Stéphane Dufau

Jeremy Yeaton

Jean-Michel Badier

Sophie Chen

Phillip J Holcomb

Jonathan Grainger
email: i.jonathan.grainger@gmail.com

Neural sentence superiority

Keywords: Sentence superiority effect, Reading, Interactive-Activation, MEG

When a sequence of written words is briefly presented and participants are asked to identify just one word at a post-cued location, then word identification accuracy is higher when the word is presented in a grammatically correct sequence compared with an ungrammatical sequence. This sentence superiority effect has been reported in several behavioral studies and two EEG investigations. Taken together, the results of these studies support the hypothesis that the sentence superiority effect is primarily driven by rapid access to a sentence-level representation via partial word identification processes that operate in parallel over several words. Here we used MEG to examine the neural structures involved in this early stage of written sentence processing, and to further specify the timing of the different processes involved. Source activities over time showed grammatical vs. ungrammatical differences first in the left inferior frontal gyrus (IFG: 325-400 ms), then the left anterior temporal lobe (ATL: 475-525 ms), and finally in both left IFG and left posterior superior temporal gyrus (pSTG: 550-600 ms). We interpret the early IFG activity as reflecting the rapid bottomup activation of sentence-level representations, including syntax, enabled by partly parallel word processing. Subsequent activity in ATL and pSTG is thought to reflect the constraints imposed by such sentence-level representations on on-going word-based semantic activation (ATL), and the subsequent development of a more detailed sentence-level representation (pSTG). These results provide further support for a cascaded interactive-activation account of sentence reading.

Introduction

The sentence superiority effect reflects the literate brain's ability to better recall a grammatically correct sequence of words compared with an ungrammatical sequence. Since its discovery by [START_REF] Cattell | The time it takes to see and name objects[END_REF][START_REF] Scheerer | Early German approaches to experimental reading research: The contributions of Wilhelm Wundt and Ernst Meumann[END_REF], it has been used to study the relational binding of words according to certain syntactic and grammatical rules (see [START_REF] Roverud | Examining the sentence superiority effect for sentences presented and reported in forwards or backwards order[END_REF], for an overview). Typically, in such studies, words are presented one after another (serially) either in a visual or auditory modality.

Recently Snell and Grainger (2017) asked whether or not such an effect could be observed using a brief visual parallel-rather than serial-presentation of a sentence. In this paradigm-the Rapid Parallel Visual Presentation (RPVP) procedure-a word would be better recognized when it is embedded in a syntactically correct (grammatical) sequence. Such a finding would echo the word superiority effect, whereby a letter is better identified when presented embedded in a word rather than a random string of letters [START_REF] Cattell | The time it takes to see and name objects[END_REF][START_REF] Reicher | Perceptual recognition as a function of meaningfulness of stimulus material[END_REF][START_REF] Wheeler | Processes in word recognition[END_REF]). Snell and Grainger asked their participants to identify a single target word that was embedded either in a 4-word sentence or in an ungrammatical scrambled sequence of the same words. The authors were interested in knowing whether a 200-millisecond display of a word sequence was sufficient for a coarse syntactic structure to emerge, which could influence word identification accuracy via feedback to on-going word identification processes. The results showed that words-in-sentences were recognized with better accuracy (+20%) than words in random sequences, regardless of their position in the sequence. This effect, initially studied in literate adults, was also found in 9-year-old primary school children who showed a smaller but significant accuracy gain of around 10% [START_REF] Massol | The sentence superiority effect in young readers[END_REF].

The sentence superiority effect observed with the RPVP paradigm raises the question as to the mechanisms driving this effect. Snell and Grainger interpreted their finding in an Interactive-Activation framework [START_REF] Mcclelland | An interactive activation model of context effects in letter perception: I. An account of basic findings[END_REF] applied to sentence reading, whereby a fast bottom-up activation of sentence-level structures (when these are available) would reinforce word identification processes via top-down feedback from sentence-level structures to words. However, an alternative explanation for the sentence superiority effect obtained with the RPVP paradigm could appeal to the sophisticated-guessing explanation of the word superiority effect [START_REF] Johnston | A test of the Sophisticated Guessing Theory of word perception[END_REF]. That is, identification of part of the sentence context (one or two words and their association) would enable participants to make informed guesses about the identity of the post-cued target word given partial information about that word (e.g., one or two letters). To put these two interpretations to test, an electroencephalography (EEG) experiment was conducted by [START_REF] Wen | Parallel, cascaded, interactive processing of words during sentence reading[END_REF] Snell and Grainger (2017). The authors found a difference in the eventrelated potential (ERP) traces that started at around 270 ms with the bulk of the sentence superiority effect occurring in an early onsetting N400 component. Such an effect, emerging within the classical time-window of lexical access as revealed by EEG studies (see [START_REF] Grainger | Watching the word go by: On the time-course of component processes in visual word recognition[END_REF], for a review), pointed to online, automatic processing of linguistic information (word identities and sentence-level structures) rather than offline guessing procedures as the source of the sentence superiority effect (see [START_REF] Wen | Fast syntax in the brain: Electrophysiological evidence from the rapid parallel visual presentation paradigm (RPVP)[END_REF], for a similar timing of ERP effects in the RPVP paradigm but with a grammatical decision task).

Here, we conducted a similar experiment to [START_REF] Wen | Parallel, cascaded, interactive processing of words during sentence reading[END_REF] but using magnetoencephalography (MEG) rather than EEG in order to associate the timing of the visual sentence superiority effect with the underlying neural structures. MEG affords much higher spatial resolution than EEG and is thus the gold standard technique for cortical source reconstruction combined with millisecond timing. For the present experiment, we used the same materials and procedure as the two preceding studies (Snell & Grainger, 2017;[START_REF] Wen | Parallel, cascaded, interactive processing of words during sentence reading[END_REF], however our participants had to report the post-cued target word by reading it out loud rather than typing it using a computer keyboard.

Current findings suggest that the sentence superiority effect is primarily, but not exclusively, driven by syntactic processing [START_REF] Declerck | Unified syntax in the bilingual mind[END_REF]Massol et al., 2021). In line with our theoretical work on sentence reading (Snell et al., 2017;2018), the first phase of processing driving this effect would involve the rapid association of word identities (partially performed in parallel) with the corresponding parts-of-speech, hence enabling the construction of a primitive (or "good-enough":

Ferreira & Lowder, 2016) syntactic structure. The processing of semantic information associated with words would provide further input to the construction of a sentence-level representation (see e.g., Massol et al., 2021) that provides feedback to on-going word identification processes. The overall processing driving the sentence superiority effect is therefore thought to involve partially parallel word identification, the association of word identities with parts-of-speech, the activation of semantic representations associated with word identities, and the construction of a sentence-level representation on the basis of this syntactic and semantic information. Crucially, this processing is thought to be cascaded and interactive in nature, hence allowing a sentence-level representation to be generated rapidly enough to modulate the ease with which a single word in the sentence can be identified.

It is important to note that much prior research investigating the neural structures associated with syntactic processing has focused on spoken language (see [START_REF] Friederici | The brain basis of syntactic processes: functional imaging and lesion studies[END_REF]Matchin & Hickok, 2020, for reviews), whereas the present study investigates written sentence processing with simultaneous presentation of words. This parallel word presentation contrasts with the serial presentation typically used with sequences of written words in EEG studies (e.g., [START_REF] Hagoort | ERP effects of listening to speech compared to reading: the P600/SPS to syntactic violations in spoken sentences and rapid serial visual presentation[END_REF][START_REF] Osterhout | Event-related brain potentials elicited by syntactic anomaly[END_REF], MEG studies (e.g., [START_REF] Halgren | N400-like magnetoencephalography responses modulated by semantic context, word frequency, and lexical class in sentences[END_REF][START_REF] Marinkovic | Spatiotemporal dynamics of modality-specific and supramodal word processing[END_REF][START_REF] Matchin | The temporal dynamics of structure and content in sentence comprehension: Evidence from fMRI-constrained MEG[END_REF], and fMRI studies (e.g., Wilson et al., 2018;Udden et al., 2022;see Walenski et al., 2019, for a review of written sentence processing). Therefore, the regions of interest (ROIs) for data analysis were primarily determined by prior research using written materials presented simultaneously and not serially. Nevertheless, it should be noted that the ROIs selected for the present study do form part of the more general "language network", highlighted by extensive fMRI research (e.g., [START_REF] Fedorenko | Functional specificity for high-level linguistic processing in the human brain[END_REF][START_REF] Hagoort | Nodes and networks in the neural architecture for language: Broca's region and beyond[END_REF][START_REF] Price | A review and synthesis of the first 20 years of PET and fMRI studies of heard speech, spoken language and reading[END_REF] and confirming the picture already painted by lesion studies in the 19 th century [START_REF] Broca | Remarks on the seat of the faculty of articulated language, following an observation of aphemia (loss of speech)[END_REF][START_REF] Wernicke | Der aphasische Symptomencomplex: eine psychologische Studie auf anatomischer Basis[END_REF]. This language network encompasses neural structures in the left frontal and temporal cortices that include Broca's area and Wernicke's area.

In sum, the key contribution of the present work consisted in investigating the processing of written word sequences presented simultaneously (as opposed to the serial presentation typically used in prior EEG, MEG, and fMRI investigations of written sentence processing) and under data-limited presentation conditions (i.e., brief stimulus exposures). We expected these specific testing conditions to help isolate the neural structures involved in the rapid elaboration of a primitive sentence-level representation during written sentence comprehension. On the basis of prior fMRI work with simultaneously presented sequences of written words (e.g., [START_REF] Bonhage | Combined eye tracking and fMRI reveals neural basis of linguistic predictions during sentence comprehension[END_REF][START_REF] Pallier | Cortical representation of the constituent structure of sentences[END_REF][START_REF] Schuster | Words in context: The effects of length, frequency, and predictability on brain responses during natural reading[END_REF] we expected to find activation related to sentence-level processing predominantly in the left inferior frontal gyrus (IFG) and left posterior temporal regions (pSTG, pMTG), as well as activity in the anterior temporal lobe (ATL: [START_REF] Carter | Linguistic networks associated with lexical, semantic and syntactic predictability in reading: A fixation-related fMRI study[END_REF][START_REF] Henderson | Language structure in the brain: A fixation-related fMRI study of syntactic surprisal in reading[END_REF]. On the basis of our prior ERP work [START_REF] Wen | Parallel, cascaded, interactive processing of words during sentence reading[END_REF], we expected to observe differences between sentences and ungrammatical sequences to first emerge around 300 ms post-stimulus onset (i.e., onset of the complete sequence of words), and to cover the epochs traditionally associated with the N400 and P600 ERP components. Given the hypothesized early role for syntax in our proposed account of the sentence superiority effect, we expected to observe the earliest differences in left IFG given the evidence that IFG plays a key role in processing syntactic information during spoken language comprehension (e.g., [START_REF] Hagoort | Nodes and networks in the neural architecture for language: Broca's region and beyond[END_REF][START_REF] Friederici | Neurobiology of syntax as the core of human language[END_REF][START_REF] Friederici | The brain basis of syntactic processes: functional imaging and lesion studies[END_REF]) and text comprehension under conditions of parallel presentation [START_REF] Constable | Sentence complexity and input modality effects in sentence comprehension: an fMRI study[END_REF]Henderson et al., 6 2016). This was then expected to be followed by activity in the left anterior temporal lobe (ATL) where it is thought that syntactico-semantic integration occurs (Carter et al., 2016), and where sentence-level representations begin to constrain on-going word identification [START_REF] Carter | Linguistic networks associated with lexical, semantic and syntactic predictability in reading: A fixation-related fMRI study[END_REF][START_REF] Henderson | Language structure in the brain: A fixation-related fMRI study of syntactic surprisal in reading[END_REF]. We then expected to observe differences in left posterior temporal regions (pSTG, pMTG), regions that are traditionally associated with the elaboration of more detailed syntactic representations for speech and text [START_REF] Constable | Sentence complexity and input modality effects in sentence comprehension: an fMRI study[END_REF]. Finally, we expected to observe later differences in pSTG, driven by processes that combine lexico-semantic and syntactic information for sentence comprehension (e.g., [START_REF] Friederici | The brain basis of syntactic processes: functional imaging and lesion studies[END_REF], as thought to be reflected in the P600 ERP component [START_REF] Osterhout | Event-related brain potentials elicited by syntactic anomaly[END_REF].

Materials and methods

Material and methods exposed below closely follow those of the reported studies conducted in the MEG center of La Timone Hospital, Marseille, France.

Participants.

Twenty native French speaking adults (mean age= 23.65 years; range= [20;30]; SD= 3.01; 14 women; 6 men) were included in this research and earned EUR 50 for participation. All the participants were recruited from Aix-Marseille University (France). In order to prevent adding unnecessary noise in the MEG signal, participants with non-removable piercings or surgical implants were not included in this study. Participants were asked not to wear metallic or magnetic jewelry and to avoid using cosmetic products. Participants were monolingual French native speakers, had no record of neurological or psychiatric disorders and had regular or corrected hearing and vision. This study was carried out in accordance with the recommendations of the French ethics committee (Comité de Protection des Personnes SUD-EST IV No. 17/051). All participants gave written informed consent in accordance with the Declaration of Helsinki.

Stimuli and Design.

Materials from Snell and Grainger (2017) were used in this experiment. They consisted of fourhundred 4-word sequences that were either syntactically correct or not (word length, average= 4.03, SD= 0.82 letters; word frequency, average= 5.66, SD= 1.08 Zipf units 1 ; 200 syntactically correct sequences). The set of sequences was built in the following way. First, two hundred syntactically correct sequences were constructed and tested using cloze probability measures to ensure that they were semantically neutral (for details, see Snell & Grainger, 2017). Syntactically incorrect sequences were then derived from the correct pool: each correct sequence was associated with an incorrect sequence that was composed of the same words but in a different order. Within each pair of sequences, a word was chosen as the target (used in the identification task; counter-balanced across trials using the 4 possible word positions) and remained in the same position within the two sequences while the three other words were randomly permuted to form the incorrect sequence. This procedure ensured that the obtained scrambled sequence was not syntactically correct while keeping intact the properties of the individual words composing the pair of sequences. To avoid within-participant stimulus repetition, two counterbalanced lists of 200 sequences each (100 syntactically correct and 100 syntactically incorrect sequences) were created, and participants were assigned to one of them. As such, all stimuli were presented in both syntactic conditions (across two participants) and each participant was presented a single member of a sequence pair. Each list of 200word sequences was divided into 4 blocks of 50 sequences. A 2 (Grammatical vs. Ungrammatical) x 4 (within-sequence word target position) factorial design was used, and stimulus order was randomized within blocks.

Procedure.

We used E-Prime software (Psychology Software Tools, 1999) to implement a visual post-cued wordin-sequence identification task where 4 horizontally aligned words were briefly presented (RPVP paradigm). Each trial began with the visual presentation of a fixation sign in the centre of the screen consisting of 2 bars vertically separated by an empty space (see Figure 1). After 500 milliseconds (ms), the empty space was filled by a sequence (either syntactically correct or incorrect) for 200 ms. The sequence's letters were then replaced by a post-mask formed of hash symbols (#) preserving between-word spaces and inter-letter spacing. The post-mask consisted of two colors: the hashes replacing the target word were displayed in yellow and the three other locations were displayed in white. The task consisted of naming the word which had appeared in the cued (yellow) position. The stimulus remained on screen until a verbal response was given. When E-Prime detected an oral response, the visual sequence disappeared, leaving an empty screen for 100 ms followed by a stimulus composed of a single letter "C" (from the French word cligner -to blink) in the middle of the screen presented for 1500 ms. A trial ended with an empty screen for 100 ms. Participants were instructed to fixate the empty space in between bars at the beginning of the trial and to blink only during the "C" phase of a trial. Vertical bars remained on screen from the start of the trial to the display of the letter C. All visual stimuli were displayed using a white 18pt bold Courier New font on a black background. After an initial fixation stimulus was displayed for 500 milliseconds (ms; see Methods section 2.3), a word sequence appeared that was either syntactically correct (cet ours vit ici -this bear lives here) or not (cet ici vit ours -this here lives bear). The word sequence was visible for 200 ms.

Data acquisition and analysis.

The data were acquired on a 248-channel whole-head 4D Neuroimaging MEG system at La Timone Hospital in Marseille, France (4D Neuro-imaging, San Diego, CA). Sampling rate was 2035 Hz.

Individual head shapes, consisting of the forehead, nose, and the location of the head-position coils were digitized using a 3D Polhemus Fastrak device (Polhemus Inc., Colchester, VT, USA). Five headpositioning coils were attached to the forehead and periauricular points to determine the position of the head. Head position was captured at the beginning of the first and third block to check and possibly compensate for differences in head position between the first and second half of the stimuli set. Participants were lying inside a magnetically shielded room and were instructed to move as little as possible. Stimuli were presented on an 1024x768 resolution video-projector screen placed about 40 cm in front of participants. The exact timing of stimuli onset was captured using a photodiode that detected brightness changes on the presentation screen. A non-magnetic microphone (Sennheiser MO2000) was placed nearby the participant's mouth (without skin contact) to capture verbal responses. A stereo audio file was recorded during the entire experiment, with one channel dedicated to the participant's verbal output and another one to record clicks corresponding to the word sequence display onset timing. Verbal output was coded off-line for the correctness of the response.

Following [START_REF] Gross | Good practice for conducting and reporting MEG research[END_REF], an electrooculogram was recorded throughout the experiment to capture the activity of eye movements as well as an electrocardiogram for heartbeats. For all participants but three, cortical surface extraction was performed on individual high-resolution 3D T1-weighted MRI structural images using BrainSuite [START_REF] Shattuck | BrainSuite: an automated cortical surface identification tool[END_REF]. For the three participants having a non-conclusive extraction, sources were constrained to a cortical surface mesh template obtained from the MNI ICBM152 brain. Brainstorm was used with default parameters to warp the template to each participant's digitized head shape. Cortical surface was defined with 15,000 vertices and the MRI was realigned on three fiducials (nasion, left and right pre-auricular points).

Continuous data processing was performed using Anywave [START_REF] Colombet | AnyWave: A cross-platform and modular software for visualizing and processing electrophysiological signals[END_REF] for visual rejection of channels showing excessive noise, muscle, or SQUID jump artifacts, for filtering (1-to 300-Hz bandpass) and independent component analysis (runica algorithm) to identify and remove the heartbeat and blink artifacts. Brainstorm [START_REF] Tadel | Brainstorm: a user-friendly application for MEG/EEG analysis[END_REF] was used for additional filtering (25-Hz lowpass), for epoching signal segments time-locked to word sequence onset (-200 to 800 ms), trial rejection and averaging. Data processing and source reconstruction were performed independently for each participant. Artifact-free epochs were averaged separately for each experimental condition to obtain event-relate fields (ERFs) for each participant. This average was then projected onto the cortical surface using a free orientation, cortically constrained minimum norm estimation (MNE) procedure [START_REF] Hauk | Keep it simple: a case for using classical minimum norm estimation in the analysis of EEG and MEG data[END_REF][START_REF] Hämäläinen | Interpreting magnetic fields of the brain: minimum norm estimates[END_REF]. The MNE was weighted by a sample estimate of sensor noise covariance matrix obtained from empty room recordings for each of the participants and used for improved data modelling as is typical in MNE approaches [START_REF] Baillet | Electromagnetic brain mapping[END_REF][START_REF] Dale | Dynamic statistical parametric mapping: combining fMRI and MEG for high-resolution imaging of cortical activity[END_REF]. The MEG forward model was obtained from overlapping spheres fitted to each participant's scalp points [START_REF] Huang | A sensor-weighted overlapping-sphere head model and exhaustive head model comparison for MEG[END_REF]. For all participants but three, cortical surface extraction was performed on individual MRI images using BrainSuite [START_REF] Shattuck | BrainSuite: an automated cortical surface identification tool[END_REF]. For the three participants having a non-conclusive extraction, sources were constrained to a cortical surface mesh template obtained from the MNI ICBM152 brain. Brainstorm was used with default parameters to warp the template to each participant's digitized head shape. The norm of the three source time series at each cortical voxel (i.e., conversion of orientation-unconstrained sources to flat maps, taking the norm of the three elementary dipoles at each time step, yielding only one value by vertex) was extracted. The difference was taken between the source projections for the two conditions for each participant, then the difference was z-scored with respect to the [-200, 0] milliseconds baseline, and absolute value transformed, thus yielding a single value corresponding to the difference in activation between conditions by vertex at each time point.

We defined four regions of interest (ROIs; see Figure 2) corresponding to the left hemisphere brain areas discussed in the Introduction: inferior frontal gyrus (IFG; x=47 , y=143, z=85, N=287 vertices), posterior superior temporal gyrus (pSTG; x=30 , y=90, z=93, N=227 vertices), anterior temporal lobe (ATL; x=28 , y=121, z=58, N=184 vertices), and posterior middle temporal gyrus (pMTG; x=29, y=92, z=70, N=144 vertices). To obtain the time course of activity in each ROI, the average difference between the conditions (described above) was taken over all the vertices comprising that region. This ROI time course was then compared to the baseline using a t-test procedure 1 (one-tail, positive) with an alpha level of 0.05, and a Bonferroni correction for multiple comparisons over the time dimension.

Only regions with at least 50ms continuously above the alpha threshold were considered significant.

Results

Accuracy results of the word-in-sequence identification task are shown in Figure 3. The accuracy data were analyzed in a 2 (Grammatical, Ungrammatical) x 4 (Positions) logistic mixed-effects regression modeling [START_REF] Jaeger | Categorical data analysis: Away from ANOVAs (transformation or not) and towards logit mixed models[END_REF]. Overall, individual words were better identified when the words were presented in a syntactically-correct sequence (62.4% for Grammatical, 50.3% for Ungrammatical), 11 b=0.83, SD=0.36645, z=2.26 (Pr(>|z|) = 0.024. The effect of Position and the interaction between Position and Grammaticality were not significant.

Figure 3. Mean accuracy per position of the target word (1: most leftward word of the horizontal sequence; 4: most rightward). Responses in the Grammatical condition (words in grammatical sequences) had a higher accuracy than responses in the Ungrammatical condition (words in ungrammatical sequences) at all positions. Error bars correspond to the bootstrapped by-participant 95% confidence interval standardized for participants across positions [START_REF] Cousineau | Confidence intervals in within-subject designs: A simpler solution to Loftus and Masson's method[END_REF].

Cortical source activities for the Grammatical and Ungrammatical conditions in 4 ROIs in left hemisphere (inferior frontal gyrus, IFG; posterior superior temporal gyrus, pSTG; anterior temporal lobe, ATL; posterior middle temporal gyrus, pMTG) were compared using a Bonferroni-corrected ttest at each time point. Three out of the four ROIs revealed a significant difference (see Figure 4

Discussion

Presented in the context of a sequence of written words, a word is better recognized when the sequence is grammatical (i.e., within a sentence or a phrase) rather than an ungrammatical word sequence. This sentence superiority effect was recently brought to light in a series of behavioral and EEG experiments that used post-cued partial report with the Rapid Parallel Visual Presentation (RPVP) procedure (e.g., Snell & Grainger, 2017;[START_REF] Wen | Parallel, cascaded, interactive processing of words during sentence reading[END_REF]. In these studies, enhanced word identification was observed in the grammatical compared with the ungrammatical sequences. Here, we used the materials and procedure of Snell and Grainger (2017) in an MEG setting to study the spatio-temporal dynamics of the sentence superiority effect. As expected, behavioral results revealed the standard sentence superiority effect (i.e., word identification accuracy was greater in the grammatical sequences compared with the ungrammatical sequences). This pattern is evidence that some form of primitive sentence-level representation had been computed even with the very brief (200 ms) stimulus exposures used in the RPVP paradigm. Based on prior fMRI research investigating the processing of simultaneously presented sequences of written words [START_REF] Bonhage | Combined eye tracking and fMRI reveals neural basis of linguistic predictions during sentence comprehension[END_REF][START_REF] Carter | Linguistic networks associated with lexical, semantic and syntactic predictability in reading: A fixation-related fMRI study[END_REF][START_REF] Henderson | Language structure in the brain: A fixation-related fMRI study of syntactic surprisal in reading[END_REF][START_REF] Pallier | Cortical representation of the constituent structure of sentences[END_REF][START_REF] Schuster | Words in context: The effects of length, frequency, and predictability on brain responses during natural reading[END_REF] we identified 4 ROIs as brain regions likely to be involved in written sentence processing, and we examined the sentence superiority effect in each of these ROIs (all located in the left hemisphere). Source activations over time revealed grammatical vs. ungrammatical differences in three of these ROIs: first in IFG, then ATL, and finally in both IFG and pSTG (see Figure 4). We now examine in more detail the potential role of these three ROIs in subserving the sentence superiority effect, and we discuss the implications of the relative timing of the activities that we observed.

Early IFG activity. A previous EEG investigation using the same materials as in the present study found an early onsetting effect in the N400 time-window (274-410 ms) where ungrammatical sequences produced more negative-going waveforms than the grammatical condition [START_REF] Wen | Parallel, cascaded, interactive processing of words during sentence reading[END_REF]. This timing aligns with the first significant activity we found in the IFG (321-406 ms), and therefore, following our interpretation of the EEG findings, points to a key role for IFG in rapidly generating a primitive sentence-level representation upon presentation of a sequence of written words. We suspect that this primitive sentence-level representation takes the form of a "good-enough" syntactic structure, as hypothesized by [START_REF] Ferreira | Prediction, information structure, and good-enough language processing[END_REF]see Friederici, 2011, for an earlier proposal for spoken language comprehension). Further evidence for an early role of IFG in written sentence processing, albeit obtained with a serial presentation procedure, was reported in the MEG study of congruous or incongruous word. In line with our results, activity in IFG was found starting around 300 ms post final word onset. ATL activity. We expected to see activity in ATL given prior fixation-related fMRI work suggesting that this could be the locus of the initial phase of integration of syntactic and semantic information for sentence comprehension (Carter et al., 2016) and where sentence-level representations begin to constrain on-going word identification [START_REF] Carter | Linguistic networks associated with lexical, semantic and syntactic predictability in reading: A fixation-related fMRI study[END_REF][START_REF] Henderson | Language structure in the brain: A fixation-related fMRI study of syntactic surprisal in reading[END_REF]. The timing of the observed activity in ATL (475-525 ms) fits with this interpretation, according to which an early primitive syntactic representation is first computed by IFG (as attested by the present study and the EEG study of [START_REF] Wen | Parallel, cascaded, interactive processing of words during sentence reading[END_REF] followed by the combination of this syntactic information with ongoing semantic processing in ATL as a key step in interpreting the sentence being read. Once again, these findings fit with the earlier MEG study of [START_REF] Halgren | N400-like magnetoencephalography responses modulated by semantic context, word frequency, and lexical class in sentences[END_REF]see also Brennan& Pylkkänen, 2012;[START_REF] Matchin | The temporal dynamics of structure and content in sentence comprehension: Evidence from fMRI-constrained MEG[END_REF] that revealed activity in ATL in the time-window of the N400 ERP component, hence pointing to a role for semantics in generating activity in ATL. pSTG activity co-occurring with a second phase of activity in IFG.

The final structures that were revealed by our sentence superiority manipulation (pSTG and IFG, between 550-600ms) point to an interactive process involving the primitive syntactic structure generated by IFG and the construction of a more detailed sentence-level representation that combines both syntactic and semantic information, but going beyond what is already achieved by ATL. As already proposed by [START_REF] Friederici | Neurobiology of syntax as the core of human language[END_REF], on the basis of spoken language studies, pSTG could well be the host of syntactic reanalysis and repair. In other words, activity in IFG followed by ATL would enable the generation of a tentative sentence-level representation combining syntax and semantics, and the later activity in both pSTG and IFG would reflect a process of verifying that the sentence-level structures computed by IFG and pSTG are compatible. The relatively late activity found in STG in the [START_REF] Halgren | N400-like magnetoencephalography responses modulated by semantic context, word frequency, and lexical class in sentences[END_REF] MEG study could also reflect such a verification process, which would be triggered by both syntactic and semantic anomalies. Very tentatively, this might well involve a top-down matching of the more detailed sentence-level representation (syntax and semantics) with the word identities and parts-of-speech that are thought to form the basis of a primitive sentencelevel structure.

Conclusions

In an MEG investigation of the sentence superiority effect, we found source activities revealing significant differences between the grammatical and ungrammatical word sequences first in the left IFG, then the left ATL, and finally in both left IFG and left pSTG. We interpret the early IFG activity as reflecting the rapid bottom-up activation of sentence-level representations from a sequence of written words, and in particular a primitive syntactic representation of the sequence of words when that is available. Subsequent activity in ATL and pSTG is thought to reflect the constraints imposed by the early sentence-level representation on on-going word-based semantic activation and the integration of semantic and syntactic sentence-level information (ATL), followed by the subsequent development of a more detailed sentence-level representation (pSTG), including verification that this more detailed representation is compatible with the early primitive representation computed by IFG.

Figure 1 .

 1 Figure 1. Trial description for the grammatical (left panel) and ungrammatical (right panel) word sequences.After an initial fixation stimulus was displayed for 500 milliseconds (ms; see Methods section 2.3), a word sequence appeared that was either syntactically correct (cet ours vit ici -this bear lives here) or not (cet ici vit ours -this here lives bear). The word sequence was visible for 200 ms.

Figure 2 .

 2 Figure 2. Illustration of the ROIs selected for their noted role in sentence-level processing in the left hemisphere of the human brain. IFG: inferior frontal gyrus; pSTG: posterior superior temporal gyrus; ATL: anterior temporal lobe; pMTG: posterior middle temporal gyrus.

  ): IFG in 2 time-windows (321-406 ms; 549-602 ms), ATL in a single time window (466-531 ms), and pSTG in a single time window as well (553-622 ms).

Figure 4 .

 4 Figure 4. Results of a t-test comparing the Grammatical-Ungrammatical difference against baseline for the 4 ROIs in the [-100; 700] ms time window across participants (Bonferroni-corrected; minimum of 50 milliseconds of continuous positivity). Y-axis scale for t-values was cut at significance value (~4.1).

  using the same (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

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Funding: This research was supported by grant ERC 742141 awarded to J. Grainger.

Disclosure of competing interests:

The authors declare no competing interests.

Author contributions: Conceptualization (JG); Data curation (SD, JY, JMB, SC); Formal analysis (SD, JY); Funding acquisition (JG); Investigation (SD, SC); Methodology (SD, JMB, SC, JG); Project administration (SD, JMB); Resources (JMB, SC); Software (SD, JY); Supervision (SD, JMB, SC, JG); Validation (SD, JY, JMB, SC, PH, JG); Visualization (JY); Roles/Writing -original draft (SD, JY, JG); Writing -review & editing (SD, JY, PH, JG).

(which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Data/code availability statement: Stimuli and data are available at https://osf.io/wzq5y/.

Ethics statement: Ethics approval was given by the French ethics committee (Comité de Protection des Personnes SUD-EST IV) agreement No. 17/051. All participants gave written informed consent in accordance with the Declaration of Helsinki.


Nicolas Torres

Antun Skuric

Lucas Joseph

Vincent Padois

David Daney

Model Predictive Control for robots adapting their task space motion online

Robots require the ability to autonomously and continuously react to unexpected online changes in the task definition and in the environment, especially those cohabited with humans. To react to these changes, the task, from the current state up to the finish, must instantly be re-considered. This implies a prohibitive re-computation cost. This paper proposes a modular control architecture based on Model Predictive Control, that offers a good compromise between optimally achieving the task and the required computation time, by only re-considering the near future. This framework offers a generic way to formulate task-related objectives and constraints that dissociates the planning from the execution, which depends mainly on the robot dynamics. The proposal exploits a linear formalization of the MPC in SE(3) to implement this architecture in a high-frequency closed-loop controller, achieving task re-planning at the control rate.

The pertinence of the proposed control architecture is demonstrated using experiments with the Franka Emika robots in scenarios where the task to be achieved is modified on the fly.

I. Introduction

Collaborative robotics has increasingly gained interest over the last decades. From an application point of view, it crystallises early expectations [START_REF] Makinson | Research and development prototype for machine augmentation of human strength and endurance: Hardiman i project[END_REF], [START_REF] Colgate | Intelligent assist devices in industrial applications: a review[END_REF] around the improvement of human working conditions [START_REF] Maurice | Human-oriented design of collaborative robots[END_REF], [START_REF] Schoose | Evolution of the biomechanical dimension of the professional gestures of grinders when using a collaborative robot[END_REF]. From a research stand point, it raises several essential questions, for example regarding the prediction of human motion [START_REF] Mainprice | Predicting human reaching motion in collaborative tasks using inverse optimal control and iterative re-planning[END_REF] or the underlying cognitive principles behind the concept of collaboration [START_REF] Chen | Handbook Of Human Factors And Ergonomics[END_REF]. Among these research questions, the more general one of controlling robots in dynamic environments1 is a recurring yet unsolved problem. A large amount of literature has been dedicated to this problem but the most recent works in the domain of collaborative robotics have focused on human avoidance [START_REF] Eckhoff | An MPC framework for planning safe and trustworthy robot motions[END_REF], [START_REF] Merckaert | Real-time motion control of robotic manipulators for safe human-robot coexistence[END_REF]. Another essential feature when considering dynamic environments is the general ability to autonomously and continuously adapt to unplanned task updates. The work in this paper proposes a control approach to address this problem.

Assuming there exist means to update online the state of the robot with respect to its environment and its task goal (i.e. perception), the most obvious way to endow robots with adaptation capabilities is based on online motion re-planning towards the goal. Actually, even though finding a feasible path to the target and updating it continuously is mandatory to get a chance to realize the task at 1 Stellantis, Centre Technique Vélizy, France nicolas.torres@stellantis.fr 2 Inria, AUCTUS Team, Talence, France antun.skuric@inria.fr, lucas.joseph@inria.fr, vincent.padois@inria.fr, david.daney@inria.fr This work is funded by the ANRT and Stellantis Group and performed within the framework of the OpenLab AI on the one hand and by BPI France through the LICHIE project in collaboration with Airbus on the other hand.

1as opposed to static ones where robots are physically separated from anything that could provoke a change in either the task or the surrounding objects during execution some point, re-planning the temporal aspects of the motion from start to finish is actually debatable. Indeed, the whole planned trajectory is unlikely to be achieved when facing unplanned changes in the task. Moreover, despite the advances in the domain of motion planning [START_REF] Coleman | Reducing the barrier to entry of complex robotic software: a moveit! case study[END_REF] and optimal trajectory computation [START_REF] Pham | A new approach to time-optimal path parameterization based on reachability analysis[END_REF], online re-planing of both the path and the time profile over the whole motion remains an intractable problem. It is all the more true when the planning approach includes the computation of a control input trajectory. Re-planning over a receding horizon appears as a good alternative [START_REF] Ghazaei Ardakani | Model predictive control for real-time point-to-point trajectory generation[END_REF] to maintain the control problem computationally feasible online while still aiming at an optimal achievement of the task. This is a compromise solution between approaches where an optimal trajectory is computed once offline and approaches where adaptation to changes in the environment are performed without full adaptation of that trajectory [START_REF] Joseph | Online velocity constraint adaptation for safe and efficient human-robot workspace sharing[END_REF].

A second implication of the evolution of robots in dynamic environments is related to the impossibility to verify a priori that all constraints, either task-related or linked to the intrinsic robot capabilities, will be met. In other words, on-line re-planning definitely provides some level of guarantee that the generated motion will not lead the robot to violate constraints. However, it is unlikely to be the case if unplanned events occur while executing the motion. This leads to situations where emergency stops are the only way to prevent an accident. To avoid these control exceptions, controllers must be formulated as optimization problems where constraints are explicitly considered as such in the resolution [START_REF] Liu | Generalized hierarchical control[END_REF].

Model Predictive Control (MPC) appears as a good candidate as it features both the ability to optimally re-plan the motion given the current state of the system (i.e. in closed-loop) over a receding horizon while accounting for constraints. Beyond local optimality, reasoning over a horizon provides an implicit way to render the set of considered constraints compatible which is much more likely to succeed with respect to approaches attempting to perform this compatibilisation once off-line [START_REF] Rubrecht | Constraints compliant control: constraints compatibility and the displaced configuration approach[END_REF], [START_REF] Meguenani | Safe control of robotic manipulators in dynamic contexts[END_REF], [START_REF] Del Prete | Joint position and velocity bounds in discrete-time acceleration/torque control of robot manipulators[END_REF].

Assuming that MPC can keep up with the robot control rate, one can rely on a control architecture where, given an operational space target, a joint velocity or torque control input horizon is re-computed at each control time step and fed to the robot. This is what is proposed in [START_REF] Kleff | High-Frequency Nonlinear Model Predictive Control of a Manipulator[END_REF] where MPC is performed through Differential Dynamic Programming which can be solved efficiently [START_REF] Mastalli | Crocoddyl: An efficient and versatile framework for multi-contact optimal control[END_REF] at each control step over a receding horizon. This approach couples two underlying sub-problems and solves them together: 1) the computation of an operational space trajectory towards the target and 2) the resolution of the operational to joint space mapping. Even though constraints are implemented in the cost function and cannot be formally guaranteed, this approach presents the advantage of considering joint and actuation constraints directly at the MPC problem level. As a consequence, except for the joint torque servoing performed at the actuation level, no inner control loop is required.

Yet, alternative approaches can be considered where an MPC is solved at the operational space level to incrementally generate a feasible task-space trajectory given some task-related constraints [START_REF] Faroni | An mpc framework for online motion planning in human-robot collaborative tasks[END_REF], [START_REF] Eckhoff | An MPC framework for planning safe and trustworthy robot motions[END_REF]. The output of the MPC is then fed to a joint level controller which can, for example, solve for the joint torques at each control instant using a constrained optimization approach [START_REF] Liu | Generalized hierarchical control[END_REF]. This approach does not allow to account for the joint level limits of the robot at the MPC level. However, it provides a modular architecture closing the loop on the current operational state of the robot, independently from the nature of the controller provided with a given robot. The later approach is retained in this work.

The formulation of both control levels, task-space MPC and constrained inverse dynamics, is described in Section II. Section III provides a description of the experiments performed with a 7 DoF manipulator to validate the pertinence of the proposed control approach. These results, exemplified in Fig. 1, illustrate several interesting characteristics of the approach ranging from: 1) constraints compliance; 2) the ability to manage instantaneously switching targets; 3) the ability to continuously adapt the trajectory towards a continuously moving target; 4) the effect of the modification of velocity constraints in the MPC on the generated motion horizon at each control step. An accompanying video offers a compact summary of all these results. Finally, Section IV discusses several aspects of the obtained results and potential improvements. Section V concludes the paper by summarizing the contributions and describing potential future research work on this topic.

II. Control Architecture

The control architecture drafted in Section I and summarily described on Figure 2 consists of two main components. The first one is a linear MPC, formulated at the operational space level. It generates an optimal trajectory in SE(3) over a receding horizon, given a reference target and constraints on the admissible motion, both provided on the fly. These constraints can be both related to the real joint space motion capabilities of the robot and to restrictions relative to the performed task. The second one computes the robot control input leading to the execution of the planned trajectory while respecting the joint-level constraints using an inverse dynamics solver. These two components are described hereafter. In all, the architecture enables the robot to re-plan a trajectory at the control frequency of the robot while accounting for pose constraints on the end-effector up to the jerk order.

A. MPC-based motion re-planning over a receding horizon

Assuming jerk to be a control input at the task level, the evolution of the state -pose, twist and acceleration -of the end-effector of a robot can be described using a discrete time linear system (indexed with n):

𝒙 𝑛+1 = 𝑨𝒙 𝑛 + 𝑩𝒖 𝑛 (1)
where 𝑨 and 𝑩 represent its state and input matrices whereas 𝒙 and 𝒖 are the state and input vectors. Discretizing time in h time steps of duration T starting at time 𝑡 𝑛 (total duration hT) yields the propagated state space equation:

𝑿 0...h =          𝒙 𝑘 |𝑛 𝒙 𝑘+1|𝑛 . . . 𝒙 𝑘+h|𝑛          𝑼 0...h-1 =          𝒖 𝑘 |𝑛 𝒖 𝑘+1|𝑛 . . . 𝒖 𝑘+h-1|𝑛          (2) 𝑿 = 𝑿 1...h 𝑿 = 𝑿 0...h-1 𝑼 = 𝑼 0...h-1
where the discretized time steps inside the horizon are defined as 𝑡 𝑘 = 𝑡 𝑛 + 𝑘T for 𝑘 = 0, 1, . . . , h. This allows extending the definition in (1) for the horizon:

𝑿 = 𝑨 ′ 𝑿 + 𝑩 ′ 𝑼 (3)
where 𝑨 ′ , 𝑩 ′ are matrices constructed from on 𝑨, 𝑩 via the recursive propagation of (1) over the time horizon. 𝑿 and 𝑼 respectively correspond to the future states and control inputs of the system. From these propagated linear dynamics, one can formulate a linear optimal control problem over a receding horizon as a Quadratic Program (QP) with the following cost function:

min 𝒖 𝑿 𝑡 -𝑿 (𝑼) 2 𝑷 + 𝑼 2 𝑹 (4)
subject to equality (3) and polyhedral constraints:

𝒙 𝑚 ≤ 𝑪 𝒙 𝒙 𝑘 ≤ 𝒙 𝑀 𝒖 𝑚 ≤ 𝑪 𝒖 𝒖 𝑘 ≤ 𝒖 𝑀 (5)
where 𝑿 𝑡 is a vector of the target state 𝒙 𝑡 over the horizon. This is an important detail and one of the original features of the proposed approach: no pre-interpolation or a priori trajectory generation is required. The goal at each discrete step of the horizon is to reach the final target with a null velocity and a greater importance is given to the last step of the horizon to favour convergence towards the target. The resulting motion is solely shaped by the considered constraints on the admissible task space motions. This implicitly yields a time optimal trajectory over the horizon. 𝑷, 𝑹 respectively represent symmetric positive definite weighting matrices for the input and state throughout the horizon. Finally, 𝑪 𝒙 , 𝑪 𝒖 allow formulating linear constraints with respect to the state and input, while 𝒙 𝑚 , 𝒙 𝑀 and 𝒖 𝑚 , 𝒖 𝑀 respectively designate the state and input bounds. This formulation at the jerk level can be almost equivalently reformulated at the acceleration level through the inclusion of a discretized constraint on the derivative of the acceleration. This is the choice made in this work and throughout the remainder of this paper. Equation (4) under constraints (3) and ( 5) constitutes the general linear MPC formulation as a QP. It progressively provides a trajectory to track for the end-effector as the optimization window evolves. The linearity of this MPC formulation is a mandatory feature for the corresponding optimization problem to be solved at each control time step. However, writing this problem in a linear fashion, given that the Cartesian pose of the end-effector X belongs in SE( 3) is far from simple. The next sub-section describes the proposed formulation.

B. Pose integration over a horizon

To understand how (1) can express the pose dynamics in linear form, consider the first-order integration of a pose in SE(3) subject to an infinitesimal body twist increment for a single time step [START_REF] Solà | A micro lie theory for state estimation in robotics[END_REF]:

X 𝑘+1 = X 𝑘 𝑒 𝝂 𝑘 T (6)
where 𝑒 𝝂 𝑘 𝑇 is the homogeneous matrix equivalent to the application of 𝝂 𝑘 for a duration of T. It is computed using the matrix exponential, that surjectively maps the Lie algebra 𝔰𝔢(3) onto the Lie group SE(3). Its inverse operation is known as the logarithmic map. This relation can be exploited to represent X in a vector form. In fact, given some initial pose X 0 , there exists a function 𝝍 (referred to as the lift function) that maps X 𝑘 to a vector 𝝃 𝑘 (further explained in [START_REF] Forster | On-manifold preintegration theory for fast and accurate visual-inertial navigation[END_REF] [20] [START_REF] Torres Alberto | A linearization method based on Lie algebra for pose estimation in a time horizon[END_REF]):

𝝍 :SE(3) → 𝔰𝔢(3) log(X -1 0 X 𝑘 ) → 𝝃 𝑘 𝝍 -1 :𝔰𝔢(3) → SE(3) X 0 𝑒 𝝃 𝑘 → X 𝑘 (7)
The lift function maps elements from the SE(3) manifold M to its tangent space at X 0 : 𝝍 : M → T X 0 M. Notice that the pose delta ΔX 𝑘 ∈ SE(3) (or equivalent homogeneous matrix) needed to move from X 0 to X 𝑘 can be computed with ΔX 𝑘 = 𝑒 𝝍 ( X 𝑘 ) . The function 𝝍 -1 is referred to as the retract function.

Using the lift and retract functions, the geodesic path between X 0 and X 𝑡 can be interpolated for any pose in between with a parameter 𝛼 ∈ [0, 1]:

X(𝛼) = 𝝍 -1 (𝛼 𝝍(X 𝑡 )) = X 0 𝑒 𝛼 𝝍 ( X 𝑡 ) (8)
It is straightforward to notice that X(0) = X 0 and X(1) = X 𝑡 .

This geodesic path is obtained by minimizing the Log-Euclidean ( [START_REF] Jayasumana | Kernel methods on the riemannian manifold of symmetric positive definite matrices[END_REF], Table 1) distance metric between the lifted pose 𝝃 𝑘 and the lifted target pose 𝝃 𝑡 = 𝝍 (X 𝑡 ):

𝑒 𝑘 = 𝝃 𝑡 -𝝃 𝑘 2 2 (9)
The target related cost in (4) corresponds to the discretized version of (9) over the MPC horizon. Hence, it leads to minimizing the geodesic distance towards the target pose over the horizon.

Furthermore, in order to link [START_REF] Chen | Handbook Of Human Factors And Ergonomics[END_REF] with (1) (with linear MPC in mind), it needs to be linearized. Computing 𝝍(X 𝑘+1 ) yields:

𝝍(X 𝑘+1 ) = log(ΔX 𝑘 𝑒 T𝝂 𝑘 ) (10)
In fact, this expression has a first-order approximation in its Lie algebra (see Equations 68-74 in [START_REF] Solà | A micro lie theory for state estimation in robotics[END_REF])

𝝃 𝑘+1 = log(𝑒 𝝃 𝑘 𝑒 T𝝂 𝑘 ) ≈ 𝝃 𝑘 + T.J log R (𝝃 𝑘 )𝝂 𝑘 (11)
where J log R (𝝃 𝑘 ) is the right-trivialized Jacobian of the logarithmic map, evaluated at 𝝃 𝑘 . A closed-form expression can be found in [START_REF] Barfoot | Associating uncertainty with threedimensional poses for use in estimation problems[END_REF] [20] and is implemented in [START_REF] Carpentier | The Pinocchio c++ library, a fast and flexible implementation of rigid body dynamics algorithms and their analytical derivatives[END_REF]. It relates the additive increments in T 0 M (the tangent map at the origin) to the rightmultiplied increments in SE(3).

Equation ( 11) offers the missing piece to have a concrete definition for (1) when considering acceleration 𝒂 𝑘 as the control input:

𝑨 𝑘 = I 6 T.J log R (𝝃 𝑘 ) 0 6 I 6 𝑩 𝑘 = T 2 2 J log R (𝝃 𝑘 ) T.I 6 (12)
for which the discretized state and input vectors at each 𝑘, and the target state 𝒙 𝑡 are:

𝒙 𝑘 = 𝝃 𝑇 𝑘 𝝂 𝑇 𝑘 𝑇 𝒙 𝑡 = 𝝃 𝑇 𝑡 0 𝑇 6 𝑇 𝒖 𝑘 = 𝒂 𝑘 (13)
With the MPC being formulated in 𝔰𝔢(3), the lift operator 𝝍 is used to map the input state and the desired pose into 𝔰𝔢(3) while the retract operator 𝝍 -1 is used to map the computed trajectory to SE(3). It is also important to note that the duration of a step T in the horizon is generally longer than the robot control/re-planning period and the output of the MPC is thus interpolated using [START_REF] Merckaert | Real-time motion control of robotic manipulators for safe human-robot coexistence[END_REF].

C. Constrained task space inverse dynamics solver

Given the interpolated first step of the trajectory computed by the MPC, task space inverse dynamics has to be performed to compute the input joint torque for the robot. Feeding the inverse dynamics solver with the desired acceleration computed by the MPC may sound tempting. Nevertheless, the feedback provided by the closed-loop MPC is, partly due to the interpolated output, not efficient enough to properly reject tracking errors related to the inaccuracies in the model of the robot. These inaccuracies are mostly related to dry friction at the joint level, imperfectly rejected by the lower-level torque control loop in most robots. As a consequence, a PD controller including a feed-forward term in acceleration is used to compute a corrected desired acceleration :

𝝂 * = 𝑲 𝑝 log(X -1 X mpc ) + 𝑲 𝑑 (𝝂 mpc -𝝂) + 𝝂 mpc ( 14
)
where X mpc , 𝝂 mpc , 𝝂 mpc are respectively the interpolated desired pose, twist and acceleration outputted by the MPC; X, 𝝂 are the measured pose and twist of the robot and 𝐾 𝑝 and 𝐾 𝑑 are positive proportional and derivative gains. Given this control acceleration, one can solve task-space inverse dynamics under constraints through a quadratic programming formulation, similar to the one in [START_REF] Joseph | Towards x-ray medical imaging with robots in the open: safety without compromising performances[END_REF]. The optimal joint torque 𝝉 opt is computed at each control step as:

𝝉 opt = arg min 𝝉 𝝂 * -𝝂(𝝉) 2 2 + 𝑤 reg 𝒇 reg (𝝉) 2 2 (15) s.t. 𝝉 𝑚 ≤ 𝝉 ≤ 𝝉 𝑀 , 𝒒 ∈ Q, 𝒒 ∈ Q (16) 𝝂(𝝉) = 𝑱(𝒒) 𝑴 (𝒒) -1 𝝉 + 𝒃(𝒒, 𝒒) (17) 𝒇 reg (𝝉) = 𝝉 * -𝝉 (18)
The cost function is designed to find the optimal torque 𝝉 opt that minimizes the difference between the acceleration resulting from the applied torque through the system's dynamics in [START_REF] Kleff | High-Frequency Nonlinear Model Predictive Control of a Manipulator[END_REF] and the control acceleration 𝝂 * computed to track the desired trajectory. This optimization accounts for joint level constraints on the minimum and maximum torque 𝝉 𝑚 and 𝝉 𝑀 in [START_REF] Del Prete | Joint position and velocity bounds in discrete-time acceleration/torque control of robot manipulators[END_REF]. It also considers the admissible joint space configuration and velocity Q and Q, expressed at the current robot state {𝒒, 𝒒} as a function of 𝝉 using a Taylor expansion. In [START_REF] Kleff | High-Frequency Nonlinear Model Predictive Control of a Manipulator[END_REF], 𝑱 and 𝑴 are respectively the task space Jacobian matrix associated to the end-effector and the joint space inertia matrix. Finally, 𝒃 is a bias acceleration term which also incorporates the effect of gravity and the Coriolis/Centrifugal forces acting on the system and expressed at the end-effector level.

A regularization term is added to the cost function with a small weight (𝑤 reg ≪ 1) to ensure the uniqueness of the computed solution in case of a redundant robot while minimally affecting the main task of the robot.Moreover, 𝝉 * can be chosen in different ways mostly depending on the targeted application. Here, 𝝉 * is chosen to ensure convergence of the robot configuration to the neutral configuration 𝒒 0 . The servoing of this configuration can be expressed at the joint torque level using a proportional-derivative controller such that: 𝝉 * = 𝑲 𝒒 (𝒒 0 -𝒒) -𝑲 𝒒 𝒒 with 𝑲 𝒒 and 𝑲 𝒒 positive proportional and derivative gains.

III. Experiment description and results

The objective of the experiment is to validate the online adaptation of the robot to arbitrary target pose modifications. A three phases experiment is proposed, as illustrated in Fig. 3 and detailed in the following Section.

A. Experiment description

The robot is first requested to pick-up a plastic brick placed in the workspace over an irregular surface (to induce non-trivial orientations). The pose of the brick is not known a priori and is detected via a vision system. Given this target pose information X 𝑡 , the robot starts its motion towards the brick using the control scheme described in Fig. 2 and in Section II. Once the object is reached, the robot grasps it and releases it after a few seconds. Then, the robot goes back to its neutral position and the brick is placed at a new pose. After a few seconds, the robot starts its second motion, towards this new goal. While in motion, the brick is displaced by a human and the robots adapts its trajectory online to reach it. In the third phase, the human grasps the brick, moves it randomly around and the robot tracks it in real time.

The set-up is illustrated on Fig. 3. Perception of the plastic brick is provided by an Optitrack camera array. The experiment is carried out on a torque-controlled 7-DoF Franka Emika Panda serial robot. The control loop described in Fig. 2 runs at 1kHz. In order to keep up with the control rate, the MPC uses a 750ms control horizon (h = 5, T = 150ms), which yields an average computation time of 0.1ms (max 0.2ms) using qpOASES [START_REF] Ferreau | qpoases: A parametric active-set algorithm for quadratic programming[END_REF]. The same QP solver is used to solve the constrained task space inverse dynamics. The values of gains and weights for this inner QP controller are summarized in Table I. They are not provided in details for the MPC controller for the sake of compactness. It is just recalled here that a larger cost is assigned to the terminal step of the horizon to favour convergence towards the target. Details can be found in the source code2. Details on the motion resulting from one run of this experiment are provided in Fig. 4 for the first of the three described phases for space reasons. The two other experiments can be observed in the video attached to this paper.

B. Analysis

Fig. 4a shows the error between the current and the target poses. The discontinuity seen at the beginning corresponds to the start of the experiment sequence, where the target pose switches from the "home pose" to the plastic brick pose. This highlights the convergence towards the requested pose without any a priori knowledge on the trajectory to follow. Note that this is different from the resulting tracking error of following the trajectory planned by the MPC with the constrained task-space inverse dynamics solver. This error, shown in Fig. 4b), is bounded to 19.7mm with a mean value of 4mm ± 1 for the linear component of the motion and 5.18 × 10 -2 rad with a mean value of 2.5 × 10 -3 rad ± 5.5 × 10 -3 for the angular component. The final error is not exactly zero (2.1mm and 10.4 × 10 -2 rad) due to the relatively low gains of the PD controller and the absence of an integral term to reject disturbances due to dry friction at the joint level. Yet, the robot is able to grasp the targeted object at the end of the motion as can be seen in the accompanying video.

Fig. 4c shows both the planned (by the MPC) and the actual velocities; as opposed to the acceleration and jerk curves (respectively in Fig. 4d and4e) that only show the planned trajectories. The first thing to notice is that the constraints on the planned trajectory are met by the MPC, as can be seen on the velocity 2https://gitlab.inria.fr/auctus-team/people/nicolas-torres/lmpcpoly-ws/ and jerk curves. It is also interesting to notice that the obtained linear velocity profile is similar, to some extent, to trapezoidal acceleration profiles, maximizing the use of some of the prescribed Cartesian limits shown in Table II. This confirms the near-optimal character of the generated trajectory. Finally, one can notice that the robot's end-effector velocity surpasses the established limits. This is discussed in Section IV.

IV. Discussions

The modular nature of the proposed architecture offers the potential to generically consider different types of task inputs. It can for example be easily adapted to use a discretized pose path, which could come from an obstacle avoidance solver. Hence, the same architecture can accommodate a complex pathing solution while retaining its core structure. This architecture can also accommodate different types of task space to joint space solvers. This is an important feature as not all robots allow for joint torque control.

Moreover, the cost function of the MPC and the Cartesian constraints offer a generic way to design human-aware solutions considering the non-trivial cognitive aspects of humanrobot collaboration. For example, representing the human motion capabilities as Cartesian constraints [START_REF] Skuric | On-line feasible wrench polytope evaluation based on human musculoskeletal models: an iterative convex hull method[END_REF]. This is further enabled by decoupling the task-related definition and constraints from the joint-level controller. Indeed, some constraints are intrinsically task related and somewhat independent of the robot performing the task while others are strongly related to the whole robot motion. It is for example the case for safety-aware constraints [START_REF] Joseph | Experimental validation of an energy constraint for a safer collaboration with robots[END_REF] or the exploitation of the robot redundancy to perform secondary tasks without compromising performance. Examples of the later case are shown for safety purposes in [START_REF]Online minimization of the projected mass of a robot for safe workspace sharing with a human[END_REF], [START_REF] Mansfeld | Improving the performance of biomechanically safe velocity control for redundant robots through reflected mass minimization[END_REF] and for situation awareness in [START_REF] Camblor | Taskconsistent signaling motions for improved understanding in humanrobot interaction and workspace sharing[END_REF].

Fig. 4c shows that the real end-effector's velocity loosely follows the planned velocity, going beyond the established limits in some cases. The measured velocity is being used as the initial state of the MPC at each time step, this causes the associated QP to become infeasible. This characteristic is intrinsic to any QPbased architecture and it can be considered both a problem and an advantage: for one, it allows easily detecting ill-conditioned initial states that could result in undefined behaviors (for example, to trigger an emergency stop mode); on the other hand, it demands that a proper strategy is put in-place to deal with this situation. For this article, the inner loop controller continues to follow the last horizon that was successfully computed until the MPC can compute a new one. This allows the execution to continue smoothly.

Nevertheless, this opens one fundamental question that naturally arises from this type of cascade control architectures: the compatibility of the cascading controllers. For instance, the MPC could plan a trajectory that respects the Cartesian constraints even if it is instantaneously unfeasible in a control step or may lead to an inevitable joint-level constraint violation in the near future [START_REF] Rubrecht | Constraints compliant control: constraints compatibility and the displaced configuration approach[END_REF]. To solve this problem, joint-level constraints [START_REF] Del Prete | Joint position and velocity bounds in discrete-time acceleration/torque control of robot manipulators[END_REF] should be implicitly considered at the MPC level. This opens up one potential line of research on how to efficiently formulate the actual robot capacities as Cartesian constraints [START_REF] Skuric | On-line force capability evaluation based on efficient polytope vertex search[END_REF]. This type of constraints goes beyond classical box-like bounds and is a step towards more complex adaptation of the robot behaviour based on its motion capabilities.

V. Conclusion

This paper proposes an efficient linear model predictive control approach that can deal with the online planning of SE(3) motion in task space. The main features of the proposed control approach lie in its ability to generate optimal Cartesian motion which dynamically accounts both for targets updated on-the-fly and evolving motion constraints. This receding horizon approach endows the robot with the ability to interactively adapt to task adaptation. In the context of collaborative robotics, using this controller, the safety of a human operator sharing its workspace with the robot could be accounted for through the sensor-based adaptation of maximum velocity constraints.

Future work will focus on extending linear constraints in the MPC for the Cartesian pose state, to limit the effective position and orientation space. This work also opens doors to potentially exploiting a more efficient expression of Cartesian constraints such as convex polytopes that consider the actual robot joint space capabilities in the receding horizon optimization stage. The reduction of the computation time required to solve the MPC problem is also an important matter. It will be pursued to refine the computation of the trajectory over the horizon and potentially improve the performance of the closed-loop. 𝝂 mpc (rad/s 3 ) (e) Fig. 4: Tracking results of driving the robot towards a target pose given on the fly. For each plot, the red, green and blue curves respectively correspond to the components in the x, y and z axis. For the velocity plots, the plain lines correspond to the actual robots velocity and the dotted lined to the computed by the mpc (𝝂 mpc ). The shaded areas represent the bounds given to the MPC.

Fig. 1 :

 1 Fig. 1: A linear Model Predictive Controller is used to compute a position and orientation trajectory towards a target (plastic brick) in a receding horizon. (a) and (b) highlight the effects of changing the task space constraints on the fly (in this case max velocities) on the covered distance for one horizon. (c) and (d) illustrate the continuous adaptation of the trajectory when the target location is modified online.

Fig. 2 :

 2 Fig. 2: The proposed modular control architecture is based on a cascade loop composed of: 1) a task-space Model Predictive Controller (MPC) for online re-planning; 2) a constrained Task Space Inverse Dynamics solver to follow the planned trajectory.

Fig. 3 :

 3 Fig. 3: The experimental setup is composed of a torquecontrolled 7-DoF Franka Emika Panda serial robot and an Optitrack camera array. Reflective markers are placed on a plastic brick. The camera array is used to detect the position and orientation of the plastic brick relatively to the robot base.

TABLE I :

 I Values of the gains and weights retained for the described experiments.

  𝝂 ≤ 𝝂 max , -𝝂 max ≤ 𝝂 ≤ 𝝂 max , -𝝂 max ≤ 𝝂 ≤ 𝝂 max

		𝝂 max	𝝂 max	𝝂 max
	linear	0.2 (m/s)	2 (m/s 2 )	1000 (m/s 3 )
	angular	0.8 (rad/s)	5 (rad/s 2 )	3000 (rad/s 3 )
	-𝝂 max ≤

TABLE II :

 II Cartesian constraints for 𝝂, 𝝂, 𝝂 in the MPC configuration during the experiment.