Antoine Zalc

Revital Rattenbach

Frédéric Auradé

Bruno Cadot

Frédéric Relaix
email: frederic.relaix@inserm.fr

Pax3 and Pax7 play essential safeguard functions against environmental stress-induced birth defects

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SUMMARY:

Exposure to environmental teratogenic pollutant leads to severe birth defects. However, the biological events underlying these developmental abnormalities remain undefined. Here we report a molecular link between an environmental stress response pathway and key developmental genes during craniofacial development. Strikingly, mutant mice with impaired Pax3/7 function display severe craniofacial defects. We show these are associated with an upregulation of the signaling pathway mediated by the Aryl hydrocarbon Receptor (AHhR), the receptor to 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD), revealing a genetic interaction between Pax3 and AhR AHR signaling. Activation of AhR AHR signaling in Pax3-deficient embryos drives facial mesenchymal cells out of the cell cycle through the up-regulation of p21 expression.

Accordingly, inhibiting AhR AHR activity rescues the cycling status of these cells and the facial closure of Pax3/7 mutants. Together, our findings demonstrate that the regulation of AhR AHR signaling by Pax3/7 is required to protect against TCDD/AhRAHR-mediated teratogenesis during craniofacial development.

INTRODUCTION:

Exposure to environmental teratogenic pollutants is a major threat to embryonic development. In humans, among the population presenting birth defects, about 10-15% of these anomalies are due to exposure of pregnant women to environmental teratogenic pollutants (Gilbert-Barness, 2010). Among these developmental defects, craniofacial malformations represent a third of the defects observed [START_REF] Dixon | Cleft lip and palate: understanding genetic and environmental influences[END_REF]. It is postulated that most teratogens interfere with genetic programs regulating developmental processes [START_REF] Dixon | Cleft lip and palate: understanding genetic and environmental influences[END_REF]. Among environmental pollutants, TCDD commonly known as dioxin is a recognized potent teratogen [START_REF] Yonemoto | The effects of dioxin on reproduction and development[END_REF]. During historical events such as the American Vietnam war or the Seveso incident, populations were exposed to high doses of TCDD [START_REF] Mocarelli | Seveso: a teaching story[END_REF][START_REF] Stellman | The extent and patterns of usage of Agent Orange and other herbicides in Vietnam[END_REF]. It was suggested that the spreading of TCDD led to an increase in congenital defects, however rigorous epidemiologic study establishing the link with TCDD exposure are lacking. Nevertheless, TCDD exposure has been associated with developmental alterations, including impaired psychomotor functions and cognition as well as reproductive and developmental defects [START_REF] White | An overview of the effects of dioxins and dioxin-like compounds on vertebrates, as documented in human and ecological epidemiology[END_REF]. Furthermore, mouse embryos exposed to TCDD present with hydronephrosis, thymic hypoplasia, craniofacial defects such as cleft palate, reduced weight and lethality at high doses [START_REF] Courtney | Teratology studies with 2,4,5-trichlorophenoxyacetic acid and 2,3,7,8tetrachlorodibenzo-p-dioxin[END_REF][START_REF] King-Heiden | Reproductive and developmental toxicity of dioxin in fish[END_REF][START_REF] Kransler | Comparative developmental toxicity of 2,3,7,8tetrachlorodibenzo-p-dioxin in the hamster, rat and guinea pig[END_REF][START_REF] Moore | Postnatal effects of maternal exposure to 2,3,7,8tetrachlorodibenzo-p-dioxin (TCDD)[END_REF][START_REF] Pratt | 2,3,7,8-Tetrachlorodibenzo-p-dioxin-induced cleft palate in the mouse: evidence for alterations in palatal shelf fusion[END_REF][START_REF] Yamada | Cleft lip and palate in mice treated with 2,3,7,8-tetrachlorodibenzo-p-dioxin: a morphological in vivo study[END_REF][START_REF] Yonemoto | The effects of dioxin on reproduction and development[END_REF].

AhR AHR has been identified as the receptor to TCDD [START_REF] Fernandez-Salguero | Aryl-hydrocarbon receptor-deficient mice are resistant to 2,3,7,8-tetrachlorodibenzo-p-dioxin-induced toxicity[END_REF][START_REF] Hankinson | The aryl hydrocarbon receptor complex[END_REF], suggesting this signaling pathway could be involved in TCDD-mediated teratogenesis. While the etiology of TCDD-induced developmental defects remains unclear and studies linking TCDD exposure and oral cleft in human are lacking, palatal shelves from human, rat and mouse display similar response to TCDD exposure in vitro (Abbott et al., 1999a), which suggests that human exposure to TCDD may be associated with oral clefts.

In vertebrates, the face mainly derives from the Neural Crest (NC), a transient structure that arises at the dorsal tip of the closing neural tube [START_REF] Knecht | Induction of the neural crest: a multigene process[END_REF]. Cells from the neuroepithelium undergo an epithelial-mesenchymal transition prior to migrating to various regions of the embryo. Depending on their location along the anterior-posterior axis, these cells populate different structures and give rise to a large variety of cell types [START_REF] Le Douarin | Neural crest cell plasticity and its limits[END_REF]. In the most rostral part of the embryo, Cranial Neural Crest Cells (CNCC) migrate ventrolaterally to colonize facial prominences [START_REF] Birgbauer | Rhombomeric origin and rostrocaudal reassortment of neural crest cells revealed by intravital microscopy[END_REF][START_REF] Lumsden | Segmental origin and migration of neural crest cells in the hindbrain region of the chick embryo[END_REF][START_REF] Osumi-Yamashita | The contribution of both forebrain and midbrain crest cells to the mesenchyme in the frontonasal mass of mouse embryos[END_REF][START_REF] Serbedzija | Vital dye analysis of cranial neural crest cell migration in the mouse embryo[END_REF] where they participate in the formation of craniofacial bones, cartilage, connective tissue, neurons and glial cells [START_REF] Couly | The triple origin of skull in higher vertebrates: a study in quail-chick chimeras[END_REF][START_REF] Dupin | Neural crest progenitors and stem cells: from early development to adulthood[END_REF][START_REF] Knecht | Induction of the neural crest: a multigene process[END_REF][START_REF] Kontges | Rhombencephalic neural crest segmentation is preserved throughout craniofacial ontogeny[END_REF][START_REF] Noden | Relations and interactions between cranial mesoderm and neural crest populations[END_REF]. Craniofacial malformations are generally linked to anomalies in CNCC development, part of them being due to teratogen exposure. Despite the identification of gene regulatory networks (GRNs) underpinning CNCC development [START_REF] Betancur | Assembling neural crest regulatory circuits into a gene regulatory network[END_REF], little is known about how environmental pollutants interfere with these genetic networks during craniofacial development.

Central to these GRNs are genes coding for the paralogous paired-box transcription factors Pax3 PAX3 and PaxAX7. These transcriptional regulators play a key role in the integration of inputs during NC induction and in controlling the specification of NC derivatives [START_REF] Basch | Specification of the neural crest occurs during gastrulation and requires Pax7[END_REF][START_REF] Betancur | Assembling neural crest regulatory circuits into a gene regulatory network[END_REF][START_REF] Minchin | Sequential actions of Pax3 and Pax7 drive xanthophore development in zebrafish neural crest[END_REF][START_REF] Monsoro-Burq | Msx1 and Pax3 cooperate to mediate FGF8 and WNT signals during Xenopus neural crest induction[END_REF][START_REF] Sato | Neural crest determination by co-activation of Pax3 and Zic1 genes in Xenopus ectoderm[END_REF].

Although Pax3 and Pax7 function during early NC development is highly conserved among vertebrates [START_REF] Betancur | Assembling neural crest regulatory circuits into a gene regulatory network[END_REF], their function during craniofacial formation is not understood.

Here we show that mutant mice with impaired Pax3 and Pax7 function display severe facial morphogenesis defects. Using a large-scale transcriptomic approach, we identified the AhR AHR signaling pathway as specifically up-regulated during craniofacial development of Pax3/7 deficient mice, leading to precocious growth arrest in the cranial neural crest lineage. We further show that blocking AhR AHR signaling rescues facial growth and closure in mice with impaired Pax3/7 function, while exposure to AhR AHR agonists such as TCDD leads to similar defects in embryos with reduced Pax3/7 activity, genetically linking Pax3/7 with AhRAhr. Our data therefore demonstrate that important developmental genes play a key function to preserve the developing face from AhRAHR/TCDD-mediated teratogenesis.

RESULTS:

Essential role of Pax3 and Pax7 during craniofacial development

Using mouse genetic models, we first examined the dynamics of Pax3 and Pax7 expression profiles during craniofacial development. As in other vertebrates [START_REF] Nelms | ))[END_REF], Pax3 expression is detected within the anterior neural plate border prior to neural tube closure at embryonic day (E)8.5 (Figure S1A). Pax3 PAX3 expression co-localizes with the NCC marker Sox9 SOX9 [START_REF] Nelms | ))[END_REF]; Figure S1B) and persists in several CNCC-derived tissues such as the frontonasal mass (FNM) and the medial and lateral nasal processes (MNP and LNP; Figure S1C). Of note, Pax7 expression is restricted within Pax3 domain; first observed in the FNM and the LNP, and later maintained in the LNP at E10.5 and E11.5 (Figure S1C and S1D).

We next analyzed facial morphology in two distinct models with altered Pax3 PAX3 and Pax7 PAX7 functions (Figure 1A and 1B). While Pax3 is required for NC induction in the sacral, trunk and vagal regions [START_REF] Li | Transgenic rescue of congenital heart disease and spina bifida in Splotch mice[END_REF][START_REF] Van Ho | Neural crest cell lineage restricts skeletal muscle progenitor cell differentiation through Neuregulin1-ErbB3 signaling[END_REF], cranial NC induction occurs in our genetic models. However, a strong frontonasal dysplasia phenotype with a fully penetrant frontal cleft was observed in both Pax3 GFP/GFP ; Pax7 LacZ/LacZ double mutant and Pax3 Pax3-ERD/GFP mutant embryos (Figures 1A,1B and S2A). In the latter, cells in the Pax3 positive lineage expressed a potent dominant negative form of Pax3 (Pax3-ERD) which alters both Pax3 PAX3 and Pax7 PAX7 function [START_REF] Bajard | A novel genetic hierarchy functions during hypaxial myogenesis: Pax3 directly activates Myf5 in muscle progenitor cells in the limb[END_REF]. This phenotype contrasted with the medial fusion of the nasal processes observed in wild-type embryos and other compound Pax3/Pax7 mutants (Figure S2B), supporting the notion that both proteins are required for facial development in mice. In Pax3; Pax7 double mutant embryos, the phenotype was more severe than in Pax3 Pax3-ERD/GFP embryos where the medial and lateral nasal swellings were rudimentary and failed to fuse resulting to a frontonasal cleft, whereas the maxillary and mandibular prominences were less affected (Figures 1A,1B and S2A). Histological analyses at E11.5 and E13.5 revealed that the nasal process of both genetic models was severely diminished and the nasal septum bifurcated (Figure 1A and 1B).

Since CNCC reached the facial prominences and early specification is not affected in our genetic models (Figures 1C and2A-C), we next assessed whether craniofacial defects observed in Pax3 Pax3-ERD/GFP and Pax3; Pax7 double mutant embryos were due to increased cell death of CNCC. Strikingly, we observed an increased proportion of apoptotic CNCC when both Pax3 and Pax7 were missing (Figure 2D). Furthermore, we were unable to detect any cell co-expressing GFP (Pax3PAX3) and -galactosidase (Pax7PAX7) at E10.5 or E11.5 (data not shown), suggesting that the PaxAX3 + /Pax7PAX7 + cell population is lost in double mutant embryos, precluding further analysis in this genetic model.

In contrast, cell death was not increased in the CNCC of Pax3 Pax3-ERD/GFP embryos (Figure 2D). Thus, we tested whether the frontal cleft face phenotype observed in Pax3 Pax3-ERD/GFP could be a consequence of tissue misspecification. We observed impaired expression of Msx1 in the MNP and LNP (Figure 2E) associated with reduced expression of Pax7 and Pax9 in the LNP (Figure S3A), suggesting a proliferation defect in the nasal processes of Pax3 Pax3-ERD/GFP embryos [START_REF] Bhatt | Signals and switches in Mammalian neural crest cell differentiation[END_REF][START_REF] Houzelstein | Insertional mutation of the mouse Msx1 homeobox gene by an nlacZ reporter gene[END_REF][START_REF] Nelms | ))[END_REF]. In addition, Dlx2 expression in nasal pits was barely detectable (Figure 2E) implying patterning defects of the nasal process in Pax3 Pax3-ERD/GFP embryos (McKeown et al., 2005; Thomas et al., 2000). As confirmed by skeletal staining at E17.5, Pax3 Pax3-ERD/GFP fetuses primarily presented with defects in the premaxilla and the palate bones (Figure S3B). The primary palate was almost completely absent, while the secondary palate was formed but divided in two halves in Pax3 Pax3-ERD/GFP embryos (Figures 2F andS3B).

Collectively, these data demonstrate that Pax3 and Pax7 are essential in regulating morphogenesis, survival, patterning and specification of the frontonasal structures during facial development.

Up-regulation of AhR AHR signaling pathway expression is associated with impaired Pax3/7 function

In order to gain insight into the molecular mechanism by which Pax3 and Pax7 regulate craniofacial development, we performed a microarray screen to compare the transcriptomes of the facial prominences of Pax3 Pax3-ERD/GFP mutant and Pax3 GFP/+ embryos at E11.5 (Figure 3A).

We identified 76 up-regulated (Table S1) and 44 down-regulated (Table S2) genes in cells with impaired Pax3/7 function compared to the control (Figure 3A). Among the deregulated genes, the most highly represented were components of the AhR AHR signaling pathway that were upregulated in our screen, suggesting a strong activation of this signaling pathway in Pax3 Pax3- ERD/GFP embryos.

Transcripts levels for AhR Ahr and its direct target genes, including Aldh1a3, a gene coding for an enzyme involved in retinoic acid production and nasal process development [START_REF] Dupe | A newborn lethal defect due to inactivation of retinaldehyde dehydrogenase type 3 is prevented by maternal retinoic acid treatment[END_REF][START_REF] Hankinson | The aryl hydrocarbon receptor complex[END_REF] and p21 (Cdkn1a, cyclin-dependent kinase inhibitor 1A), a mediator of cell cycle exit and growth arrest [START_REF] Barnes-Ellerbe | 2,3,7,8-Tetrachlorodibenzo-p-dioxin blocks androgendependent cell proliferation of LNCaP cells through modulation of pRB phosphorylation[END_REF][START_REF] Besson | CDK inhibitors: cell cycle regulators and beyond[END_REF], were higher in Pax3 Pax3-ERD/GFP CNCC (Figure 3B). Consistently, we detected a specific up-regulation of AhR AHR protein in the nasal process mesenchyme and epithelium in both E10.5 and E11.5 Pax3 Pax3- ERD/GFP embryos compare to control (Figures 3C and S4). In situ hybridization for Aldh1a3 transcripts revealed that its expression in the nasal process was shifted to more anterior and

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Police :Italique Mis en forme : Police :Italique medial positions within this tissue (Figure 3D), reinforcing the notion that AhR AHR signaling activity domain is increased when the function of PaxAX3/7 is impaired. Furthermore, in the frontonasal ectoderm, Aldh1a3 was previously shown to regulate Fgf8 expression, a mediator of CNCC growth in the underlying mesenchyme [START_REF] Dupe | A newborn lethal defect due to inactivation of retinaldehyde dehydrogenase type 3 is prevented by maternal retinoic acid treatment[END_REF][START_REF] Hu | A zone of frontonasal ectoderm regulates patterning and growth in the face[END_REF]. Consistent with this, expression of Fgf8 was lost in the nasal epithelium of Pax3 Pax3-ERD/GFP embryos (Figure 3D), suggesting a decreased proliferation of the underlying mesenchymal CNCC that could explain the smaller nasal prominence phenotype. In addition, CNCC isolated from facial prominences exposed to TCDD, specifically induced Aldh1a3 and p21 expression (Figure 3E) demonstrating the responsiveness of AhR AHR signaling to environmental pollutants in this tissue.

Impaired Pax3/7 function leads to CNCC growth arrest and frontal cleft face

Migration of the CNCC takes place in Pax3 Pax3-ERD/GFP embryos (Figure 1C), since the number of GFP + cells was essentially the same in Pax3 Pax3-ERD/GFP and control embryos at E9.5 (Figure 4A). However from E10.5 onwards the number of GFP + cells within the frontonasal region was decreased by 30% in Pax3 Pax3-ERD/GFP embryos compared to control (Figure 4A). Moreover, by E11.5 the proportion of GFP + FACS-sorted cells over the total number of cells was severely diminished by 34%, in Pax3 Pax3-ERD/GFP embryos compared to the control (Figure 4B and4C).

This decrease in the Pax3PAX3 + /Pax7PAX7 + population correlated with a decline in the number of cycling cells in this population (Figure 4D and4F) which explains the overall reduction of nasal process size, leading to a frontal cleft face.

As we found a marked up-regulation of the growth arrest gene p21 in Pax3 Pax3-ERD/GFP embryos, we hypothesized that p21 induction might lead to growth failure of the nasal process, resulting in the frontal cleft face phenotype. Immunostaining for p21 and Pax7 PAX7 demonstrated that 43% of the cells had exited the cell cycle in Pax3 Pax3-ERD/GFP embryos, while only 26% of the cells normally did so at E10.5 (Figure 4E and 4G). This increased cell cycle exit was also seen at E11.5 (Figure 4G). In addition, this growth arrest was associated with a decrease in the CNCC progenitor pool, as shown by analysis of AP2 and Sox9 SOX9 expression (Figure 4H and4I).

Together these results suggest that during facial prominences growth, up-regulation of AhR AHR signaling pathway in CNCC drives these cells out of the cell cycle. Hence, this precocious cell cycle exit generates reduced craniofacial prominences unable to fuse together, resulting in the formation of a frontal cleft face.

Pax3/7-mediated regulation of AhR AHR signaling allows CNCC growth during craniofacial development

In order to demonstrate that up-regulation of AhR AHR signaling leads to the morphological defects observed in Pax3 Pax3-ERD/GFP embryos, we treated control and mutant embryos with naphthoflavone, an AhR AHR antagonist [START_REF] Jang | Antiteratogenic effects of alpha-naphthoflavone on 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) exposed mice in utero[END_REF]. Treatments were performed by daily gavages administrated from E8.5 to E11.5. Strikingly, a considerable proportion (39%) of treated Pax3 Pax3-ERD/GFP embryos exhibited a rescue of the frontal cleft face at E13.5 compared to the DMSO-treated Pax3 Pax3-ERD/GFP embryos, which all display a frontal cleft face (Figure 5A and 5C). Histological analysis of the rescued embryos revealed that the nasal septum was fused in its medial part (Figure 5A). In addition, we observed that AhR AHR inhibition was sufficient to rescue primary palate formation (Figure 5B). Moreover, as the alignment of the prominent rugae suggested, the fusion of the secondary palate appeared to be rescued but delayed in time when compared with DMSO-treated Pax3 Pax3-ERD/GFP embryos which present with a bifurcated secondary palate (Figure 5B).

Following inhibition of AhR AHR signaling with -naphthoflavone, the number of cycling Pax3PAX3 + /Pax7PAX7 + cells in both control and mutant embryos reached a similar level as in the DMSO-treated Pax3 GFP/+ control embryos (Figure 5D). Hence, inhibition of AhR AHR signaling in Pax3 Pax3-ERD/GFP embryos was sufficient to rescue the proportion of proliferating cells in the Pax3PAX3 + /Pax7PAX7 + population.

To demonstrate that the rescue of the number of cycling CNCC was due to the inhibition of AhR AHR signaling, we quantified the expression of p21 in these embryos. As anticipated, in nearly half of -naphthoflavone-treated Pax3 Pax3-ERD/GFP embryos the expression level of p21 within the facial prominences reached similar levels compared to that observed in DMSO-treated Pax3 GFP/+ control embryos (Figure 5E), demonstrating that the specific inhibition of AhR AHR signaling was sufficient to rescue the number of cycling CNCC in the facial prominences of these embryos.

We also tested -naphthoflavone, an AhR AHR agonist [START_REF] Swanson | Half-life of aryl hydrocarbon receptor in Hepa 1 cells: evidence for liganddependent alterations in cytosolic receptor levels[END_REF]. Strikingly, nearly all -naphthoflavone-treated Pax3 GFP/GFP or Pax3 Pax3-ERD/GFP mutant embryos presented with an early lethality by E13.5. By contrast, WT and Pax3 GFP/+ embryos all developed normally (data not shown). The cause of death was not determined but this further demonstrates a strong genetic link between Pax3/Pax7 and AhR Ahr signaling during development.

Altogether, our results demonstrate that during facial morphogenesis PaxAX3 and Pax7 PAX7 act by regulating CNCC growth through the action of AhR AHR signaling.

Pax3 and Pax7 play a safeguard function against TCDD-induced craniofacial defects.

To further establish the link between Pax3/7 function and AhR AHR signaling during craniofacial development, we speculated that they act by restricting the input of AhR AHR signaling to specific locations in the facial prominences. We therefore hypothesized that reducing Pax3PAX3/7 activity in embryos exposed to TCDD should lead to facial defects.

Accordingly, we used TCDD to stimulate AhR AHR signaling in a Pax3-null context. To this end, we intercrossed Pax3 GFP/+ mice and treated the pregnant females with a low dose of TCDD.

As expected, p21 expression was significantly up-regulated in both TCDD-treated Pax3 GFP/+ and Pax3 GFP/GFP facial prominences (Figure 6A) confirming that AhR AHR signaling was activated.

As expected, WT and Pax3 GFP/+ control embryos were normally formed. Yet, while all vehicletreated control Pax3 GFP/GFP mutant embryos displayed a normal craniofacial phenotype, 56% of the TCDD-treated Pax3 GFP/GFP mutant embryos presented with a frontal cleft face phenotype (Figure 6B and 6C). This phenotype is reminiscent to the one observed in Pax3 Pax3-ERD/GFP embryos (Figure 1B) and confirms that when Pax3 PAX3 function is impaired, activating AhR AHR signaling is sufficient to generate craniofacial defects.

Collectively, our results reveal an unexpected safeguard function for Pax3PAX3/7 during face morphogenesis whereby they restrict AhR AHR signaling input in order to allow the correct growth and maintenance of CNCC during craniofacial development.

DISCUSSION:

TCDD, and derived compounds such as Polychlorodibenzo-p-dioxines (PCDDs) and Polychlorodibenzofurane (PCDFs), are highly toxic persistent chemicals released into the environment as unintentional by-products of incomplete combustion of fossil fuels and wood, and during the incineration of municipal and industrial wastes. In humans, most of the exposure occurs through food, mainly meat and dairy products, fish and shellfish. The developing fetus is especially sensitive to TCDD exposure. It is therefore important to monitor the in vivo cellular response to TCDD during development in order to understand and prevent the molecular and cellular response following exposure to dioxins.

We show here that during morphogenesis of the face, Pax3 PAX3 and Pax7 PAX7 regulate the environmental stress response pathway mediated by AhR AHR signaling. Restriction of AhR AHR signaling by Pax3 PAX3 and Pax7 PAX7 highlights a key function for this pathway in controlling CNCC proliferation during craniofacial development.

Interestingly, inhibition of AhR AHR signaling only rescues the craniofacial phenotype of Pax3 Pax3-ERD/GFP mutant embryos. Severe defects in peripheral nervous system (PNS) and muscle development are maintained in the trunk of these embryos [START_REF] Bajard | A novel genetic hierarchy functions during hypaxial myogenesis: Pax3 directly activates Myf5 in muscle progenitor cells in the limb[END_REF][START_REF] Van Ho | Neural crest cell lineage restricts skeletal muscle progenitor cell differentiation through Neuregulin1-ErbB3 signaling[END_REF]. This suggests that fundamental divergences exist between the GRNs controlling cranial and trunk neural crest development, possibly due to the late appearance of the head during evolution. Moreover, it points out that the regulation of AhR AHR signaling by Pax3 may be a cranial neural crest-specific feature. In the trunk, Pax3 has been shown to control migration and early specification of NCC generating the PNS [START_REF] Tremblay | A transgenic neuroanatomical marker identifies cranial neural crest deficiencies associated with the Pax3 mutant Splotch[END_REF][START_REF] Van Ho | Neural crest cell lineage restricts skeletal muscle progenitor cell differentiation through Neuregulin1-ErbB3 signaling[END_REF]. In the head, our data demonstrate that Pax3 is mainly involved in regulating the cell cycle progression of CNCC via the regulation of AhR AHR signaling. This probably reflects the wider range of derivatives that CNCC normally generate compare to trunk NCC. Our study has been mainly focusing on mesenchymal cells however, analysis of cranial PNS development in Pax3 Pax3-ERD/GFP mutant embryos would further confirm or refute the divergence between trunk and cranial GRNs governing NCC development.

In addition, the study of our mouse mutants for Pax3 and Pax7 revealed that one copy of either of these two genes is sufficient to allow correct craniofacial development, demonstrating a strong functional conservation (Figure S2B). Interestingly, in Pax3 Pax7/Pax7 embryos, migration of neural crest cells is not affected [START_REF] Relaix | Divergent functions of murine Pax3 and Pax7 in limb muscle development[END_REF]. This can suggest that Pax3 and Pax7 functions are notably redundant during CNCC development or, alternatively it can also reflect that GRNs controlling formation of the face are more robust than the ones controlling muscle development.

The latter hypothesis seems more plausible when looking at the complexity of the GRNs controlling each different steps of CNCC development [START_REF] Betancur | Assembling neural crest regulatory circuits into a gene regulatory network[END_REF], with several transcription factors being described to be binding to the same target gene to ensure its correct expression during the different processes of craniofacial development.

In chick and Xenopus, Pax3 and Pax7 have been described as essential factors for the induction and formation of the neural crest [START_REF] Basch | Specification of the neural crest occurs during gastrulation and requires Pax7[END_REF][START_REF] Monsoro-Burq | Msx1 and Pax3 cooperate to mediate FGF8 and WNT signals during Xenopus neural crest induction[END_REF][START_REF] Sato | Neural crest determination by co-activation of Pax3 and Zic1 genes in Xenopus ectoderm[END_REF]. However, in our mouse genetic models, despite the absence of Pax3/7 or when their function is impaired, we observed normal NC induction and migration in the facial prominences of the embryo (Figure 1C). Whether this observation reflects a change in Pax3/7 function during evolution and how they became associated with AhR AHR latter during development remains an open question. Our data reveal that unlike chick and Xenopus, mouse Pax3/7 are not essential during early NC development but are involved in late craniofacial development to control the maintenance of a cycling CNCC population. It is plausible that the association with AhR signaling, which only starts to be significantly expressed from E10.5 [START_REF] Abbott | Developmental expression of two members of a new class of transcription factors: I. Expression of aryl hydrocarbon receptor in the C57BL/6N mouse embryo[END_REF][START_REF] Jain | Expression of ARNT, ARNT2, HIF1 alpha, HIF2 alpha and Ah receptor mRNAs in the developing mouse[END_REF], was co-opted during evolution as a mechanism to control cell cycle of CNCC derivatives. It is believed that AhR AHR protein evolved about 450 million years ago [START_REF] Hankinson | The aryl hydrocarbon receptor complex[END_REF]. Despite pre-existing natural sources of TCDD, a marked increase in environmental TCDD release occurred in the 19 th century and still persist today due to anthropogenic causes [START_REF] Weber | Dioxin--contemporary and future challenges of historical legacies. Dedicated to Prof. Dr. Otto Hutzinger[END_REF]. Thus, it is tempting to speculate that AhR AHR signaling has acquired different functions during evolution. The ability of AhR AHR to bind TCDD has made it a crucial regulator of pollution-induced teratogenesis.

In the facial prominences of the developing mouse embryo, AhrR expression is detected from E10.5 [START_REF] Abbott | Developmental expression of two members of a new class of transcription factors: I. Expression of aryl hydrocarbon receptor in the C57BL/6N mouse embryo[END_REF][START_REF] Jain | Expression of ARNT, ARNT2, HIF1 alpha, HIF2 alpha and Ah receptor mRNAs in the developing mouse[END_REF], suggesting a potential role during early craniofacial development. Yet, AhRAhr-null mouse mutants that are non-sensitive to TCDD exposure [START_REF] Fernandez-Salguero | Aryl-hydrocarbon receptor-deficient mice are resistant to 2,3,7,8-tetrachlorodibenzo-p-dioxin-induced toxicity[END_REF][START_REF] Mimura | Loss of teratogenic response to 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) in mice lacking the Ah (dioxin) receptor[END_REF], do not show craniofacial defects but present defects in liver and immune system development [START_REF] Fernandez-Salguero | Immune system impairment and hepatic fibrosis in mice lacking the dioxinbinding Ah receptor[END_REF]. Indeed AhR is involved in the maintenance of hematopoeitic stem cells quiescence [START_REF] Singh | Aryl hydrocarbon receptor-null allele mice have hematopoietic stem/progenitor cells with abnormal characteristics and functions[END_REF]. Studies demonstrating the teratogenic effect of TCDD on mouse palate development [START_REF] Abbott | Interactive regulation of Ah and glucocorticoid receptors in the synergistic induction of cleft palate by 2,3,7,8-tetrachlorodibenzo-p-dioxin and hydrocortisone[END_REF][START_REF] Abbott | AH receptor, ARNT, glucocorticoid receptor, EGF receptor, EGF, TGF alpha, TGF beta 1, TGF beta 2, and TGF beta 3 expression in human embryonic palate, and effects of 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD)[END_REF]Abbott et al., 1999b;[START_REF] Courtney | Teratology studies with 2,4,5-trichlorophenoxyacetic acid and 2,3,7,8tetrachlorodibenzo-p-dioxin[END_REF][START_REF] Moore | Postnatal effects of maternal exposure to 2,3,7,8tetrachlorodibenzo-p-dioxin (TCDD)[END_REF][START_REF] Pratt | 2,3,7,8-Tetrachlorodibenzo-p-dioxin-induced cleft palate in the mouse: evidence for alterations in palatal shelf fusion[END_REF][START_REF] Yamada | Cleft lip and palate in mice treated with 2,3,7,8-tetrachlorodibenzo-p-dioxin: a morphological in vivo study[END_REF][START_REF] Yonemoto | The effects of dioxin on reproduction and development[END_REF] are consistent with our observation that AhR AHR signaling up-regulation is associated with the occurrence of craniofacial defects. During normal development, it was shown that AhR AHR is involved during secondary palate development and is controlled by the retinoic acid (RA) signaling pathway [START_REF] Jacobs | Retinoic acid drives aryl hydrocarbon receptor expression and is instrumental to dioxin-induced toxicity during palate development[END_REF]. Mutant mice missing components of the RA signaling downregulate AhRr expression and do not present with secondary palate clefts when exposed to TCDD [START_REF] Jacobs | Retinoic acid drives aryl hydrocarbon receptor expression and is instrumental to dioxin-induced toxicity during palate development[END_REF]. Additional studies performed in zebrafish reported that TCDD-

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Police :Italique induced craniofacial defects are mediated by AhR signaling. However, the defects are uniquely found in the lower jaw, affecting the structure of the mandible and the development of the Meckel's cartilage [START_REF] Keller | Genetic differences in sensitivity to alterations of mandible structure caused by the teratogen 2,3,7,8-tetrachlorodibenzo-p-dioxin[END_REF][START_REF] Planchart | 2,3,7,8-Tetrachlorodibenzo-p-dioxin upregulates FoxQ1b in zebrafish jaw primordium[END_REF] highlighting possible evolutionary divergences between zebrafish and mammals.

TCDD-mediated activation of AhR AHR signaling in a Pax3-null context is sufficient to trigger craniofacial defects, confirming a tight regulatory interaction between Pax3PAX3/7 function and AhR AHR signaling to control growth of facial prominences in mouse. Accordingly, it was shown that AhR AHR directly binds the first intron of p21 in vitro [START_REF] Barnes-Ellerbe | 2,3,7,8-Tetrachlorodibenzo-p-dioxin blocks androgendependent cell proliferation of LNCaP cells through modulation of pRB phosphorylation[END_REF][START_REF] Dere | Integration of genome-wide computation DRE search, AhR ChIP-chip and gene expression analyses of TCDD-elicited responses in the mouse liver[END_REF][START_REF] Lo | High-resolution genome-wide mapping of AHR and ARNT binding sites by ChIP-Seq[END_REF]. Importantly, despite p21 expression being upregulated, all TCDD-treated Pax3 GFP/+ embryos developed normally, suggesting that additional GRNs downstream of Pax3 are also regulating craniofacial development. This will be investigated in future studies.

EXPERIMENTAL PROCEDURES:

Mutant mice for Pax3 and Pax7 and reagents

The mutant alleles for Pax3 (Pax3 nLacZ , Pax3 GFP and Pax3 Pax3-ERD ) and for Pax7 (Pax7 LacZ ) have been described previously [START_REF] Bajard | A novel genetic hierarchy functions during hypaxial myogenesis: Pax3 directly activates Myf5 in muscle progenitor cells in the limb[END_REF][START_REF] Mansouri | Dysgenesis of cephalic neural crest derivatives in Pax7-/-mutant mice[END_REF][START_REF] Relaix | The transcriptional activator PAX3-FKHR rescues the defects of Pax3 mutant mice but induces a myogenic gain-offunction phenotype with ligand-independent activation of Met signaling in vivo[END_REF][START_REF] Relaix | A Pax3/Pax7-dependent population of skeletal muscle progenitor cells[END_REF]. Of note, the Pax3 Pax3-ERD allele is a conditional one that drives the expression of the dominant negative form of Pax3 (Pax3-ERD) composed of the Pax3 DNA binding domain fused to the engrailed repressor domain (ERD) upon activation of a Cre recombinase. In this study, the Cre was driven by the zygote specific PGK enhancer [START_REF] Lallemand | Maternally expressed PGK-Cre transgene as a tool for early and uniform activation of the Cre site-specific recombinase[END_REF]. -naphthoflavone and -naphthoflavone (Sigma-Aldrich) were dissolved in DMSO then diluted in 1% carboxymethylcellulose (Sigma-Aldrich). -naphthoflavone (7.5mg/kg/day) and naphthoflavone (30mg/kg/day) were then administered daily, orally to pregnant female mice from E8.5 until E11.5. TCDD dissolved in toluene (Sigma-Aldrich) was diluted in Corn Oil (Sigma-Aldrich). TCDD (4g/kg/day) was then injected intra-peritoneally, to pregnant female mice from E8.5 until E11.5. All experiments were conducted according to the Institutional Animal Care and Use Committee of Paris 6 University.

Immunofluorescence and quantification

Mouse embryos from timed pregnant females were fixed by immersion in 4% paraformaldehyde/PBS for 45 min to 2 hours at 4°C. Fixed embryos were cryoprotected by equilibration in 30% sucrose/PBS, cryosectioned and processed for immunostaining as described in [START_REF] Relaix | A Pax3/Pax7-dependent population of skeletal muscle progenitor cells[END_REF]. The following primary antibodies were used: mouse anti-AhR AHR (Abcam, 1:100), rabbit anti-AP2 (Santa-Cruz, 1:200), rabbit anti-Cleaved-CaspaseCASPASE--3 (Cell Signaling, 1:100), mouse anti-Pax7 PAX7 (Developmental Studies Hybridoma Bank, 1:100), rabbit anti-Phospho-HistoneHISTONE-3 (Millipore, 1:500), rabbit anti-p21 (ProteinTech, 1:50), and goat anti-Sox9 SOX9 (R&D Systems, 1:100).

Histology, X-Gal and Dapi staining, skeletal preparation and mRNA in situ hybridization

For histology, sections of embryos prepared as for immunofluorescence were stained with Hematoxylin and Eosin [START_REF] Vandenberg | Non-canonical Wnt signaling regulates cell polarity in female reproductive tract development via van gogh-like 2[END_REF]. X-Gal staining and whole mount in situ hybridization were performed as previously reported [START_REF] Van Ho | Neural crest cell lineage restricts skeletal muscle progenitor cell differentiation through Neuregulin1-ErbB3 signaling[END_REF]. For whole mount Dapi staining, protocol described by [START_REF] Sandell | Whole mount nuclear fluorescent imaging: convenient documentation of embryo morphology[END_REF] was used. Briefly, E15.5 embryos where fixed for 2 hours in 4% paraformaldehyde/PBS and washed 3 times with PBS before incubation with 6g/ml Dapi in PBS overnight at 4°C. After 3 washes with PBS, embryos were placed under a fully automatized Nikon Ti microscope and pictures were taken every 3m in the z-axis to cover all the embryo thickness. The maximum projection of all the z-axis images using Fiji software revealed the fine topological details of the embryos. Staining of bones and cartilages of whole E17.5 embryos was performed as described in [START_REF] Depew | Specification of jaw subdivisions by Dlx genes[END_REF]. For in situ hybridization, probes against the following mRNAs were used: Dlx2 (Genbank Acc: BC094317), Fgf8 (kindly provided by Martin G.), Msx1 (Genbank Acc: BC016426), Pax9 (Genbank Acc: BC005794). Analysis was carried out using a Leica MZ16 F stereomicroscope.

Images were processed with Adobe Photoshop CS4 software (Adobe Systems). Analyses were performed on n≥3 embryos.

FACS sorting, cell culture and TCDD treatment

For FACS-sorting, CNCC were isolated from faces of E11.5 embryos initially incubated in digestion buffer [DMEM (LifeTechnologies), 0.1% Trypsin and 0.1% Collagenase D (Roche)] and purified via FACS Aria II based on gating of the GFP signal.

For TCDD treatment, CNCC were isolated from faces of E10.5 embryos, incubated in digestion buffer and purified using cell strainers (100m then 40m, BD Falcon) to obtain a single cell preparation. CNCC were then cultured in DMEM/F12 (1:1) medium for 24h with TCDD (100nM) or carrier in a collagen-plated dish.

RNA extraction, Microarray and RT-qPCR

RNA was extracted directly from dissected facial prominences or from FACS-sorted GFP + cells from facial prominences using NucleoSpin RNA II Extract kit (Macherey-Nagel) and the quality assessed with a Nanodrop ND-1000 (Thermo Scientific). cDNA was synthesized using the Transcriptor First Strand cDNA Synthesis kit (Roche).

Microarrays were performed using Affimetrix GeneChip MOE130 2.0 (PartnerChip) chips containing 45000 oligonucleotide probes (25 mers) covering the totality of the 30000 genes of the mouse genome. Briefly, two-cycle cDNA synthesis was performed using 100ng of total RNA. cDNA was then hybridized to GeneChip Mouse Genome Array. Microarray analysis was performed using GeneChip Operating Software 1.4. For statistical analysis, data from three biological replicates of each genotype were averaged then normalized using the Affymetrix Mas5.0 algorithm. Statistical analysis was performed using BioConductor software (http://www.bioconductor.org/). Gene expression comparison between Pax3 Pax3-ERD/GFP and Pax3 GFP/+ samples was performed using a statistical Student's t-test on normalized data.

Microarray data are available in the ArrayExpress database (www.ebi.ac.uk/arrayexpress) under accession number E-MTAB-3238.

RT-qPCR reactions were carried out in triplicate using the LightCycler 480 System (Roche). The expression of each gene was normalized to that of Gapdh transcripts. Results are given as mean ± standard deviation. The single (*), double (**) and triple (***) asterisks represent the P-values P<0.05, P<0.005 and P<0.0001 respectively for Student's unpaired t-tests. In Fig. 4C,naphthoflavone treated Pax3 Pax3-ERD/GFP embryos can be statistically segregated into two populations using a Ki 2 test. We compared the expression value for a given gene obtained in each -naphthoflavone treated Pax3 Pax3-ERD/GFP embryo to the mean of the expression values for all the DMSO treated Pax3 GFP/+ embryos (control set). If there were significant differences between these two values, the embryo was classified into the population described as responsive to the naphthoflavone treatment, if not the embryo was associated with the non-responsive population.

The following oligonucleotides were used:

  Altogether, our data show that during craniofacial development, Pax3 and Pax7 regulate AhR AHR signaling expression and activity. Restriction of AhR AHR signaling by Pax3 and Pax7 highlights a previously unknown key function for this pathway in controlling CNCC proliferation during craniofacial development.

  Secondary antibodies DyLight 649 donkey anti-mouse IgG (H+L), DyLight 649 donkey anti-rabbit IgG (H+L) and DyLight 649 goat anti-rabbit IgG (H+L) were purchased from Jackson Immuno Research and Alexa 594 goat anti-mouse IgG (H+L), Alexa 594 donkey anti-goat IgG (H+L) and Alexa 594 goat anti-rabbit IgG (H+L) from LifeTechnologies. Analysis was carried out using a Leica TCS SPE confocal microscope and images processed with Adobe Photoshop CS4 software (Adobe Systems). Cells were counted using ImageJ (version 1.46; National Institutes of Health, Bethesda, Maryland, USA) and Cell Counter plugin (Kurt De Vos, University of Sheffield, Academic Neurology) and were used to calculate the percentage of one cell population against another. Mean ± standard deviation was given. The single (*), double (**) and triple (***) asterisks represent P-values P<0.05, P<0.005 and P<0.0001 respectively by the Mann-Whitney non-parametric statistical test. All experiments have been performed on at least 3 independent embryos for each condition.

Figure 1 .

 1 FIGURE LEGENDS:

Figure 2 .Figure 3 .

 23 Figure 2. Pax3 Pax3-ERD/GFP embryos present a fully penetrant frontal cleft face phenotype. (A) Immunostaining showing the expression of AP2 in red, Pax7 PAX7 in white and GFP in green on transverse sections within the cranial regions of Pax3 GFP/+ and Pax3 Pax3-ERD/GFP embryos at E9.5. Scale bar, 50m. (B-C) Quantification of the proportion of AP2 + (B) and Sox9SOX9 + (C) cells within the Pax7PAX7 + population found on transverse sections through the cranial regions of E9.5 Pax GFP/+ and Pax3 Pax3-ERD/GFP embryos. Error bars represent the standard deviation. (D) Percentage of Cleaved-Caspase3CASPASE-3 + cells in the Pax3PAX3 + CNCC population of E11.5 embryos of the indicated genotype. Error bars represent the standard deviation. (E) Whole mount in situ hybridization for Msx1 and Dlx2 transcripts on E11.5 embryos of the indicated genotype. Arrowhead indicates Dlx2 expression. Star shows its absence. Dotted lines indicate nasal pit location. LNP: lateral nasal process, MNP: medial nasal process, Mx: maxillary process, Md: mandibular process. Scale bars, 500m. (F) Composite images of multiple fields showing the Ooral cavity of Dapi DAPI stained E15.5 embryos of the indicated genotype from which the lower jaw was removed. White arrowheads indicate the remaining of the primary palate. Black arrowheads indicate the bifurcated secondary palate. Mx: maxillary, pr: prominent rugae, pp: primary palate, sp: secondary palate, saf: site of apposition and fusion of palatal shelves, v: primordia of vibrissae. Scale bar, 500m.

Figure 4 .

 4 Figure 4. Impairing Pax3/7 function induces cell cycle exit of CNCC. (A) Quantification of the number of Pax7PAX7 + cells within the GFP + population on transverse sections of the cranial structures of E9.5 and E10.5 embryos of the indicated genotype (B) GFP expression in E11.5 embryos of the indicated genotypes (lateral views). Scale bar, 500m. Red boxes delineate the facial prominences dissected to perform the FACS-sorting in C. (C) Percentage of FACS-sorted GFP + cells within the total population of cells from the dissected facial prominences of E11.5 embryos of the indicated genotype. Error bars represent the standard deviation. (D-E) Phospho-Histone HISTONE-H3 (PH3) and p21 expression in red, Pax7 PAX7 in white and GFP in green on transverse sections within the cranial regions of embryos at indicated genotypes and stages. Scale bars, 50m. (F, G, H, I) Quantification of the proportion of PH3 + (F), p21 + (G) AP2 + (H) and Sox9SOX9 + (I) cells within the Pax7PAX7 + population in sections through the cranial regions of embryos with the stages and genotypes indicated. Error bars represent the standard deviation.

Figure

  Figure 6. Pax3 PAX3 plays a safeguard function against TCDD-induced craniofacial defects. (A) p21 relative expression assayed by RT-qPCR in dissected faces of independent E11.5 embryos of the indicated genotype treated with carrier or TCDD. Dotted line represents p21 expression level in carrier-treated control embryos. (B) Bright field and GFP expression of E13.5 embryos of the indicated genotype treated with carrier or TCDD (facial views, top and middle panels). Arrowheads indicate divided nasal processes. Scale bar, 500m. (C) Number of carrierand TCDD-treated E13.5 embryos presenting with a normal or a frontal cleft with the genotype indicated.

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ACKNOWLEDGEMENTS:

We are grateful to Vanessa Ribes for helpful discussions and comments. We thank Margaret Buckingham, Carmen Birchmeier, Shahragim Tajbakhsh, Robert Kelly and Sonia Alonso Martin for helpful comments. We thank Catherine Bodin for histology. We also wish to acknowledge the animal care facilities at UPMC and CDTA, and Catherine Blanc from Flow Cytometry Core CyPS. This work was supported by funding to FR from INSERM Avenir Program, Association Française contre les Myopathies (AFM), Association Institut de Myologie (AIM), Labex REVIVE, the European Union Sixth and Seventh Framework Program in the project MYORES and ENDOSTEM (Grant # 241440), Fondation pour la Recherche Médicale (FRM; Grant FDT20130928236 and DEQ20130326526), Agence Nationale pour la Recherche (ANR) grant Epimuscle, Bone-muscle-repair and BMP-biomass, and the Agence Nationale pour la Recherche Maladies Rares (MRAR) grant Pax3 in WS. This work was also funded by the German Research Foundation (DFG; Grant GK1631), French-German University (UFA-DFH; Grant CDFA-06-11) and the AFM as part of the MyoGrad International Research Training Group for Myology. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

AUTHOR CONTRIBUTIONS:

AZ, designed and performed experiments, analyzed data and wrote the manuscript. RR, designed and performed experiments, analyzed data. FA and BC, designed and performed experiments.

FR, oversaw the entire project, designed experiments, performed experiments, analyzed data and wrote the manuscript.


Matteo Guerra

Gang Zheng

Denis Efimov

Wilfrid Perruquetti

An ISS based Solution to avoid Local Minima in the Potential Field Method

This work solves the obstacle avoidance problem extending the Potential Field (PF) method for a mobile robot. The usual definition of the PF has been modified to have a field which is continuous everywhere. It is shown that the system has an attracting equilibrium at the target point, repelling equilibriums in the centers of the obstacles and saddle points on the borders. Those unstable equilibriums are avoided capitalizing on the established Input-to-State Stability (ISS) property of this multistable system. To escape a local minima this work makes the most of ISS property that is not lost for perturbations. And for small properly designed disturbances the global attractivity of the target point is proved. The proposed modification of the PF method is shown to be effective by simulation and then applied for unicycle-like mobile robots with additive input disturbances.

I. INTRODUCTION

Within the most studied topics in the robotic field there is, of course, obstacles avoidance due to the obvious necessity for the platforms to avoid collisions while executing a task. Many solutions have been proposed to tackle the problem. One of the most used is the Potential Field Method, firstly introduced by Khatib in [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF]; for motion planning, the Potential Field (PF) drives the robot along a potential field whose minimum is at the target and each obstacle generates an additional repellent force which drives the robot away. It has been shown that this solution, even though mathematically elegant and quite effective practically, has some drawbacks when special events occur [START_REF] Koren | Potential field methods and their inherent limitations for mobile robot navigation[END_REF]. In its standard formulation local minima could appear resulting in a trap for the robot that gets blocked (the easiest example is the case in which the obstacle center belongs to the straight line connecting the target and the robot center); other issues arise if several obstacles are too closely placed or when the size of them are considerably bigger than the robot (i.e. a wall) resulting in oscillations in both cases. A solution was proposed by Koditschek et al. in [START_REF] Rimon | Exact robot navigation using artificial potential functions[END_REF], they proposed navigation functions in a n-dimensional space in which the adopted potential field had no other local minima than the target specified, supposing though that the complete environment was known a priori. Other solutions use a harmonic potential field proposed in [START_REF] Connolly | Path planning using laplace's equation[END_REF], [START_REF] Sato | Deadlock-free motion planning using the laplace potential field[END_REF], and the more recent [START_REF] Aa Masoud | A harmonic potential field approach for navigating a rigid, nonholonomic robot in a cluttered environment[END_REF], in which the method computes solutions to Matteo Guerra, Gang Zheng, Denis Efimov and Wilfrid Perruquetti are with Non-A team @ Inria, Parc Scientifique de la Haute Borne, 40 av. Halley Laplace's Equation in arbitrary n-dimensional domains, and results in a weak form of [START_REF] Rimon | Exact robot navigation using artificial potential functions[END_REF].

Last flaw of the method is the possibility to miss the target in case of an obstacle too close to it. This problem is called Goals nonreachable with obstacles nearby (GNRON). Treated in [START_REF] Ge | New potential functions for mobile robot path planning[END_REF], it deals with the case in which the repulsive force generated by an obstacle close to the target generate a force higher than the attractive one, preventing the robot to accomplish the task.

There are also methods which do not eliminate unwanted equilibriums but generate local forces, Virtual Hill, to escape the disturbing minimum as in [START_REF] Gyu | Real-time path planning in unknown environment and a virtual hill concept to escape local minima[END_REF]. Many other modifications have been proposed in order to overcome all the cited shortages also in the case of the collision avoidance for multiple robots [START_REF] Okamoto | Novel potential-functionbased control scheme for non-holonomic multi-agent systems to prevent the local minimum problem[END_REF], [START_REF] Kim | Decentralized control of autonomous swarm systems using artificial potential functions: analytical design guidelines[END_REF], [START_REF] Yingchong | Motion planning for non-holonomic mobile robots using the i-pid controller and potential field[END_REF].

This work presents a modification of the standard Potential Field method in which first of all, the field is continuous everywhere, property that is not common in literature; considering a particle in 2D dimension, the integrator (that has the gradient of such a field as input) under proper assumptions is shown to be an ISS stable system with respect to decomposable invariants sets using the results presented in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF], while local minima are avoided adding an input which plays on the fact that the appearance of any bounded perturbation does not compromise the ISS property. Simulations are shown to support the theoretical results and the effectiveness for motion planning and collision avoidance considering a particle in 2D dimension. Moreover simulations have been carried for an unicyle-like non-holonomic Wheeled Mobile Robots (WMR). In the case of the WMR a finite-time controller has been designed to regulate the orientation to guarantee a fast and robust response.

II. PRELIMINARIES

For an n-dimensional C 2 connected and orientable Riemannian manifold M without boundary (0 ∈ M ), let the map f : M × R m → T x M be of class C 1 (T x M is the tangent space), and consider a nonlinear system of the following form:

ẋ(t) = f (x(t), d(t)) (1)
where the state x ∈ M and d(t) ∈ R m (the input d(•) is a locally essentially bounded and measurable signal) for t ≥ 0. We denote by X(t, x; d(•)) the uniquely defined solution of (1) at time t fulfilling X(0, x; d(•)) = x. Together with (1) we will analyze its unperturbed version:

ẋ(t) = f (x(t), 0). (2)
A set S ⊂ M is invariant for the unperturbed system (2) if X(t, x; 0) ∈ S for all t ∈ R and for all x ∈ S. For a set S ⊂ M define the distance to the set |x| S = min a∈S δ(x, a) from a point x ∈ M , where the symbol δ(x 1 , x 2 ) denotes the Riemannian distance between x 1 and x 2 in M , |x| = |x| {0} for x ∈ M or a usual euclidean norm of a vector x ∈ R n . For a signal d : R → R m the essential supremum norm is defined as d ∞ = ess sup t≥0 |d(t)|.

A. Decomposable sets

Let Λ ⊂ M be a compact invariant set for (2).

Definition 1.

[10] A decomposition of Λ is a finite and disjoint family of compact invariant sets Λ 1 , . . . , Λ k such that

Λ = k i=1 Λ i .
For an invariant set Λ, its attracting and repulsing subsets are defined as follows:

W s (Λ) = {x ∈ M : |X(t, x, 0)| Λ → 0 as t → +∞}, W u (Λ) = {x ∈ M : |X(t, x, 0)| Λ → 0 as t → -∞}. Define a relation on W ⊂ M and D ⊂ M by W ≺ D if W s (W) ∩ W u (D) = . Definition 2. [10] Let Λ 1 , . . . , Λ k be a decomposition of Λ, then 1. An r-cycle (r ≥ 2) is an ordered r-tuple of distinct indices i 1 , . . . , i r such that Λ i1 ≺ . . . ≺ Λ ir ≺ Λ i1 . 2. A 1-cycle is an index i such that [W u (Λ i ) ∩ W (Λ i )] - Λ i = .
3. A filtration ordering is a numbering of the Λ i so that

Λ i ≺ Λ j ⇒ i ≤ j.
As we can conclude from Definition 2, existence of an rcycle with r ≥ 2 is equivalent to existence of a heteroclinic cycle for (2) [START_REF] Guckenheimer | Structurally stable heteroclinic cycles[END_REF]. And existence of a 1-cycle implies existence of a homoclinic cycle for (2) [START_REF] Guckenheimer | Structurally stable heteroclinic cycles[END_REF]. Definition 3. The set W is called decomposable if it admits a finite decomposition without cycles, W = k i=1 W i , for some non-empty disjoint compact sets W i , which form a filtration ordering of W, as detailed in definitions 1 and 2.

Let a compact set W ⊂ M be containing all αand ω-limit sets of (2).

B. Robustness notions

The following robustness notions for systems in (1) have been introduced in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF]. Definition 4. We say that the system (1) has the practical asymptotic gain (pAG) property if there exist η ∈ K ∞ and a non-negative real q such that for all x ∈ M and all measurable essentially bounded inputs d(•) the solutions are defined for all t ≥ 0 and the following holds:

lim sup t→+∞ |X(t, x; d)| W ≤ η( d ∞ ) + q. (3)
If q = 0, then we say that the asymptotic gain (AG) property holds.

Definition 5. We say that the system (1) has the limit property (LIM) with respect to W if there exists µ ∈ K ∞ such that for all x ∈ M and all measurable essentially bounded inputs d(•) the solutions are defined for all t ≥ 0 and the following holds:

inf t≥0 |X(t, x; d)| W ≤ µ( d ∞ ).
Definition 6. We say that the system (1) has the practical global stability (pGS) property with respect to W if there exist β ∈ K ∞ and q ≥ 0 such that for all x ∈ M and all measurable essentially bounded inputs d(•) the following holds for all t ≥ 0:

|X(t, x; d)| W ≤ q + β(max{|x| W , d ∞ }).
It has been shown in [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF] that to characterize (3) in terms of Lyapunov functions the following notion is appropriate:

Definition 7. A C 1 function V : M → R is a practical ISS-Lyapunov function for (1) if there exists K ∞ functions α 1 , α 2 ,
α and γ, and scalar q ≥ 0 and c ≥ 0 such that

α 1 (|x| W ) ≤ V (x) ≤ α 2 (|x| W + c),
the function V is constant on each W i and the following dissipative property holds:

DV (x)f (x, d) ≤ -α(|x| W ) + γ(|d|) + q.
If the latter inequality holds for q = 0, then V is said to be an ISS-Lyapunov function.

Notice that the existence of α 2 and c follows (without any additional assumptions) by standard continuity arguments.

The main result of [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF] connecting these robust stability properties is stated below: Theorem 1. Consider a nonlinear system as in (1) and let a compact invariant set containing all α and ω limit sets of (2) W be decomposable (in the sense of Definition 3). Then the following facts are equivalent.

1. The system admits an ISS Lyapunov function; 2. The system enjoys the AG property; 3. The system admits a practical ISS Lyapunov function; 4. The system enjoys the pAG property; 5. The system enjoys the LIM property and the pGS.

A system in (1), for which this list of equivalent properties is satisfied, is called ISS with respect to the set W [START_REF] Angeli | On input-to-state stability with respect to decomposable invariant sets[END_REF].

III. POTENTIAL FIELD METHOD WITH STATIC OBSTACLES

First, let us consider a simplified model of a mobile agent represented by the integrator dynamics:

ẋ = u x , (4
) ẏ = u y , where x ∈ R and y ∈ R are the coordinates of the agent in the plane, z = [x y] T , u x ∈ R and u y ∈ R are the corresponding controls. It is necessary to design the controls u x , u y providing the agent regulation to the origin under avoidance of collisions with isolated point-wise obstacles, which are defined by their coordinates ζ i = (x i , y i ) and safe distances d i around them for i = 1, . . . , N , where N > 0 is a finite number of obstacles.

We will assume that |ζ i -ζ j | > max{d i , d j } and |ζ i | > d i for all 1 ≤ i = j ≤ N , i.e. the obstacles are separated and the origin is not occupied by an obstacle.

The problem will be solved using the potential field method, whose idea consists in defining a repulsion potential U r with respect to the obstacles and attraction potential U a with respect to the origin, next the controls can be designed proportional to the "forces" generated by the total potential U , [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF]. The main drawback of that approach consists in appearance of local conditional extrema, which theoretically do not allow a global problem solution to be guaranteed by the method. In this work we will use the results presented in the previous section to design the agent dynamics that is C 1 and ISS with respect to the set W composed by equilibriums, among them the equilibrium at the origin is attractive, the equilibriums related to the obstacles are repulsing, while ones corresponding to the local extrema are saddle. Next, applying specially designed small perturbations to that ISS system we will avoid the unstable equilibriums.

To design the attraction potential U a we would like to impose the following constraints: it should be twice continuously differentiable with respect to x and y, and its gradient should be globally bounded (the velocity of movement out the origin in the collision-free case should be approximately constant in a robotic application). The following potential yields these constraints:

U a (z) =      |z| 2 if |z| ≤ υ, |z| if |z| ≥ Υ, λ(|z|)|z| 2 + [1 -λ(|z|)]|z| otherwise, λ(s) = 2s 3 -3(υ + Υ)s 2 + 6Υυs + Υ 2 (Υ -3υ) Υ 2 (Υ -3υ) + υ 2 (3Υ -υ) 2 ,
where 0 < υ < Υ < +∞ are the design parameters. Thus, the potential U a is quadratic in z close to the origin, it has a linear growth rate far enough and the function λ ensures a smooth transition between these zones.

The repulsion potential U r should be also twice continuously differentiable with respect to x and y, and it should be active only on a small zone around the obstacle (the agent can detect the obstacle presence only locally in an uncertain environment in a robotic application, for example):

U r (z) = α N i=1 max{0, d 2 i -|z -ζ i | 2 } 2 ,
where α > 0 is a tuning parameter. The total potential U has the form:

U (z) = U a (z) + U r (z),
with the gradient

∇ z U (z) = ∇ z U a (z) + ∇ z U r (z), ∇ z U a (z) =      2z if |z| ≤ υ, z|z| -1 if |z| ≥ Υ, ϕ(z) otherwise, ∇ z U r (z) = -4α N i=1 (z -ζ i ) max{0, d 2 i -|z -ζ i | 2 }, where ϕ(z) = ∇ z λ(|z|)|z| 2 + [1 -λ(|z|)
]|z| is the corresponding C 1 function ensuring a continuous transition between 2z and z|z| -1 . Note that by construction ∇ z U (z) is a C 1 function of z. As usual in the potential field method we assign:

u x u y = -∇ z U (z) + v, (5)
where v ∈ R 2 is an auxiliary bounded input to be designed later, then the closed-loop system (4), ( 5) takes the gradient form:

ż = -∇ z U (z) + v. (6)
Next, we are going to show that for v = 0 this system has an attracting equilibrium at the origin, repulsing equilibriums in a vicinity of ζ i for each i = 1, . . . , N and a saddle equilibrium in the border of the repulsion zone around ζ i for each i = 1, . . . , N . Therefore, a compact invariant set W containing all αand ω-limit sets of ( 6) for v = 0 is decomposable in the sense of Definition 3, and that Theorem 1 can be applied to establish ISS with respect to the set W for the input v.

A. Equilibrium at the origin

Under the restrictions |ζ

i | > d i for all 1 ≤ i ≤ N , the system (6) is reduced to ż = -2z
for |z| ≤ υ for some 0 < υ ≤ υ, which is obviously locally attractive. For simplicity of presentation below we will assume that the constants υ and Υ are selected in a way to provide

|ζ i | ≥ Υ + d i for all 1 ≤ i ≤ N .

B. Equilibriums around the obstacles

Since the obstacles are separated from one another and from the origin, around each obstacle the system (6) takes a reduction (|z| ≥ Υ):

ż = -z|z| -1 + 4α(z -ζ i ) max{0, d 2 i -|z -ζ i | 2 } + v for some 1 ≤ i ≤ N . Clearly, if d i < |z-ζ i | then ż = -z|z| -1
and there is no equilibrium, thus we may restrict the attention to the case |z -

ζ i | ≤ d i and ż = -z|z| -1 + 4α(z -ζ i )(d 2 i -|z -ζ i | 2 ) + v.
On this set the equilibriums of (6) satisfy the vector equation

z = 4α(z -ζ i )(d 2 i -|z -ζ i | 2 )|z| or the corresponding scalar equation |z| 2 -8α(d 2 i -|z -ζ i | 2 )|z| z T (z -ζ i ) +16α 2 |z -ζ i | 2 (d 2 i -|z -ζ i | 2 ) 2 |z| 2 = 0.
Introducing parametrization z = κζ i + η, where κ ∈ R and η ∈ R 2 , and substituting it in the last equation it is tedious but straightforward to obtain that for any |η| = 0 the equality is not satisfied. Therefore setup η = 0, then under substitution z = κζ i we have

κ 2 |ζ i | 2 {1 -4α[d 2 i -(κ -1) 2 |ζ i | 2 ]|ζ i |(κ -1)} 2 = 0,
the equation for equilibriums is reduced to

1 -4α[d 2 i -s 2 |ζ i | 2 ]|ζ i |s = 0 for s = κ -1, or s 3 - d 2 i |ζ i | 2 s + 1 4α|ζ i | 3 = 0
that is a depressed cubic equation, which by the Cardano's method has only real roots if

αd 3 i > 3 √ 3 8 . (7)
Next, by the Routh-Hurwitz stability criterion the equation has 2 roots with positive real parts. Therefore, for |z -ζ i | ≤ d i the system (6) has two equilibriums z i,1 0 and z i,2 0 under the condition [START_REF] Kim | Decentralized control of autonomous swarm systems using artificial potential functions: analytical design guidelines[END_REF]. The Cardano's method also provides the expressions of exact solutions and, hence, the coordinates of the equilibriums z i,1 0 , z i,2 0 (not given here for compactness, both equilibriums are located farther from the origin than the obstacle ζ i on the line connecting the origin and the point (x i , y i )). Finally, the system ( 6) is continuously differentiable, then the linearization shows that the equilibrium z i,1 0 (closer to ζ i ) is purely repulsing, and another one z i,2 0 is saddle (the corresponding local minimum).

To evaluate the zone of repulsion around ζ i a Lyapunov function for linearization around z i,1 0 can be used, or let us consider a Lyapunov function V (e) = |e| 2 for e = z -ζ i and v = 0:

V = 2e T [-z|z| -1 + 4αe(d 2 i -|e| 2 )] = -2e T z|z| -1 + 8αV (d 2 i -V ) ≥ -2|e| + 8αV (d 2 i -V ). Note that |e| = √ V then V ≥ [4α √ V (d 2 i -V ) -1]2 √ V .
The Cardano's method can be used to find the solutions of the equation 4α

√ V (d 2 i -V ) = 1, which determines the sign definiteness of V . The expression in the square brackets 4α √ V (d 2 i -V )-1 reaches its maximum 2 √ 3 8 3 √ 3 αd 3 i -1 d i for V = 1 3 d 2 i
, which is positive if the condition ( 7) is fulfilled (note that since the value of d i is constrained by the physical dimensions of the agent, then ( 7) is a condition for α to satisfy). Thus the repulsion zone around the obstacle exists and it can be easily estimated.

C. Robustness with respect to v

The conditions on existence of the equilibriums, established above, are as follows:

Assumption 1. Let the condition (7) be satisfied, |ζ i -ζ j | > max{d i , d j } and |ζ i | ≥ Υ + d i for all 1 ≤ i = j ≤ N .
Now we would like to show that the set W = {{0}, z 1,1 0 , z 1,2 0 , . . . , z i,1 0 , z i,2 0 , . . . , z N,1 0 , z N,2 0 }, which is composed by the equilibrium at the origin and N pairs of equilibriums z i,1 0 , z i,2 0 associated with each obstacle, contains all αand ω-limit sets of ( 6) for v = 0 and it is decomposable in the sense of Definition 3. The system (6) has a Lyapunov function U (z), by construction α 1 (|z|) ≤ U (z) ≤ α 2 (|z|) for all z ∈ R 2 and some α 1 , α 2 ∈ K ∞ , whose derivative has the form:

U = -|∇ z U (z)| 2 + ∇ T z U (z)v ≤ -0.5|∇ z U (z)| 2 + 0.5|v| 2 (8)
and the total potential stops to decrease for v = 0 only on the set where ∇ z U (z) = 0, but by consideration above it is W: i.e. there exist

γ 1 , γ 2 ∈ K ∞ such that γ 1 (|z| W ) ≤ |∇ z U (z)| ≤ γ 2 (|z| W ) for all z ∈ R 2 .
There is no cycle in the decomposition of W due to the same property U ≤ 0 for v = 0 (indeed the obstacles are separated and to pass from one saddle equilibrium around the obstacle ζ i to another one around ζ j it is necessary to cross the zone where ∇ z U (z) = ∇ z U a (z) and U < 0, therefore a trajectory cannot return back). Thus, W is decomposable and contains all αand ω-limit sets of ( 6) for v = 0. Further, U ≤ -0.5γ 2 1 (|z| W ) + 0.5|v| 2 , then U is an ISS Lyapunov function and by Theorem 1 the following result has been proven. Lemma 1. Under Assumption 1 the system ( 6) is ISS with respect to the set W for the input v.

D. Design of the input v to escape the local minimums

The advantage of the ISS property is that appearance of any bounded disturbance v does not lead to the system instability.

In our case the total potential function U is also an ISS Lyapunov function for the system [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF]. If v = 0 and the agent in ( 6) is approaching an unstable equilibrium, then according to the expression of U the velocity of the agent is decreasing proportionally to |∇ z U (z)|. Thus, if |∇ z U (z)| ≤ for some predefined > 0 and we are far from the origin, it can be a signal of closeness to a saddle equilibrium, then an input v = 0 can be generated to shift the movement direction.

The input v can be selected bounded and pushing the system in an arbitrary direction with a uniform distribution, by ISS property the solutions asymptotically will stay close to W. However, using the Lyapunov function U the input v always can be designed in order to additionally guarantee a decreasing of U . From ( 8)

v =      ρ ∇ y U (z) -∇ x U (z) if |∇ z U (z)| ≤ and |z| > υ, 0 otherwise, (9)
ρ = sgn y - y i x i x , i = arg inf 1≤j≤N |z -ζ j |, sgn(s) = 1 if s ≥ 0, -1 otherwise ensures that U ≤ 0 for all t ≥ 0 ( U = -|∇ z U (z)| 2 while v = 0
) and for U = 0 as well, and for the case of agent velocity dangerous decreasing (|∇ z U (z)| ≤ ) far from the origin (|z| > υ) the proposed input v generates an orthogonal disturbance to the current direction of movement. The variable ρ defines the orientation of this orthogonal perturbation, in [START_REF] Aa Masoud | A harmonic potential field approach for navigating a rigid, nonholonomic robot in a cluttered environment[END_REF] it points out from the line connecting the origin and the point (x i , y i ) (that is the coordinate of the closest obstacle) and where we have the unstable equilibriums.

Assumption 2. Let z(0) / ∈ D = ∪ N i=1 D i with D i = {z ∈ R 2 : |z -ζ i | ≤ d i }.
The meaning of Assumption 2 is straightforward: the robot starts in the collision-free condition.

Theorem 2. Under Assumptions 1 and 2 the system (6) with the avoidance control (9) has the origin attractive from all initial conditions z(0) / ∈ W \ {0}.

Formally the control (9) does not use ISS property of the set W, it is designed from a pure Lyapunov approach. We may modify [START_REF] Aa Masoud | A harmonic potential field approach for navigating a rigid, nonholonomic robot in a cluttered environment[END_REF] as follows in order to make the attractiveness of the origin global:

v =      ρ ε |z| y -x if |∇ z U (z)| ≤ and |z| > υ, 0 otherwise, (10)
ρ = sgn y - y i x i x , i = arg inf 1≤j≤N |z -ζ j |, sgn(s) = 1 if s ≥ 0, -1 otherwise
where ε > 0 is a design parameter. It is easy to check that v T z = 0 for all z ∈ R 2 and |v| = ε if |∇ z U (z)| ≤ and |z| > υ in [START_REF] Nitecki | Filtrations, decompositions, and explosions[END_REF]. Theorem 3. Under Assumption 1 and 2 the system (6) with the avoidance control [START_REF] Nitecki | Filtrations, decompositions, and explosions[END_REF] has the origin globally attractive provided that ε > 2 and > 0 is selected sufficiently small.

E. More complex situations

Of course in reality the assumption about separation between obstacles may be not satisfied, but even for this case the approach can be easily extended. Application of perturbation v with the amplitude ε does not destroy boundedness of the system trajectories by ISS property. If ε has been selected sufficiently small, then asymptotically z(t) enters A, as it has been defined above, whose separated subsets contain a single isolated extremum point of U . The function |∇ z U (z)| is C 1 by construction, then ∇ z |∇ z U (z)| can be calculated and locally v can be selected proportional to ∇ z |∇ z U (z)| in order to maximize |∇ z U (z)|, which is equivalent of the extremum point avoidance. In the simple case presented above the calculation of ∇ z |∇ z U (z)| may be avoided.

F. Results of simulation

For υ = 0.1, Υ = 0.5, α = 2, N = 1 and (x 1 , y 1 ) = (2, 2) with d 1 = 1, the results of the system (6) simulation for different initial conditions with v = 0 are shown in the left part of Fig. 1. The results of the system (6) simulation with [START_REF] Nitecki | Filtrations, decompositions, and explosions[END_REF]

IV. WHEELED MOBILE ROBOT REGULATION WITH COLLISION AVOIDANCE

Consider a wheeled mobile robot, whose model is given by a unicycle:

qx = cos(θ)u(1 + δ 1 ), qy = sin(θ)u(1 + δ 1 ), (11) qθ
= ω(1 + δ 2 ),
where (q x , q y ) ∈ R 2 is the robot position, q θ ∈ (-π, π] is the robot orientation, |u| ≤ u max and |ω| ≤ ω max are linear and angular velocities of the robot respectively (u max and ω max are given bounds), δ k ∈ [δ min , δ max ], k = 1, 2 are exogenous bounded disturbances [START_REF] Guerra | Finite-time supervisory stabilization for a class of nonholonomic mobile robots under input disturbances[END_REF] , which are introduced in order to represent the dynamical model uncertainties/dynamics (they are not taken into account in the classical unicycle model), -1 < δ min < δ max < +∞.

In order to apply the approach developed for the system (6) with the control (9) or [START_REF] Nitecki | Filtrations, decompositions, and explosions[END_REF], as usual in the potential field method, the trajectory generated by ( 6), (10) (or with [START_REF] Aa Masoud | A harmonic potential field approach for navigating a rigid, nonholonomic robot in a cluttered environment[END_REF]) can be used as a reference for [START_REF] Okamoto | Novel potential-functionbased control scheme for non-holonomic multi-agent systems to prevent the local minimum problem[END_REF], defining

θ d = atan2 (∇ y U (z), ∇ x U (z)) -q θ : u = u max 1 + ε u 2 x + u 2 y , (12)
ω = ω max |θ d |sign(θ d ).
The results of simulation of the system (11), [START_REF] Gyu | Real-time path planning in unknown environment and a virtual hill concept to escape local minima[END_REF] are shown in Fig. 2 for u max = 1 and ω max = 3 in the case of single obstacle (Fig. 2a) and multiple obstacles (Fig. 2b). In both figures of Fig. 2 the obstacle is the zone filled in violet while the safe distance is the black circle around the obstacle itself. The method is called NON-Apf to emphasize the non asymptotic (finite time) behavior of the controller acting on the orientation of the WMR. Moreover the proposed modification has been compared with the standard APF [START_REF] Rimon | Exact robot navigation using artificial potential functions[END_REF] to show how the improvement of this work allows the WMR also to keep a bigger distance from the obstacle(s).

A. More complex scenarios

Several simulations were run also for more complex scenarios, in which the features of the equipped sensor for an In Fig. 3b is shown the path followed by the WMR using the proposed modification of the potential field method, while in Fig. 3a the strategy to decide which is the "point" to use as reference for the obstacle. Basically, the chosen point ζ, green star in Fig. 3a, is the averaging on the x and y coordinates of the LIDAR sensed points in a predefined range, in the test 1 m; the radius is the distance among ζ itself and the farthest sensed point of the scan, which leads to the definition of the influence distance d that is the radius augmented of the radius of the robot.

V. CONCLUSIONS

The modification of the PF method based on the ISS property for decomposable invariant sets was shown to be able to avoid local minima, main issue of the standard solution. Furthermore, it has been formally shown how the introduction of the v input does not introduce new equilibriums for the system making the origin (target) the only attractive point. When coupled with the finite time control for the orientation for a non-holonomic WMR it showed a nice behavior avoiding single obstacles, multiple obstacles and in complex environments.

The authors intend to improve the method to cancel any oscillations in future formulations and to extend it to be used in the case of multi-agent systems.

Figure 1 :

 1 Figure 1: The results of the system (6) simulation

Figure 2 :

 2 Figure 2: The results of simulation of (11),[START_REF] Gyu | Real-time path planning in unknown environment and a virtual hill concept to escape local minima[END_REF]

Figure 3 :

 3 Figure 3: Results on complex environment

  , 59650 Villeneuve d'Ascq, France and CRIStAL (UMR-CNRS 9189), Ecole Centrale de Lille, BP 48, Cité Scientifique, 59651 Villeneuve-d'Ascq, France. (matteo.guerra@inria.fr) D. Efimov is with Department of Control Systems and Informatics, Saint Petersburg State University of Information Technologies Mechanics and Optics (ITMO), 49 av. Kronverkskiy, 197101 Saint Petersburg, Russia.

ACKNOWLEGMENT

This work was supported in part by Région Nord-Pas de Calais, by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031).