insecure reference environments', also prone to try-markers.

Verbal/visual correspondences favour a global model of coherence across modalities.

Interactional reconstruction [START_REF] Fox | Interactional Reconstruction in Real-Time Language Processing[END_REF]) is achieved through an interplay of verbal, visual, and vocal cues. Devices such as conjunction signal the type of context space being developed and their relation to preceding context spaces. This can be seen as extending [START_REF] Reichman | Conversational Coherency[END_REF] model: an ongoing conversation is parsed on-line, based on verbal but also co-verbal and nonverbal cues, the results of this parsing are passed to a database, which acts as a model of the ongoing interaction.

It is in this sense that the multimodal cues in A and in P are substrates in Goodwin's terms (2012), forming a basis for transformative operations preserving the structure. 'Action uses as its input the structure and resources of a current substrate and produces as its output a new, transformed substrate' as a point of departure for a next action (p. 17). Re-using material throughout an APR triggers reanalysis, as a specific recontextualisation changing interpretation.

V-CONCLUSIONS

One of the issues raised by multimodality is how the verbal, the vocal, and the visual modalities articulate with one another in the construction of meaning and interaction. In the production process of parentheticals, speakers increase communicative efficiency by simultaneously conveying different information in various modalities, at several levels. We have mainly analysed APRs from the point of view of production. However, much finer, detailed work remains to be done on the multimodal cues for the reception and the perception of APRs, for instance through an extensive study of backchannels as a multimodal addressee response embedded in the inserted action, that could influence the speaker's communicative choices.

Furthermore, multimodal data are known to provide rich insight for discourse phenomena and language production mechanisms. However, a thirty-minute dyadic conversation provides a qualitative analysis for APRs, in which hypotheses could be generalised only through their testing on larger multimodal corpora.

APPENDIX: TRANSCRIPTION CONVENTIONS

  based on dynamic pitch contours.The Momel-Intsint algorithm[START_REF] Hirst | A Praat Plugin for Momel and INTSINT with Improved Algorithms for Modelling and Coding Intonation[END_REF][START_REF] Bigi | SPPAS: A Tool for the Phonetic Segmentation of Speech[END_REF] was used for the automatic annotation of the F0 target points in the signal. Annotations are made in two respects: the algorithm notes pitch height (in Hz) on target syllables, which then allowed us to calculate mean F0 values for specific segments. The algorithm also codes symbolic (relative) values of intonation, in which each measured F0 value is compared to preceding ones, i.e. significant changes in the F0 curve either regarding the speaker's pitch range (Top, Bottom) or regarding the neighbouring tones or sequences of tones (Upstep, Downstep, Same, Low, High). In this work, we are particularly interested in values which indicate a significant pitch reset (Top, Bottom), or a significant change in pitch key (Upstep -change towards higher pitch range, Downstep -towards lower pitch range). We are also interested in the value 'Same' which, if

  forming independent clauses, Ps were classified as non-introduced or introduced by conjunctive morphemes. From a macro-syntactic point of view, they are close to parataxis (McCawley 1982; Debaisieux & Martin 2010), i.e. a system of interdependency between two juxtaposed elements of equal status. Most of the Ps are in a suffix position to the main (or nuclear) construction, respectively identified as [SUF] and [NUCL] in example (2), featuring marked syntactic structures and verbs with low transitivity, prone to identification and subject focus.

  Figure 1. Gradual autonomy of parentheticals

  and it's a four-year course cause when you do # (P1) I don't think it's not-right it's not the same thing in France (P2) you know when you do languages (R) you have to do well we had to do a third year abroad Ps were also organised as either essential or non-essential to the sequential development. The APR shown in example (5) below is essential to the narrative, adding some necessary information for a correct reception and interpretation by the co-speaker.(5) Alex (A) (h) but hem # and then this other time quite recently i came over (h) and # (P) cause we were meant to London-Tours (R) and the weather was really really bad it was really foggy and overcastConversely, example (6) shows a non-essential APR, indexing the speaker's processing, without which the sequence develops in the same way. (6) Alex (A) yeah you should try on like hem le bon coin # there's hem that (P) and what else did I use I used hem # (R) (h) # hem # like vivastreet or something like that (h) Three kinds of break are observed in Ps, in regard to temporal rupture, change in subject, or overstep of sequential expectations (i.e. contradicting the traditional schema for a given sequential type). Example (7), uttered in a narrative, features a temporal switch from past tense to present, the utterance going out of the narrative frame. (7) Alex (A) cause I was working last summer in the train station (P) (h) you know in the kind of like small café things like these ones south and north (h) (R) like cause I did that last year

Figure 2

 2 Figure 2 below shows the observed pitch height difference. Pitch is lower in P than in A and R (mean pitch in A: 161.1 Hz; mean pitch in P: 84.2 Hz; mean pitch in R: 176.4 Hz).The final contours in A and P can however be rising and indicate further speech to come (Figure2, Figure3).

Figure 2 .

 2 Figure 2. Lower pitch in a metadiscursive P in Praat (first transcription track shows segments -A, P, or R -in the APR sequence, second transcription track shows Intsint's coded values)For both speakers, R shows a greater prosodic break than A and P in featuring more demarcative tones values (hence pitch movement) than the other two segments.

Figure 3

 3 Figure 3 below shows Alex's resumption as a downstepped plateau, pitch decreasing from 202 Hz at the end of P to 190 Hz during R. R shows a flat pitch curve as the S-value signals same pitch height after a downstep (D).

Figure 3 .

 3 Figure 3. Resumption as a downstepped plateau in a discourse APR in Praat (first transcription track shows segments -P, or R -in the APR sequence, second transcription track gives Momel's corrected F0 values in Hz, and third transcription track shows Intsint's coded values) Not only do discourse parentheticals show a greater prosodic break in gathering more tone values, but they also gather more S points than the other types, i.e. they are flatter, as shown in Table6below. The parentheticals acting on discourse are downstepped and

Figure 4 .

 4 Figure 4. Gestures emerging in an explanatory APR sequence Figure 4 above shows different moments from Alex's APR: (a) corresponds to a moment in A, (b) and (c) show moments in P, (d) is in R. Each gestural activity illustrated in a picture is represented in between brackets.

Figure 5 .

 5 Figure 5. Series of abstract pointing gestures sharing the same configuration and value throughout an APR Ps are also loci for emerging small pointing gestures, all correlated with specific lexical affiliates. This metadiscursive P in a narrative about giving English lessons features for instance two pointing gestures correlated with 'advertisements', the key lexical element of the sequence: the appropriation of the term is commented upon as an insecure repair to the disfluencies in A. Alex freezes an adaptator and produces a first pointing gesture with her right hand while proposing a potential repair, still showing hesitation (b). She then ends the adaptator to reinforce the pointing gesture she is holding with a similar sign of her left hand (c), appealing to the co-speaker. She returns to a rest position as soon as the lexical retrieval is achieved (d).

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Pointing gestures during a metadiscursive APR sequence

Figure 9 .

 9 Figure 9. Change in modal configuration during APRs

  The empirical annotation shows that the identification and description of Ps in the verbal mode are based on a whole cluster of features, not only discontinuity. The hindsight brought by Ps on mainline speech can be layered in three different scopes.Ps open a focus of attention on some object of discourse, which has to be reanalysed by the co-speaker from a new perspective because speakers think their previous or following speech might be ambiguous. Speakers mark the unexpected shift of frames with specific prosodic patterns suggesting a break in linearity (demarcative function), while gestural patterns 'tie' the parenthetical segment to mainline discourse in establishing coherence for the whole sequence (referential function). It is in the resumption part that local, editing action on discourse is indexed: parenthetical segments are signaled as such retrospectively.Both verbal and nonverbal signals are related to each other, and contribute differently to the elaboration of the message. The vocal, verbal and visual modes are dynamic, adjustable resources for indexing Ps and their particular action.

Table 1

 1

	the antecedent.
	(8) Alex	(A) so i did the second year of # hem # like English # here
			(P) (h) (laughs) which was interesting
			(R) (h) hem yeah and then i just graduated now #
			Metadiscourse	Discourse	Representations
		lexical/word search	explanation	check common ground
			code-switching	concession	add common ground
			disambiguation	evaluation	recall common ground
			Table 1. Action level of parentheticals depending on their pragmatic function
	4	Working hypotheses
	Based on the theoretical background defined by the literature and on our observations from
	the verbal annotation, three main hypotheses are addressed.

below shows that Ps offer hindsight for (re)contextualisation at three different levels. Some Ps act on single words and have a metadiscursive function. Some others have a wider scope in commenting upon a whole stretch of discourse as explanations, concessions or evaluations. A last group of Ps act at the interactional level and are concerned with the state of shared knowledge between participants. The most widespread type is that of discourse Ps. The following APR within a narrative works for instance as an evaluation of the event described in If parentheticals signal a break regarding A and R, a difference should be expected in terms of pitch, speech rate, and pauses. A new gesture unit, different in form, should begin,

  Table 2 below shows the mean speech rate and length of each APR segment.

		Metadiscursive	Discourse	Representation
	Speech rate	7.73	5.81	6.25
	Length	1.64	4.39	2.19
		A	P	R
	Speech rate	4.27	6.60	4.44
	Length	3.41	3.15	4.50
		Table 2. Mean speech rate (in syll/sec.) and length (in seconds) for each part of the
		parenthetical sequence
	Example (9) is a metadiscursive P that is shorter and faster than A and R, as a word
	search in an argumentation:
	(9) Alex	(A) yeah you should try on like hem le bon coin # there's hem that	3.00 syll/sec
		(P) and what else did I use I used hem #		6.69 syll/sec
		(R) (h) # hem # like vivastreet or something like that (h)	2.75 syll/sec

If metadiscursive Ps are the fastest and shortest, discourse Ps are the slowest and longest as seen in Table

3

below.

Table 4 .

 4 Mean pitch height (in Hz) in function of APR segment and P type This metadiscursive P is produced by Alex in a descriptive sequence about the student loans she was allocated in the UK:

	F0/ APR segment	A	P	R
	F0	192.6	182	185.7
	F0/ P type	Metadiscursive	Discourse	Representation
	F0	189.3	177.1	188.1

Table 5 .

 5 Number of occurrences of symbolic relative values of intonation for each segment of the parenthetical sequences extracted from the INTSINT-MOMEL algorithm

	Tones/APR	A	P	R
	Bottom	8	15	11
	Top	0	2	1
	Downstep	49	45	76
	Upstep	42	41	53
	Total	99	103	141
	Same	37	33	69

Table 5

 5 In the context of a dialogic sequence about University, Alex produces a representation P, checking whether UCL is part of the common ground.

	VIDEO 2: UNIVERSITY COLLEGE LONDON
	(13) Alex (A) (h) hem I just finished my deg # I did a four-year degree in London
	(P)

above also shows flat contours detected by the value 'Same' (S): resumptions are prototypically downstepped plateaus for both speakers.

uh University College London I don't know if you've heard of it UCL #

(R) and hem

Table 6 .

 6 Table 6 below. The parentheticals acting on discourse are downstepped and upstepped plateaus in majority. If discourse Ps are flat, representation Ps show modulation (few S points). Number of occurrences of symbolic relative values of intonation for each type of parenthetical extracted from the INTSINT-MOMEL algorithm

	Tones/P type	Metadiscourse	Discourse	Representation
	Bottom	4	10	1
	Top	1	1	0
	Downstep	12	30	3
	Upstep	8	28	5
	Total	25	69	9
	S	9	21	3

Transcription conventions are provided in the Appendix at the end of the paper.

The discrepancy between the two speakers may be explained by the fact that one of them is more familiar with the principal investigator of this study and therefore feels more in charge of the interaction.

ACKNOWLEDGEMENTS

We would like to thank our two participants Alexandra and Rhianna, as well as the staff at the Audio-Visual Centre at the University of Nantes. We are also indebted to two anonymous reviewers for their valuable comments and suggestions on a previous version of this paper.


Hervé Molique
email: herve.molique@ires.in2p3.fr

Jerzy Dudek
email: jerzy.dudek@ires.in2p3.fr

EXACT AND NEARLY EXACT PAIRING TREATMENT FOR LARGE SCALE CALCULATIONS *

Keywords: numbers: numbers: 21.60.-n, 21.45.+v, 21.60.Fw

In this article we present an analysis of the practical applicability of the earlier introduced PSY-MB method in solving the nuclear pairing hamiltonian. In particular, we illustrate the convergence properties of the groundstate correlation energy, as well as the first excitation energy, in the case of the so-called picket-fence model where 32 particles are distibuted over 64 equispaced, doubly-degenerated levels. In order to illustrate the ability of the method, we compare the correlation energies of the ground-state to the exact solutions obtained with the Richardson formalism, as well as the BCS approach, in function of the increasing monopole pairing strength parameter.

The PSY-MB Method

Some years ago the so-called PSY-MB (P -SYmmetries oriented Many Body) method has been introduced (cf. Ref. [1]); its aim is to solve the eigenvalue problem of the state-dependent pairing hamiltonian composed of the mean-field term Ĥmf and the pairing term Ĥpair , which can be written

G αβ P + α Pβ , (1)
where Nα = a + α a α + a + ᾱ a ᾱ and P + α = a + α a + ᾱ represent the number-and the pair-operators, respectively. In this expression, the states α and ᾱ denote a pair of conjugated single-particle levels with respect to time-reversal, signature, simplex, or any other dichotomic symmetry.

The PSY-MB method is a procedure based on the analytical blockdiagonalization of the original Hamiltonian. More precisely, three mutually commuting operators, say P 1 , P 2 and P 12 , can be built up out of the appropriately constructed unitary-group generators. They can be shown also to commute with the Hamiltonian of the problem, the latter conveniently expressed in terms of the same generators. The implied blocks of the Hamiltonian can be treated using the Lanczos method up to the largest limits acceptable by the computer at hand. Today the spaces corresponding to 18-20 particle on 36-40 levels can be treated. Above that limit, an extremely effective pre-selection of some many-body configurations adapted to the nature of the pairing interaction, in conjunction with a given energy cut-off are used before the final diagonalization employing the Lanczos method.

The configurations are classified as n-pair configurations if, starting from the ground-state configuration, one excites n pairs of particles from below to above the Fermi level. For simplicity, we concentrate here on calculations of the seniority zero solutions, but other seniorities can be studied in the same way. In order to be useful in realistic nuclear physics calculations, a singleparticle space needs to be chosen sufficiently large, and the convergence of the obtained solution has to be investigated with respect to the increase of the selected many-body configurations. In this paper we focus our attention on the space composed of 32 particles distributed over 64 equispaced (the level-spacing being equal to 1 MeV) single-particle states (picket-fence model). By construction, the PSY-MB method can be applied in the case of any state-dependent pairing, but in this paper we shall restrict ourselves to the case of a constant pairing interaction.

Convergence of the Ground-State Correlation Energy and the First Excitation Energy

In this Section, we study the convergence properties of the correlation energy of the ground state for two different values of the pairing strength parameter G. The results are plotted in Figs. 1 and2 for G = 0.345 and G = 0.375, respectively. On the left hand sides of the Figures, the groundstate basis configuration and all the 256 1-pair states are considered, and then the number of 2-pair configurations is increased until the maximum value of 14 400 configurations has been reached. On the right hand sides of the Figures we plot the convergence in function of the number of 3-pair states. In this case the number of 1-pair configurations is also equal to 256, and the number of 2-pair configurations is kept equal to 14 400 all the way through. We compare the results with the exact solution given by the Richardson procedure [2]. We can see from these figures that in both cases the convergence seems to be very similar: the calculated correlation energy decreases steadily to come very close to the exact results.

First Excitation Energy (MeV) PSY-MB Exact

Fig. 4. The same as in Fig. 3, but for the pairing strength G=0.375 MeV.

illustrate the convergence properties of the first excitation energy in Figs. 3 and4. From these Figures, it is clear that one cannot generaly conclude that a monotone convergence can be achieved as the number of 2-pair configurations is increased. In the case of G = 0.345 MeV, we see that if all the 2-pair configurations are taken into account, the solution may be further away from the exact value than if not all the configurations are included. Furthermore, if all the 2-pair configurations are included, adding some 3pair configurations may lead to a result again further away from the exact value.

Comparison between the PSY-MB, BCS and the Exact Results

As an illustration for the use of the PSY-MB method, we plot in Fig. 5 the ground-state correlation energies in function of the pairing strength parameter G, for the PSY-MB, the BCS and the exact calculations. In the PSY-MB case, the ground-state configuration, all the 256 1-pair, all the 14 400 2-pair and 76 113 3-pair configurations are taken into account, which means that less than 0.02 % of the 601 080 390 seniority zero states are considered. (Remark: the number of 3-pair states taken here, namely 76 113, comes as a result of the fact that there are many basis configurations which have the same energy because of the somehow trivial structure of the picket-fence model. In the selection procedure of the configurations one is fixing a given energy cut-off, and it seems a priori natural to select all the configurations with the same energy). We see from this Figure that the agreement of the PSY-MB results with the exact ones is very satisfying, whereas the BCS solutions lie too far above. Of course, other quantities of interest such as single-particle occupation probabilities are accessible within the PSY-MB framework, and the overall agreement with the exact quantities has been checked in the same way.

Conclusions and Outlook

In this paper we have illustrated the use of the PSY-MB method in the case of the so-called picket-fence model composed of 32 particles distributed over 64 doubly-degenerated equidistant single-particle levels. Special attention has been payed to the convergence properties of the ground-state correlation energy and the energy of the first excited state in function of the increase of the pre-selected many-body configurations.

In recent detailed investigations we have performed a comparison, cf. Ref. [3], with other many-body techniques such as the Self-Consistent RPA treatment (see Ref. [4]) and a natural interpretation of the PSY-MB procedure that goes beyond a simple TDA treatment of the pairing correlation has been underlined.

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

 123 Fig. 1. Correlation energy of the ground state solution of a system of 32 particles distributed over 64 equispaced doubly degenerate levels, in function of the number of 2-pair configurations (left) and the number of 3-pair configurations (right). The value of the pairing strength parameter is G=0.345 MeV.

Fig. 5 .

 5 Fig. 5. Correlation energy of the ground state solution of a system of 32 particles distributed over 64 equispaced doubly degenerate levels, in function of the pairing interaction strength parameter.


Hector Garcia De Marina

Bayu Jayawardhana

Ming Cao

Distributed scaling control of rigid formations

Recently it has been reported that biased rangemeasurements among neighboring agents in the gradient distance-based formation control can lead to predictable collective motion. In this paper we take advantage of this effect and by introducing distributed parameters to the prescribed interdistances we are able to manipulate the steady-state motion of the formation. This manipulation is in the form of inducing simultaneously the combination of constant translational and angular velocities and a controlled scaling of the rigid formation. While the computation of the distributed parameters for the translational and angular velocities is based on the wellknown graph rigidity theory, the parameters responsible for the scaling are based on some recent findings in bearing rigidity theory. We carry out the stability analysis of the modified gradient system and simulations in order to validate the main result.

I. INTRODUCTION

The use of teams of autonomous agents has attracted a lot of interest in recent years. This is due to the fact that in many tasks, such as the transportation of objects or area exploration & surveillance, robotic teams can effectively accomplish tasks with robustness against uncertain environment and offer new functionalities, e.g. enhanced sensing instrumentation [START_REF] Sheng | Distributed multi-robot coordination in area exploration[END_REF]. One of the key tasks in coordinating a team of agents is the formation and motion control, where the former refers to keeping a prescribed shape while the later refers to the steering of it. In particular, a very active topic regarding formation control is the distance-based control for rigid shapes, where the combination of potential-gradient control and rigidity graph theory allows us to achieve (locally) a prescribed shape by only controlling the inter-distances between neighboring agents [START_REF] Krick | Stabilization of infinitesimally rigid formations of multi-robot networks[END_REF], [START_REF] Anderson | Rigid graph control architectures for autonomous formations[END_REF]. It is a very appealing approach since the agents can work with only local information, such as their own frame of coordinates and the relative positions of their neighbors. In addition, the equilibrium at the prescribed shape is exponentially stable and it can be made robust against sensor's biases [START_REF] Cao | Controlling a triangular formation of mobile autonomous agents[END_REF], [START_REF] Sun | Generalized controllers for rigid formation stabilization with application to event-based controller design[END_REF], [START_REF] Garcia De Marina | Controlling rigid formations of mobile agents under inconsistent measurements[END_REF].

In this paper we propose a novel distributed control algorithm for achieving the following three tasks simultaneously: i) Formation scale-free shape keeping.

ii) Steering the scale-free formation as a whole with the combination of a constant translation velocity and a constant angular velocity applied at its centroid.

Hector Garcia de Marina is with the Ecole de la Aviation Civile, Toulouse, France. Bayu Jayawardhana and Ming Cao are with the Engineering and Technology Institute of Groningen, University of Groningen, 9747 AG Groningen, The Netherlands. (e-mail: {h.j.de.marina, b.jayawardhana, m.cao}@rug.nl). This work was supported by the the EU INTERREG program under the auspices of the SMARTBOT project and the work of Cao was also supported by the European Research Council (ERC-StG-307207) and the Dutch Technology Foundation (STW-vidi-14134).

iii) Precise scaling of the formation, i.e. controlling precisely the rate of growing or shrinking between two desired scaled shapes. The proposed control law even allows the changing between two different shapes.

The findings of our work employ the recent results in [START_REF] Zhao | Bearing rigidity and almost global bearingonly formation stabilization[END_REF] on bearing rigidity theory. Roughly speaking, bearing rigidity theory is employed for controlling a shape instead of focusing on maintaining constant distances or positions between neighbors, so one is interested in maintaining constant inner angles of the shape which can be obtained from the unit vectors between neighbors of a scale-free rigid shape. In fact these findings have been recently employed to control the translational motion of a rigid formation with a precise scaling rate in [START_REF]Translational and scaling formation maneuver control via a bearing-based approach[END_REF]. The approach presented in this paper has several advantages over [START_REF]Translational and scaling formation maneuver control via a bearing-based approach[END_REF]. Firstly, it does not require a common frame of coordinates for the agents. Secondly it is estimator free and it does not require global information such as the position of the centroid and its desired velocity. Lastly, the distance-based approach also allows rotational motion, a feature lost in the position-based control since the steadystate orientation is globally fixed by design.

The strategy employed in this paper is based on assigning motion parameters to the prescribed distances of a desired rigid formation. It has been reported in [START_REF] Mou | Undirected rigid formations are problematic[END_REF] that when two neighboring agents differ in the prescribed distance to maintain, collective motion of the formation occurs. More precisely, the formation converges to a distorted version of the desired rigid shape and at the same time it undergoes a constant translation together with a rotation about its centroid as has been described in detail in [START_REF] Sun | Formation movements in minimally rigid formation control with mismatched mutual distances[END_REF]. It has been shown in [START_REF] Garcia De Marina | Distributed rotational and translational maneuvering of rigid formations and their applications[END_REF] that if these mismatches in the prescribed distance are taken as distributed motion parameters, one can maintain a desired rigid shape and at the same time control precisely a combination of a constant translation of the formation with a constant rotation about its centroid. By unifying the aforementioned results on motion control employing distributed motion parameters and bearing rigidity theory, one can control simultaneously the motion of the prescribed rigid shape and its scale in a precise way, i.e. not distorting a scale-free shape for a desired rate of growing/shrinking.

The rest of the paper is organized as follows. In Section II we introduce the notation and background for bearing rigid formations. Section III explains the design of the motion controller with precise scaling/morphing of the formation by introducing changing-motion parameters in a distance-based controller. In Section IV we demonstrate the exponential convergence of our proposed algorithm. Numerical simulations validate the main results in this paper in Section V.

II. PRELIMINARIES

We start by introducing some notation employed throughout the paper. For a given matrix A ∈ IR n×p , define A ∆ = A⊗ I m ∈ IR nm×pm , where the symbol ⊗ denotes the Kronecker product, with m = 2 for the 2D formation case or m = 3 for the 3D case, and I m is the m-dimensional identity matrix. For a stacked vector/matrix

x ∆ = x T 1 x T 2 . . . x T k T with
x i ∈ IR n×l , i ∈ {1, . . . , k}, we define the block diagonal matrix D x ∆ = diag{x i } i∈{1,...,k} ∈ IR kn×kl . We denote by |X | the cardinality of the set X , by ||x|| the Euclidean norm of a vector x and by x = x ||x|| the unit vector of a non-zero x. We define the orthogonal projector operator as

P ⊥ x ∆ = I m -xk xT k
or more generally P ⊥ X over an orthogonal subspace of X . Finally we use 1 n×m and 0 n×m to denote the all-one and all-zero matrix in IR n×m respectively and will drop the subscript if the dimensions are clear from the context.

A. Graphs and rigidity theory

We consider a formation of n ≥ 2 agents whose positions are denoted by p i ∈ IR m for i ∈ {1, . . . , n}. The agents can measure their range and directions with respect to their neighbors. The representation of this sensing topology is given by an undirected graph G = (V, E) with the vertex set V = {1, . . . , n} and the ordered edge set E ⊆ V × V. The set N i of the neighbors of agent i is defined by N i ∆ = {j ∈ V : (i, j) ∈ E}. We define the elements of the incidence matrix A framework is defined by the pair (G, p), where p = p T 1 . . . p T n T is the stacked vector of the agents' positions. The available relative positions of the agents in the framework are given by the following stacked vector

B ∈ IR |V|×|E| for G by b ik ∆ =      +1 if i = E tail k -1 if i = E head k 0 otherwise,
z = B T p,
where each vector z k = p i -p j in z corresponds to the relative position associated with the edge E k = (i, j).

Let us now briefly recall the notions of distance infinitesimally rigid framework and minimally rigid framework from [START_REF] Anderson | Rigid graph control architectures for autonomous formations[END_REF]. Define the edge function

f G by f G (p) = col k z k 2
where the operator col defines the stacked column vector and we denote its Jacobian, also known as the rigidity matrix, by

R(z) = D T z B T . A framework (G, p) is infinitesimally rigid if rankR(z) = 2n -3 when it is embedded in R 2 or if rankR(z) = 3n-6 when it is embedded in R 3 . Additionally, if |E| = 2n -3 in the 2D case or |E| = 3n -6 in the 3D
case then the framework is called minimally rigid. Roughly speaking, the only motions that one can perform over the agents in a minimally rigid framework, while they are already in the desired shape, are the ones defining translations and rotations of the whole shape. The stacked vector of relative positions z

* = [ z * 1 T z * 2 T ... z * |E| T ]
T defines a desired infinitesimally and minimally rigid shape with ||z * k || = d k for all k ∈ {1, . . . , |E|} where d k is the desired inter-distance. The resulting set Z of the possible formations with the same shape is defined by

Z ∆ = I |E| ⊗ R z * , ( 1
)
where R is the set of all rotational matrices in 2D or 3D. Roughly speaking, Z consists of all formation positions that are obtained by rotating z * . Consider a scale-free shape based on an infinitesimally and minimally rigid shape, for example the collection of all regular squares with an internal diagonal. It is obvious that this collection can be distinguished from other (infinitesimally and minimally rigid) scale-free shapes by looking at its inner angles or equivalently by looking at all the scalar products ẑT l ẑn where l and n are two edges sharing a node. This fact has been explained in more detail in [START_REF] Zelazo | Rigidity theory in se(2) for unscaled relative position estimation using only bearing measurements[END_REF]. Bearingbased rigid frameworks are related to the distance-based ones where the bearing-based shape can be defined by the inner angles, instead of the distances. Let us review some basic concepts in bearing rigidity.

Definition 2.1:

[7] Frameworks (G, p) and (G, p ) are bearing equivalent if P ⊥ z k z k = 0 for all k ∈ {1, . . . , |E|}. Definition 2.2: [7] Frameworks (G, p) and (G, p ) are bearing congruent if P ⊥ (pi-pj ) (p i -p j ) = 0 for all i, j ∈ V. Definition 2.3: [7] The bearing function is defined by f B G (p) ∆ = ẑ ∈ IR m|E|
, where1 ẑ is the stacked vector of ẑk for all k ∈ {1, . . . , |E|}. Similar to the rigidity matrix one can define the bearing rigidity matrix by computing the Jacobian matrix of the bearing function

R B (z) = ∂f B G (p) ∂p = D T z D T P ⊥ ẑ B T ,
where P ⊥ ẑ ∈ IR m|E|×m is the stacked matrix of operators P ⊥ ẑk and z ∈ IR |E| is the stacked vector of 1 ||z k || for all k ∈ {1, . . . , |E|}. The non-trivial kernel of R B (z) includes the scalings and translations of the framework [START_REF] Zhao | Bearing rigidity and almost global bearingonly formation stabilization[END_REF], leading to the following definition Definition 2.4: [START_REF] Zhao | Bearing rigidity and almost global bearingonly formation stabilization[END_REF] A framework is infinitesimally bearing rigid if the kernel of its bearing rigidity matrix only includes scalings and translations. In order words, if a scale-free shape can be determined uniquely by its inner angles, then it belongs to the infinitesimally bearing rigid framework.

Consider a given shape defined by Z, we define the scalefree Z S by taking Z rescaled by all the possible scale factors s ∈ IR + such that ||z k || = sd k for all k ∈ {1, . . . , |E|}. This leads to the following definition Definition 2.5: The shapes defined by Z within the set Z S are infinitesimally and minimally congruent rigid.

The name comes from the fact that all the scales of an infinitesimally and minimally rigid shape are bearing congruent.

B. Frames of coordinates

In order to describe and design motions for the desired scale-free formation defined by Z S , it will be useful to attach a frame of coordinates to the centroid of the shape. We denote by O g the global frame of coordinates fixed at the origin of IR m with some arbitrary fixed orientation. In a similar way, we denote by O b the body frame fixed at the centroid p c of the desired scale-free rigid formation. Furthermore, if we rotate the scale-free rigid formation with respect to O g , then O b is also rotated in the same manner. Note that p c is invariant with respect to Z S . Let b p i denote the position of agent i with respect to O b . In order to simplify notation, whenever we represent an agent's variable with respect to O g , the superscript is omitted, e.g. p i ∆ = g p i .

III. MOTION AND SCALING OF RIGID FORMATIONS

We consider the n agents in the framework (G, p) to be governed by single integrator dynamics

ṗi = u i , (2)
where u i ∈ IR m is the control action for all i ∈ {1, . . . , |V|}.

For each edge E k in the framework one can associate a potential function V k (z k ) whose minimum corresponds to the desired configuration of the associated edge, for example, in order to (locally) stabilize Z we can employ the classical elastic potential function from physics for controlling the length of the edges

V (p) = |E| i=1 V k (z k ) = 1 2 |E| i=1 (||z k || -d k ) 2 . ( 3
)
It has been reported in [START_REF] Mou | Undirected rigid formations are problematic[END_REF] that in undirected gradientbased controlled formations if at least two neighboring agents differ about the prescribed distance to maintain, i.e. they have a mismatch, then a steady-state collective motion with a distorted shape occurs. The collective motion, illustrated in Figure 1, is described by the combination of two constant velocity vectors:

• A linear velocity b v * pc of the centroid with respect to the steady-state distorted shape.

• An angular velocity b ω * that rotates the steady-state distorted shape. It has been shown in [START_REF] Garcia De Marina | Distributed rotational and translational maneuvering of rigid formations and their applications[END_REF] that if we replace mismatches by distributed motion parameters, then we can control both, a non-distorted desired shape and a desired motion of the formation with respect to O b . Throughout this section we will show that such an approach can also be employed to scale the formation shape precisely over time while simultaneously to guide the formation to travel with a desired b v * pc and b ω * . This approach is easier and more effective in several aspects than the one presented in [START_REF]Translational and scaling formation maneuver control via a bearing-based approach[END_REF], since we do not need estimators for traveling at a constant speed and we can also rotate the desired formation with respect to O g by controlling only one agent. Moreover, using our proposed approach, the agents can use only local coordinates since we are using the distance-based control strategy.

We introduce the motion and changing parameters to the gradient-based control and show the steady-state motion, including the scaling of the shape Z, is related to its unit vectors. The control inputs derived from the gradient of the distance-based potential (3) for the agents i and j on the edge E k = (i, j) are as follows

u k i = -ẑ k ||z k || -d k u k j = ẑk ||z k || -d k , (4)
where the superscript k denotes the contribution of the edge k to the total control input u i and u j . Introduce a pair of parameters µ k and μk to the prescribed distance d k as follows

u k i = -ẑ k ||z k || -d k -µ k = -ẑ k ||z k || -d k ) + ẑk µ k u k j = ẑk ||z k || -d k + μk = ẑk ||z k || -d k ) + ẑk μk .
(5) The structure in [START_REF] Sun | Generalized controllers for rigid formation stabilization with application to event-based controller design[END_REF] allows us to write the complete control law u in the following compact form

u = -cBD ẑ e + A(µ, μ)ẑ, (6)
where u ∈ IR m|V| is the stacked vector of control actions u i , c ∈ IR + is a constant gain, e ∈ IR |E| is the stacked vector of all the distance errors e k = ||z k || -sd k where all the sd k 's have been taken from Z s , the parameters µ ∈ IR |E| and μ ∈ IR |E| are the stacked vectors of µ k and μk for all k ∈ {1, . . . , |E|} and the elements of A are defined as follows

a ik ∆ =      µ k if i = E tail k μk if i = E head k 0 otherwise. (7)
Note that the elements of A are related to the incidence matrix B because of (5), and hence we still have a distributed control law. We can identify two terms at the right hand side of ( 6). The first one is clearly related to the gradient distance-based controller and its purpose is to form and keep the prescribed shape given by Z s . The second term corresponds to the steady-state collective motion and changing induced by the parameters µ k and μk and the actual shape of the formation given by the unit vectors in ẑ. We will see that in order to guarantee the stability of the system we will make use of the exponential convergence of the self-contained error system in the original gradient-based controller. By choosing c in (6) sufficiently large, we can make the gradient-based term dominate the second term. Therefore the team of agents will converge to the desired shape Z s , where s can be timevarying i.e. we will scale the shape within Z S , and the steady-state motion will be given by the parameters and the unit vectors in z ∈ Z S .

A. Design of the distributed motion and changing parameters

Suppose that the formation is at the prescribed shape, i.e. e = 0. In this case if A(µ, μ)ẑ defines translations and rotations of the infinitesimally and minimally congruent rigid family Z S , then the desired scaled shape Z s will be invariant under such an additional control term. Note that from ( 6) when e = 0 the control law for the agent i becomes

b u i = |E| k=1 a ik b ẑ * k , (8)
where b ẑ * ∈ Z S . We recall that the elements a ik of A are related to µ and μ as in [START_REF] Zhao | Bearing rigidity and almost global bearingonly formation stabilization[END_REF]. In an infinitesimally and minimally congruent rigid formation, the minimum number of neighbors for agent i is two (resp. three) in 2D (resp. 3D) shapes with its corresponding z * k 's not being in nongeneric degenerated configurations, e.g. all of them collinear (resp. coplanar), then b u i can span the whole IR 2 (resp . IR 3 ). In other words, we can design a pair of arbitrary constant velocities b v * pc and b ω * for the desired scale-free formation Z S by choosing appropriately µ and μ. For choosing such µ and μ, let us decompose them into µ = µ v + µ ω + µ s and μ = μv + μω + μs , where each term in this decomposition can be used to define the translation, rotation and scaling of the group motion. Here, the subscript 'v' refers to the motion parameters responsible for b v * pc , 'ω' refers to b ω * and finally 's' refers to the changing parameters which are responsible for dilating/contracting the shape within Z S . As shown in [START_REF] Garcia De Marina | Distributed rotational and translational maneuvering of rigid formations and their applications[END_REF] the motion parameters of µ v , μv , µ ω and μω can be determined by imposing restrictions on the dynamics of z and e in order to keep invariant e at e = 0, i.e.

B

T A(µ, μ)ẑ = 0 (9)

D ẑ B T A(µ, μ)ẑ = 0, (10)
where ( 9) stands for translations and (10) stands for rotations and translations. Let us write the following identity

A(µ, μ)ẑ = S1 D ẑ S2 D ẑ µ μ = T (ẑ) µ μ , (11)
where S 1 is constructed by setting all the 1 elements in the incidence matrix B to zero and S 2 ∆ = S 1 -B. In order to compute the distributed motion parameters µ v , μv for the translational velocity b v * pc we eliminate the components of µ and μ responsible of non-motion, i.e. A(µ, μ)ẑ = 0 in [START_REF] Garcia De Marina | Controlling rigid formations of mobile agents under inconsistent measurements[END_REF], by

O b Og z * 1 z * 2 z * 3 z * 4 z * 5
Fig. 2: The velocity of the agents at the desired shape is the linear combination of the unit vectors from their associated relative positions (black dashed lines) given by [START_REF]Translational and scaling formation maneuver control via a bearing-based approach[END_REF]. The common velocity b v * pc of the centroid is marked in blue and it is given by µ v and μv . The rotational velocity about the centroid defining b ω * is marked in red and is given µ ω and μω . The velocity responsible for scaling the formation has been marked in green and it is given by µ s and μs . Note that these velocities are constant with respect to O b .

projecting the kernel of B T T ( b ẑ * ) (derived from ( 9)) over the orthogonal space of the kernel of ( 11)

µ v μv ∈ Û ∆ = P ⊥ Ker{T ( b ẑ * )} Ker{B T T ( b ẑ * )} ( 12
)
where we have employed b ẑ * in order to define the translational velocity of the desired formation with respect to O b as in Figure 1. In a similar way, by removing the components responsible of non-motion and translational velocity, the computation of the distributed motion parameters µ ω , μω for the rotational motion of the desired shape is obtained from [START_REF] Sun | Formation movements in minimally rigid formation control with mismatched mutual distances[END_REF] and [START_REF] Zelazo | Rigidity theory in se(2) for unscaled relative position estimation using only bearing measurements[END_REF] as

µ ω μω ∈ Ŵ ∆ = P ⊥ Û Ker{D T b ẑ * B T T ( b ẑ * )} . ( 13
)
In order to compute the distributed changing parameters µ s , μs we need to look at the bearing rigidity matrix R B (z).

It has been shown in [START_REF] Zhao | Bearing rigidity and almost global bearingonly formation stabilization[END_REF] that the meaning of the kernel of the bearing rigidity matrix stands for translations and scalings of the desired shape. Therefore in a similar way as before the following condition

µ s μs ∈ S ∆ = P ⊥ Û Ker{D T P ⊥ b ẑ * B T T ( b ẑ * )} , (14)
will give us the space of changing parameters responsible for the scaling of the formation. We would like to remark that the presented motion and changing parameters have been designed for a family of infinitesimally and minimally congruent shapes Z S , i.e. for a scale-free version of a desired infinitesimally and minimally rigid shape. Note that the three spaces U, W and S have been computed in a centralized way while the parameters µ and μ are applied in a distributed fashion. This computation can be done off-line during the design stage, e.g. at the same time that one designs the corresponding d k for the desired shape Z. An example of the three spaces U, W and S for an infinitesimally and minimally congruent regular squares is given in Figure 2.

B. Design of the controller for precise motion and changing of the formation

Here by precise scaling we mean the control of agents such that the inter-distances follow a time varying d(t) but with the shape in Z S . More precisely, we set e k (t) = ||z k (t)||-d k (t) for k ∈ {1, . . . , |E|} with Z ∈ Z S in a family of infinitesimally and minimally congruent rigid shapes. For simplicity we set the following relation in the edge

E k d k (t) = s(t)d * k + d * k , (15)
where s(t) ∈ IR is a time varying scaling signal which is assumed to be at least C 1 and d * k is defined for a particular Z. We remark here that the form used in ( 15) is for convenience of design. One can of course choose d k (t) = s(t)d * k . Without loss of generality, we assume that s(0) = 0. Obviously, for well-posedness we also impose that s(t) is defined properly such that d k (t) > 0 for all t and k ∈ {1, . . . , |E|}.

It is clear that the desired linear speed || b v * pc || and the desired angular speed || b ω * || are related to the norms of µ v , μv and µ ω , μω respectively. It can also be easily checked that the speed d dt d k (t) is related to the norms of µ s and μs . We derive the dynamics of z and e from (6) but consider the time varying desired distances

ż = -cB T BD ẑ e + B T A(µ, μ)ẑ (16) ė = -cD T ẑ B T BD ẑ e + D T ẑ B T A(µ, μ)ẑ -ḋ, (17)
where we have rewritten e as the stacked vector of e k (t) = ||z k (t)|| -d k (t) for k ∈ {1, . . . , |E|}, and d is the stacked vector of d k (t) also for k ∈ {1, . . . , |E|}.

In a similar way as in ( 9) and [START_REF] Sun | Formation movements in minimally rigid formation control with mismatched mutual distances[END_REF], in order to compensate ḋ in (17) we impose the following condition for keeping invariant e for e = 0, i.e. the formation shape is always in

Z s ḋ = D T b ẑ * B T S 1 Db ẑ * S 2 Db ẑ * µ μ , (18)
so that the last two terms of the right hand side of ( 17) is zero when e = 0. Note that the solution to (18) for µ and μ includes the spaces U and W. Therefore the distributed changing parameters that we are looking for scaling the desired shape with a desired scaling speed are [ µs μs ] ∈ S such that (18) holds.

For the constant growing case, i.e. s(t) = st, where s ∈ IR + is a common constant scaling speed among all the agents, we have that ḋ = ds dt d * = sd * and therefore the solution of (18) gives constant µ s and μs . Considering the periodic scaling case we have that s(t) = s sin(ωt), which obviously satisfies ḋ = ds dt d * = sω cos(ωt)d * . Therefore the changing parameters µ s and μs for the periodic case are the same as we have seen previously calculated for the constant growing case but multiplied by the periodic signal ω cos(ωt), which is obviously independent of the actual shape.

IV. STABILITY ANALYSIS

Before presenting the main result, we need to show first that the error system in ( 17) is an autonomous system. Indeed, the second term at the right hand side of (17) depends on the dot products of the form ẑT i ẑj for i, j ∈ {1, . . . , |E|}. It has been shown in [START_REF] Mou | Undirected rigid formations are problematic[END_REF] that all the scalar products z T i z j for i, j ∈ {1, . . . , |E|} can be written as smooth functions of the inter-distances ||z k || for k ∈ {1, . . . , |E|}. Since the errors e k for k ∈ {1, . . . , |E|} are functions of only the inter-distances ||z k || and ẑk for k ∈ {1, . . . , |E|}, we have that

ẑT i ẑj = g ij (e), i, j ∈ {1, . . . , |E|}, (19)
where g ij is a local smooth function around the shape z ∈ Z s . Note that when z ∈ Z s , the second and third terms on the right hand side of (17) vanish because of (18), therefore we can write the following local function

f (e) = D T ẑ B T A(µ, μ)ẑ -ḋ, f (0) = 0 ⇐⇒ z ∈ Z s .
(20) Employing the same argument, the matrix in the first term of the right hand side of (17) can be rewritten as

Q(e) = D T ẑ B T BD ẑ , (21)
where it has been shown in [START_REF] Garcia De Marina | Distributed rotational and translational maneuvering of rigid formations and their applications[END_REF] that Q(0) with z ∈ Z s is positive definite. 12), ( 13) and ( 14) respectively. Then, there exist constants ρ, c * > 0 such that the origin of the system (17), corresponding to z ∈ Z s with time-varying s(t) as in [START_REF] Garcia De Marina | Taming interdistances mismatches for formation-motion control of rigid formations in second-order agents[END_REF], is locally exponentially stable for all c ≥ c * in the compact set Q ∆ = {e : ||e|| 2 ≤ ρ}. In particular, the formation will converge exponentially fast to the time-varying shape defined by Z s with the speed ds(t) dt satisfying (18) and the agents' velocities

b ṗi (t) → b ṗ * i , t → ∞, i ∈ {1, . . . , |V|}, (22)
with f (e) and Q(e) as in ( 20) and ( 21) respectively. We have that Q(0) is positive definite and that in a neighborhood of Z s the formation is still infinitesimally and minimally rigid, therefore Q(e) is positive definite in the compact set Q for some small positive ρ. Furthermore, f (e) is locally Lipschitz in the compact set Q and f (0) = 0 with z ∈ Z s , therefore there exists a constant q ∈ IR + such that

dV dt ≤ (-cλ min + q)||e|| 2 , (25)
where λ min is the minimum eigenvalue of Q(e) in Q. Thus if one chooses c ≥ c * > q λmin , then the exponential stability of the origin of (17) follows from Theorem 4.10 from [START_REF] Khalil | Nonlinear systems[END_REF] for non-autonomous systems. Therefore we have that the formation shape converges exponentially to Z s . Now we substitute e(t) → 0 and ẑ(t) → Z s as t goes to infinity into (6) and ( 2), which gives us

ṗ(t) -Ā(µ, μ)ẑ(t) → 0, t → ∞. (26)
In other words, the velocity of the formation converges exponentially fast to the desired velocity given as a superposition of b v * pc and b ω * with the scaling speed ds(t) dt satisfying (18).

Remark 4.2: The magnitude of the positive constant ρ only depends on the desired shape, i.e. if one chooses a shape where all the agents are far away from each other and far away from a collinear (2D) or coplanar (3D) configuration, then one should expect a bigger ρ for such desired shape than for the one that does not meet such requirements. In some sense ρ is measuring (in a conservative way) how the desired formation can be distorted without falling into a degenerated configuration. Examples about how to compute c * and q can be found in the PhD thesis [START_REF] De | Distributed formation control for autonomous robots[END_REF].

V. SIMULATION RESULTS

In this section 2 we validate the correctness of Theorem 4.1. We have four agents with a scale-free regular square as the prescribed shape. The objective of this simulation is to design the distributed motion-changing parameters µ and μ in the control law [START_REF] Garcia De Marina | Controlling rigid formations of mobile agents under inconsistent measurements[END_REF] such that the square spins around its centroid and at the same time we vary periodically the scale of the square. We define the sensing topology of the agents by B = 1 0 -1 0 -1 -1 1 0 0 0 0 -1 1 -1 0 0 0 0 1 1 and define the regular square that we will periodically scale with side-length d * 1 = 15 pixels. In order to induce the spinning motion we design the following µ ω and μω satisfying [START_REF] Khalil | Nonlinear systems[END_REF] µ ω = [ -w -w 0 w -w ] T , μω = [ -w -w 0 w -w ] T , (27) with w = 1. We want to vary periodically the size of the square following the desired time-varying distances d i (t) = d * i + 2hd * i sin(ω s t), i = {1, . . . , 5},

where one can deduce that s(t) = 2h sin(ω s t) and we set h = 2 and ω s = 1.5 rads/sec. The desired µ s and μs satisfying ( 14) and ( 18) are

[µs(t) T μs(t)

T ] = hω s cos(ω s t)[ 1 1 0 1 1 -1 -1 0 -1 -1 ].

(29) Finally we choose c = 5 for (6), which in numerical checking is much smaller than the conservative gain in Theorem 4.1. The numerical results are shown in Figure 3.

VI. CONCLUSIONS

In this paper we have modified the popular distancebased controller by adding distributed parameters at their prescribed inter-distances in order to control the steady-state motion while at the same time controlling precisely the scaling rate of the formation. This approach is compatible with higher order agent dynamics [START_REF] Garcia De Marina | Taming interdistances mismatches for formation-motion control of rigid formations in second-order agents[END_REF] and is applicable to 2 Video footage from actual mobile robots can be found at www.youtube.com/c/HectorGarciadeMarina with their explanations in [START_REF] De | Distributed formation control for autonomous robots[END_REF].

Fig. 1 :

 1 Fig. 1: The resultant motion of the tetrahedron is the composition of the two constant velocities b v * pc and b ω * . The described trajectory has been split in two red curves.

Theorem 4 . 1 :

 41 Consider the distributed parameters [ µv μv ], [ µω μω ] and [ µs μs ] belonging to the spaces (

  where the b ṗ * i 's are given by the desired b ṗ * c , b ω and ds(t) dt , that are determined by [ µv μv ], [ µω μω ] and [ µs μs ]. Proof: Consider the following candidate Lyapunov function V ė = -ce T Q(e)e + e T f (e),

Fig. 3 :

 3 Fig. 3: The left plot shows the trajectories of the agents with the 'x' denoting the initial points. The dashed lines show two different scales of the prescribes square. The right plot shows the evolution of the inter-distances.

In order not to overload the notation, here by ẑ we mean exclusively the vector-element wise normalization of z.