Aude Rochefort

Martial Briand

Coralie Marais

Marie-Hélène Wagner

Anne Laperche

Patrick Vallée

Matthieu Barret

Alain Sarniguet
email: alain.sarniguet@inra.fr

Marie- Hélène Wagner

Patrick Vallee

Influence of environment and host plant genotype on the structure and diversity of the Brassica napus seed microbiota

Keywords: Brassica napus, environment, germination, host genotype, seed microbiota, selfpollination

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Plants live in association with a diverse and complex set of microorganisms also known as microbiota. The plant microbiota can modify several host traits, such as plant nutrition [START_REF] Paredes | Design of synthetic bacterial communities for predictable plant phenotypes[END_REF], flowering kinetics [START_REF] Panke-Buisse | Selection on soil microbiomes reveals reproducible impacts on plant function[END_REF] and resistance against plant pathogens and herbivores [START_REF] Haney | Rhizosphere-associated Pseudomonas induce systemic resistance to herbivores at the cost of susceptibility to bacterial pathogens[END_REF][START_REF] Kwak | Rhizosphere microbiome structure alters to enable wilt resistance in tomato[END_REF]. Owing to such impacts on plant growth and health, it is of interest to decipher the biological and ecological processes involved in the assembly of the plant microbiota. Several studies have notably investigated the impact of environmental factors and host genetic variation on the composition of plant microbiota (e.g. [START_REF] Bulgarelli | Revealing structure and assembly cues for Arabidopsis root-inhabiting bacterial microbiota[END_REF][START_REF] Peiffer | Diversity and heritability of the maize rhizosphere microbiome under field conditions[END_REF][START_REF] Horton | Genome-wide association study of Arabidopsis thaliana's leaf microbial community[END_REF][START_REF] Edwards | Structure, variation, and assembly of the root-associated microbiomes of rice[END_REF][START_REF] Carrión | Linking rhizosphere microbiome composition of wild and domesticated Phaseolus vulgaris to genotypic and root phenotypic traits[END_REF][START_REF] Fitzpatrick | Assembly and ecological function of the root microbiome across angiosperm plant species[END_REF]. These studies have revealed a major impact of environment on the taxonomic structure of plant microbiota and a limited but significant contribution of the host genotype. Accordingly, it has been recently highlighted that community heritability (H 2 C , Opstal & Bordenstein, 2015) of the maize rhizosphere microbiota is low, even if the abundance of some operational taxonomic units (OTUs) is significantly affected by host genetics [START_REF] Walters | Large-scale replicated field study of maize rhizosphere identifies heritable microbes[END_REF]. This broad-sense heritability (H 2 ) of some bacterial taxa is probably linked to host genes involved in the recruitment of its microbiota, for instance genes related to specific plant metabolites and to plant immunity [START_REF] Hacquard | Interplay Between Innate Immunity and the Plant Microbiota[END_REF][START_REF] Sasse | Feed Your Friends: Do Plant Exudates Shape the Root Microbiome?[END_REF].

In contrast to other plant habitats, diversity of seed microbial assemblages is restrained [START_REF] Barret | Emergence Shapes the Structure of the Seed Microbiota[END_REF][START_REF] Klaedtke | Terroir is a key driver of seed-associated microbial assemblages: Terroir shapes the seed microbiota[END_REF][START_REF] Leff | Plant domestication and the assembly of bacterial and fungal communities associated with strains of the common sunflower, Helianthus annuus[END_REF][START_REF] Rybakova | The structure of the Brassica napus seed microbiome is cultivardependent and affects the interactions of symbionts and pathogens[END_REF][START_REF] Rodríguez | Commentary: seed bacterial inhabitants and their routes of colonization[END_REF][START_REF] Adam | The Cucurbita pepo seed microbiome: genotype-specific composition and implications for breeding[END_REF][START_REF] Bergna | Tomato Seeds Preferably Transmit Plant Beneficial Endophytes[END_REF][START_REF] Rezki | Assembly of seedassociated microbial communities within and across successive plant generations[END_REF], perhaps as a consequence of a seed bottleneck for microbial transmission [START_REF] Paredes | Giving back to the community: microbial mechanisms of plantsoil interactions[END_REF][START_REF] Newcombe | A Hypothetical Bottleneck in the Plant Microbiome[END_REF]. In fact considering that rhizosphere and phyllosphere are plant habitats with a highly diversified and abundant microbiota, seeds act as a filter and select smaller sized and less diversified microbial assemblages.Therefore, the relative importance of vertical transmission from maternal plant to its progeny over horizontal transmission from the environment is probably limited in plants [START_REF] Leff | Plant domestication and the assembly of bacterial and fungal communities associated with strains of the common sunflower, Helianthus annuus[END_REF][START_REF] Newcombe | A Hypothetical Bottleneck in the Plant Microbiome[END_REF].

Nevertheless, seeds represent the initial inoculum of the plant microbiota and can thus potentially impact plant fitness especially during its early development stages.

Rapidity and uniformity of seed germination and seedling growth together with emergence ability under adverse environmental conditions and performance after storage are collectively referred to as seed vigor [START_REF] Finch-Savage | Seed vigour and crop establishment: extending performance beyond adaptation[END_REF]. Seed vigor is determined by seed dormancy [START_REF] Bentsink | Seed Dormancy and Germination[END_REF] and seed longevity [START_REF] Leprince | Late seed maturation: drying without dying[END_REF]; both traits being influenced by genetic and environmental perturbations. For instance, seed dormancy in Brassica napus is affected by the genotype and pre-harvest environmental conditions [START_REF] Gulden | Relative contribution of genotype, seed size and environment to secondary seed dormancy potential in Canadian spring oilseed rape (Brassica napus)[END_REF][START_REF] Gruber | Classification of canola (Brassica napus) winter cultivars by secondary dormancy[END_REF]. In contrast to physiological and environmental factors, the impact of seed microbiota composition on seed vigor has not been yet explored. However, experimental evidences indicated that some plant-associated bacteria can repress germination through the production of oxyviniglycines [START_REF] Mcphail | 4-Formylaminooxyvinylglycine, an Herbicidal Germination-Arrest Factor from Pseudomonas Rhizosphere Bacteria[END_REF][START_REF] Chahtane | The plant pathogen Pseudomonas aeruginosa triggers a DELLA-dependent seed germination arrest in Arabidopsis (CS Hardtke and DJ Kliebenstein[END_REF] or decrease dormancy through free cytokinins production (Rodrigues [START_REF] Pereira | Cytokinins and the bacterial symbiosis of Ardisia species[END_REF][START_REF] Goggin | A potential role for endogenous microflora in dormancy release, cytokinin metabolism and the response to fluridone in Lolium rigidum seeds[END_REF]. Hence, one may argue that changes in seed microbiota structure could ultimately result in differences of seed vigor.

The few studies that have investigated the importance of ecological processes involved in the assembly of the seed microbiota have reported that the environment significantly impacts the structure of the seed microbiota, and especially fungal assemblages [START_REF] Klaedtke | Terroir is a key driver of seed-associated microbial assemblages: Terroir shapes the seed microbiota[END_REF]. Part of these studies has also highlighted that the host genotype could have a weaker but significant influence on bacterial endophytes of oilseed rape ; however, it is difficult to eliminate environmental variations that could be confounding [START_REF] Rybakova | The structure of the Brassica napus seed microbiome is cultivardependent and affects the interactions of symbionts and pathogens[END_REF]. In addition, neutral-based processes such as dispersal and ecological drift are also involved in the assembly of seed microbial assemblages [START_REF] Rezki | Assembly of seedassociated microbial communities within and across successive plant generations[END_REF].

Brassica napus (oilseed rape) is a cropping plant cultivated worldwide for its oilseed and for protein cattle feed production. B. napus is sensitive to a range of soilborne plant pathogens including Plasmodiophora brassicae, Rhizoctonia solani and Verticillium longisporum, which can cause clubroot, seedling damping-off and Verticillium wilt, respectively [START_REF] Hwang | Plasmodiophora brassicae: a review of an emerging pathogen of the Canadian canola (Brassica napus) crop[END_REF][START_REF] Sturrock | Effects of dampingoff caused by Rhizoctonia solani anastomosis group 2-1 on roots of wheat and oil seed rape quantified using X-ray Computed Tomography and real-time PCR[END_REF][START_REF] Depotter | Verticillium longisporum , the invisible threat to oilseed rape and other brassicaceous plant hosts: Verticillium longisporum , causal agent of stem striping[END_REF]. In absence of efficient control methods to restrict the populations of these phytopathogenic microorganisms, some biocontrol-based methods based on seed inoculation with microbial consortia have been proposed [START_REF] Kataria | Rhizoctonia solani damping-off and root rot in oilseed rape and canola[END_REF][START_REF] Müller | Impact of formulation procedures on the effect of the biocontrol agent Serratia plymuthica HRO-C48 on Verticillium wilt in oilseed rape[END_REF]. However, employing these seed inocula requires more basic knowledge on processes involved in assembly and dynamics of the seed microbiota [START_REF] Barret | Plant microbiota affects seed transmission of phytopathogenic microorganisms: Plant microbiota affects seed transmission[END_REF]. In this study, we estimated the impact of host genotypes and environment on the structure of the seed microbiota of nine genotypes representing the diversity of winter oilseed rape (WOSR)

during two consecutive years. In addition, we explored the potential effect of seed microbiota composition on seed germination of these nine genotypes.

MATERIALS AND METHODS

Selection of winter oilseed rape (WOSR) genotypes.

A total of 116 cultivars of winter oilseed rape (Brassica napus), registered in France between the 1950's and the 2000's, were gathered from the CRB BraCySol (INRA, IGEPP, France) to represent the diversity of inbred lines cultivated in Western Europe. All cultivars were genotyped with the infinium Brassica 60K SNP array [START_REF] Clarke | A high-density SNP genotyping array for Brassica napus and its ancestral diploid species based on optimised selection of single-locus markers in the allotetraploid genome[END_REF]. A set of 628 polymorphic SNPs presenting a low level of linkage disequilibrium (r²<0.2) were selected to analyze the structure of the genetic diversity within the accessions.

The structure of the collection was studied using admixture [START_REF] Alexander | Fast model-based estimation of ancestry in unrelated individuals[END_REF] and ten cross-validations. The genetic diversity of the collection was analyzed using Darwin software [START_REF] Perrier | Methods of data analysis[END_REF]. Sokal and Michener genetic distances were estimated between each couple of cultivars, and then clustering was performed using UPGMA. Nine B.

napus genotypes, namely Astrid, Aviso, Boston, Colvert, Express, Major, Mohican, Tenor and Zorro, were chosen among the collection (Supplementary Fig. S1, Table S1) as representative of B. napus diversity.

Seed production.

Seed phenotyping.

Seed imbibition, germination, and early radicle growth were monitored in vitro with an automated phenotyping platform (Phenotic, SFR Quasav, Angers, France), which is described in detail in [START_REF] Ducournau | An image acquisition system for automated monitoring of the germination rate of sunflower seeds[END_REF][START_REF] Ducournau | Using computer vision to monitor germination time course of sunflower (Helianthus annuus L.) seeds[END_REF] and [START_REF] Wagner | From knowledge-based research towards accurate and rapid testing of seed quality in winter rape[END_REF]. Briefly, four subsamples of 25 seeds by genotype and year (Y1 and Y2) were incubated at 20°C on germination test paper (GE Healthcare, type 3644) continuously moistened for 5 days and image acquisition was performed every two hours. Image acquisition, image analysis and data analysis methods are described in [START_REF] Demilly | Digital imaging of seed germination[END_REF]. The following parameters were determined: seed area (Area; in mm 2 ), volume 8h after initiation of imbibition (Imb_Vol; in mm 3 ), radicle elongation after 8h (Elon_germ, in mm), mean germination time (germination, in hours), and time to reach 50% of germination per seed subsample (T50, in hours). Differences were assessed with Kruskal-Wallis tests and considered as significant at a p-value < 0.01.

Microbial DNA sample preparation.

DNA extraction was performed on five subsamples of 1,000 seeds (~4 g) per genotype (45 subsamples in total per year). First, seeds were soaked in 20 mL of phosphate buffered saline (PBS) supplemented with 0.05% (v/v) of Tween 20 for 2h30 under constant agitation (140 rpm) at room temperature. After this seed soak that enables the recovery of epiphytic and endophytic microorganisms [START_REF] Barret | Emergence Shapes the Structure of the Seed Microbiota[END_REF] Numerous antimicrobial compounds are associated to seed tissues and are released during grinding, which therefore can result in underestimation of seed microbial diversity.

Libraries construction and sequencing.

An initial PCR amplification was performed with the primer sets gyrB_aF64/gyrB_aR553 [START_REF] Barret | Emergence Shapes the Structure of the Seed Microbiota[END_REF] and ITS1F/ITS2 [START_REF] Buée | 454 Pyrosequencing analyses of forest soils reveal an unexpectedly high fungal diversity[END_REF], which target a portion of gyrB and the fungal ITS1 region, respectively. The cycling conditions for ITS1F/ITS2, were an initial denaturation at 94°C for 3 min, followed by 35 cycles of amplification at 94°C (30 s), 50°C (45 s) and 68°C (90 s), and a final elongation step at 68°C for 10 min. The cycling conditions for gyrB were: initial denaturation at 94°C for 3 min, 35 cycles of amplification at 94°C (30 s), 55°C (45 s) and 68°C (90 sec), and final elongation step at 68°C for 10 min. Amplicons were purified with magnetic beads (Sera-Mag TM , Merck). A second PCR amplification was performed to incorporate Illumina adapters and barcodes. PCR cycling conditions were identical for the two molecular markers: a first denaturation step at 94°C (1 min), followed by 12 cycles at 94°C (1 min), 55°C (1 min) and 68°C (1 min), and a final elongation at 68°C for 10 min. Amplicons were purified as previously described and quantified with the QuantIT PicoGreen ds DNA Assay Kit (ThermoFisher Scientific). All the amplicons were pooled in equimolar concentrations and the concentration of the equimolar pool was monitored with quantitative PCR (KAPA SYBR ® FAST, Merck). Amplicon libraries were mixed with 10% PhiX and sequenced with MiSeq reagent kit v2 500 cycles (one sequencing cartridge per year).

Sequence processing.

Fastq files were processed with DADA2 version 1.6.0 [START_REF] Callahan | DADA2: Highresolution sample inference from Illumina amplicon data[END_REF], using the parameters described in the workflow for "Big Data: Paired-end"

(https://benjjneb.github.io/dada2/bigdata_paired.html). The only modification made relative to this protocol was a change in the truncLen argument according to the quality of the sequencing run. Taxonomic affiliations for amplicon sequence variants (ASV) generated with DADA2 were assigned with a naive Bayesian classifier [START_REF] Wang | Naïve Bayesian Classifier for Rapid Assignment of rRNA Sequences into the New Bacterial Taxonomy[END_REF] on an inhouse gyrB database [START_REF] Bartoli | In situ relationships between microbiota and potential pathobiota in Arabidopsis thaliana[END_REF] and the UNITE v7.1 fungal database [START_REF] Abarenkov | The UNITE database for molecular identification of fungi -recent updates and future perspectives[END_REF].

Microbial community analyses.

Analyses of diversity were conducted with the R package Phyloseq version 1. 22.3 (McMurdie & Holmes, 2013). Data were normalized based on sequencing depth, using a rarefaction of 1,100 and 15,000 sequences per sample for gyrB and ITS1, respectively. Observed richness (number of ASV per sample), estimated richness (Chao1 index), and inverse Simpson's index were calculated with Phyloseq. Faith's phylogenetic diversity was calculated on gyrB dataset with the R package picante version 1.6-2 [START_REF] Kembel | Picante: R tools for integrating phylogenies and ecology[END_REF]. Differences in alpha-diversity estimators between years and genotypes were assessed with Kruskal-Wallis tests.

Differences were considered as significant at a p-value < 0.01.

Changes in microbial assemblage composition were assessed with Bray-Curtis (BC) index and weighted UniFrac (wUF) distance [START_REF] Lozupone | UniFrac: a New Phylogenetic Method for Comparing Microbial Communities[END_REF]. Principal coordinate analysis (PCoA) was used for ordination of BC index and wUF distance. To quantify the relative contribution of plant generation and plant genotype in microbial community profiles, canonical analysis of principal coordinates (CAP) was performed with the function capscale of the R package vegan 2.4.2 [START_REF] Oksanen | Vegan: community ecology package. R package version 2[END_REF] followed with permutational multivariate analysis of variance (PERMANOVA; [START_REF] Anderson | A new method for non-parametric multivariate analysis of variance[END_REF].

A presence/absence matrix of common ASVs was constructed as follows. Common ASV was recorded as present if detected in all subsamples (n=5) of one seed lot. One seed lot corresponded to seeds collected each year from each plant genotype. Visualization of genotype-specific ASVs and shared ASVs between plant genotypes was assessed for each year with the R package UpSetR [START_REF] Lex | UpSet: Visualization of Intersecting Sets[END_REF]. The number of common bacterial and fungal ASV between years for each individual genotype was also assessed using Venn diagrams. This binary matrix was used for assessing Jaccard and unweighted UniFrac distance (uUF) between genotypes. Comparisons of Sokal-Michener genetic distance and Jaccard/uUF distances were monitored with coefficient of determination.

Distribution of bacterial and fungal ASVs in other Brassicaceae seeds was assessed by investigating their prevalence in 479 seed samples associated to Brassica oleracea, Brassica rapa and Raphanus sativus [START_REF] Barret | Emergence Shapes the Structure of the Seed Microbiota[END_REF][START_REF] Rezki | Differences in stability of seed-associated microbial assemblages in response to invasion by phytopathogenic microorganisms[END_REF][START_REF] Rezki | Assembly of seedassociated microbial communities within and across successive plant generations[END_REF].

RESULTS

Diversity of the seed microbiota of B. napus.

Overall 4,451 bacterial and 359 fungal ASVs were detected within gyrB and ITS1 datasets.

ASV provided a finer resolution than OTU [START_REF] Callahan | Exact sequence variants should replace operational taxonomic units in marker-gene data analysis[END_REF], this descriptor was therefore used for estimating the taxonomic structure of the seed microbiota of B. napus.

According to coverage estimates, the number of gyrB sequences was not sufficient for reaching the plateau after rarefaction, while saturation was achieved for fungal reads.

Hence, predicted richness (Chao1 index) was assessed for bacterial assemblages, while observed richness (number of ASVs) was measured for fungal assemblages. Although bacterial and fungal richness were significantly different between the two harvesting years (P < 0.01), seed samples collected in Y2 contained on average fewer bacterial and more fungal ASVs than Y1 seed sample (Fig. 1A,D). The increase of fungal richness in Y2 was associated with a significant decrease in fungal diversity (Fig. 1E). While bacterial diversity did not differ between years (Fig. 1B), bacterial phylogenetic diversity (Faith's PD), was significantly higher in Y2 (P < 0.01; Fig. 1C). In contrast to the harvesting year, no significant differences in α-diversity estimators were observed between plant genotypes (P > 0.01).

Similarities in composition of bacterial and fungal assemblages were estimated with weighted UniFrac (wUF) distance and Bray-Curtis (BC) index, respectively. Ordination of wUF and BC revealed a significant clustering of seed-associated bacterial and fungal assemblages according to the harvesting year (Fig. 2, Table 1). The relative influence of the harvesting year and the plant genotype in assemblage composition was further inspected through canonical analysis of principal coordinates (CAP) followed by PERMANOVA. Based on CAP analyses, 16.4% and 65.5% of variances in bacterial and fungal assemblage compositions, respectively, were explained by the harvesting year, (P < 0.01). Changes in bacterial and fungal assemblage compositions were also significantly (P < 0.01) explained by the plant genotype, with 19.3% and 11.5% of variances driven by this factor, respectively (Table 1).

Interaction between harvesting year and genotype was significant (P < 0.01) and explained 16.6% and 13.3% of variances of bacterial and fungal assemblage compositions respectively.

When each year is analyzed separately, the impact of plant genotype (P < 0.01) ranged from 40% to 90% of explained variance (Table 1, Supplementary Fig. S3). However, similarity in assemblage compositions between genotypes were different in Y1 and Y2.

Core and specific fractions of the B. napus microbiota.

Differences in relative abundance of bacterial orders between genotypes were mostly observed within Y1 (Fig. 3A). Notably we detected a higher abundance of Sphingomonadales and Pseudomonadales in Astrid, Aviso and Boston in comparison to the other genotypes (Fig. 3A). However, these differences between genotypes were not detected in Y2, where principal variations were related to increase in relative abundance of Enterobacteriales in Express and Zorro. Regarding fungal assemblages, differences in relative abundance of fungal orders were mostly observed between harvesting years, with a decrease in relative abundance of Tremellales and Helotiales in Y2 (Fig. 3B).

The distribution of ASVs between plant genotypes for each year of production was also assessed (Fig. 4, Fig. 5). Among the 15 bacterial ASVs systematically detected in at least one genotype in Y1, none were shared between all genotypes (Fig. 4A). In Y2, six bacterial ASVs (out of 268) were shared between all genotypes (Fig. 4B). Four of these six ASVs have been already detected in more than 10% of seed samples (n = 479) collected from various Brassicaceae including Brassica oleracea, Brassica rapa and Raphanus sativus (Table 2). The inheritance of bacterial ASVs between years was quite low, since only three genotypes (Astrid, Aviso and Boston) possessed four and five ASVs in common in Y1 and Y2, respectively (Supplementary Fig. S4A). These ASVs shared between both years are related to Sphingomonas, Pseudomonas and Frigoribacterium.

Fungal assemblages shared more ASVs between all genotypes in comparison to bacterial assemblages. Indeed, 18 (out of 37) and 19 (out of 52) fungal ASVs were shared between all the genotypes in Y1 and Y2, respectively (Fig. 5). For each genotype, 35% to 40% fungal ASVs were conserved from Y1 to Y2 (Supplementary Fig. S4B). In addition, nine fungal ASVs were systematically detected during both years. These fungal ASVs are also highly prevalent in seed samples of other Brassicaceae species (Table 2). There were very few bacterial ASVs specifically associated to one genotype in Y1, while there were in general more specific bacterial ASVs per genotype in Y2. Conversely, very few genotypes harbor specific fungal ASVs in Y1 as in Y2 (Supplementary Table S3, Table S4).

Relationship between B. napus genotypes and seed microbiota composition.

To assess if the composition of the seed microbiota could be related to genetic relatedness of B. napus, we compared Sokal-Michener genetic distance to Jaccard and uUF distances calculated with ITS1 and gyrB sequences, respectively (Supplementary Fig. S5). Coefficient of determination between Jaccard/uUF distances and Sokal-Michener distance were low for bacterial (r 2 = 0.026 in Y1; r 2 = 0.009 in Y2) and fungal (r 2 = 0.005 in Y1; r 2 = 0.117 in Y2) assemblages. While a significant (P = 0.002) positive correlation between genetic distance and fungal assemblage composition was detected in Y2, the overall genetic similarities between the B. napus genotypes employed in this work is not a robust predictor of seed microbiota composition.

Impact of production year and host genotype on seed phenotypes.

There were no significant differences between years and genotypes for seed area, radicle elongation and imbibition speed (Supplementary Fig. S6). However, according to average germination rate (n=100), germination of seed samples harvested in Y1 was significantly (P < 0.01) slower than germination of seed samples harvested in Y2 (Fig. 6). Significant differences were also observed between genotypes within specific year. For instance, seeds from Astrid germinated on average earlier (P <0.01) than the other genotypes in Y1 but not in Y2 (Fig. 6). Seeds from Tenor had a slower mean germination time (P <0.01) in comparison to the other genotypes but only for samples collected in Y2 (Fig. 6). Thus, another metric, T50, varied in the same way as mean germination time (Supplementary Fig. S6D). Of note, three bacterial ASV were specifically associated with Astrid in Y1 (Supplementary Table S3).

Moreover, 3 bacterial ASVs and 5 fungal ASVs were specifically associated to Tenor in Y2 (Supplementary Table S4).

DISCUSSION

The aim of this study was to assess the relative influence of the environment (i.e. year) and the host genotype on the structure of the B. napus seed microbiota. To eliminate environmental variations within a year, we sowed seeds of nine B. napus genotypes in the same field. In addition, to ensure homogeneous genetic material and preserve genotype specificities, each individual plant was cultivated in self-pollination.

Over two years we observed a decrease in bacterial richness for seed samples collected during the second year. This loss of bacterial richness could be explained by a selection of bacterial taxa across years, in an analogous process that has been described for continuous cropping (e.g. [START_REF] Zhao | Long-Term Coffee Monoculture Alters Soil Chemical Properties and Microbial Communities[END_REF]. Loss of bacterial richness might be related to the repeated self-pollination process, which limits dispersal of individuals among local bacterial assemblages [START_REF] Vannette | Dispersal enhances beta diversity in nectar microbes[END_REF]. If this assumption was correct, a similar reduction of richness would be expected within seed fungal assemblages. However, fungal richness increased in Y2. Therefore, alteration of richness could be solely due to differences in richness of local microbial assemblages between years. Influence of the harvesting year was the main driving factor of seed fungal assemblage composition with 65.5% of explained variance. These results agree with a previous report showing that seed fungal assemblages of common bean are mostly shaped by crop management practices and geographical locations [START_REF] Klaedtke | Terroir is a key driver of seed-associated microbial assemblages: Terroir shapes the seed microbiota[END_REF]. The high impact of environmental filtering on composition of the fungal fraction of the seed microbiota could be related to local biogeographical patterns of fungal assemblages [START_REF] Peay | Dimensions of biodiversity in the Earth mycobiome[END_REF].

Diversity and composition of the B. napus seed microbiota was different between years but also between the host genotypes used in this study. Host-genotype is an important driver in the assembly of the seed microbiota, especially for the bacterial fraction. The impact of host genotypes on the structure of B. napus microbiota was already highlighted for endophytic bacteria [START_REF] Rybakova | The structure of the Brassica napus seed microbiome is cultivardependent and affects the interactions of symbionts and pathogens[END_REF]. Since seed lots employed by Rybakova et al.

(2017) were collected from different geographical regions, the influence of host genotype was therefore inseparable from the influence of the environment. Our study confirmed without any confounding factor that the structure of the B. napus seed bacterial assemblage are partly genotype-dependent. Since interaction between year and genotype was significantly influencing β-diversity, we assessed the influence of host genotypes on microbiota compositions for each year. According to these analyses, the plant genotype explained a major part of variation in seed microbial assemblage composition, therefore suggesting an important influence of host-filtering in the assembly of the seed microbiota.

However, composition of seed microbial assemblage differed greatly between harvesting years. In addition, a small part of the B. napus seed bacterial assemblage is inherited from one generation to the next. Overall, our study highlighted a low inheritance of bacterial and fungal assemblages across B. napus generations. Indeed no bacterial ASV were consistently detected between plant generations, while nine fungal ASVs were conserved between years.

The absence of conserved bacterial ASV between year and genotypes could be explained by a stronger impact of host-filtering on assembly of the bacterial fraction of the seed microbiota. Alternatively, the absence of generalist bacterial ASV observed between years could be partly explained by the weak number of bacterial ASVs detected in Y1. The inheritance of seed-borne microbial taxa is quite low, therefore confirming previous observations performed on R. sativus over three consecutive generations [START_REF] Rezki | Assembly of seedassociated microbial communities within and across successive plant generations[END_REF] and Solanum lycopersicum over two generations [START_REF] Bergna | Tomato Seeds Preferably Transmit Plant Beneficial Endophytes[END_REF]. Hence, the main acceptable conclusion is that few microbial taxa associated with seeds are inherited and that most of seed-borne taxa represented cases of horizontal transmission [START_REF] Leff | Plant domestication and the assembly of bacterial and fungal communities associated with strains of the common sunflower, Helianthus annuus[END_REF].

Despite these changes in assemblage compositions, the ASVs detected on seed samples corresponded to microbial taxa frequently associated with other seed samples. For instance, four of the bacterial generalists detected in Y2, namely Pantoea agglomerans, Pseudomonas, Sphingomonas and Frigoribacterium, were already detected in more than 10% of other Brassicaceae seed lots [START_REF] Barret | Emergence Shapes the Structure of the Seed Microbiota[END_REF][START_REF] Rezki | Assembly of seedassociated microbial communities within and across successive plant generations[END_REF]. Hence, these taxa might be adapted to the seed habitat. The case was different for fungi with a significant number of fungal ASVs recovered through generations. However this does not mean that these taxa are vertically-transmitted. Assessing the relative importance of vertical and horizontal transmission will required an in-depth analysis of the seed transmission pathways (internal, floral and external) employed by these fungal taxa. Therefore, it is tempting to speculate that these differences in seed microbiota composition between years are the result of fluctuations of the local microbial reservoir. In this way, specific microbial recruitment and selection by the genotypes are modulated among years. Consequently, the relative impact of the host genotype on the structure of seed microbial assemblages is difficult to predict, as it is dependent of fluctuations of the local reservoir.

Previous works performed on the rhizosphere and phyllosphere of multiple maize genotypes have highlighted a relatively low H 2 of microbial taxa [START_REF] Wallace | Quantitative Genetics of the Maize Leaf Microbiome[END_REF][START_REF] Walters | Large-scale replicated field study of maize rhizosphere identifies heritable microbes[END_REF]. Although the number of genotypes employed in our study precluded the measurement of this broad sense heritability, we did not establish any significant link between the genetic distance of the nine genotypes and their microbial distances. While considering the variability between subsamples collected from plants of the same genotype and cultivated on the same field, in addition to the relative influence of environment and host genotype, there is a contribution of other neutral-based processes on the assembly of the seed microbiota [START_REF] Rezki | Assembly of seedassociated microbial communities within and across successive plant generations[END_REF]. However, some specific plant functions and traits, and especially genes implicated in plant immune system, shape the structure and diversity of their own microbiota [START_REF] Horton | Genome-wide association study of Arabidopsis thaliana's leaf microbial community[END_REF][START_REF] Li | Plant Phenotypic Traits Eventually Shape Its Microbiota: A Common Garden Test[END_REF]. Thus, it would be necessary to use a larger number of genetically diverse genotypes of WOSR to explore this hypothesis at a finer resolution (i.e. intra-species). To date, some studies failed to establish this link even when microbial diversification due to genetic diverse plant species was demonstrated [START_REF] Bouffaud | Is diversification history of maize influencing selection of soil bacteria by roots?[END_REF][START_REF] Schlaeppi | Quantitative divergence of the bacterial root microbiota in Arabidopsis thaliana relatives[END_REF]. However, establishing a direct correlation between genotypes and their microbiota would be of great interest for selecting genotypes with a specific microbiota in which certain taxa would ensure positive functions for the plant fitness.

Among different observed seed phenotypes, only the mean germination time and T50 (half-mean germination time) varied between years. Seed germination progresses through three phases [START_REF] Bewley | Seed Germination and Dormancy[END_REF]. The first phase, known as imbibition, is a physical process driven by difference in water potential between the inside and the outside of the seed. Based on seed volume monitored during 8 hours after initiation of imbibition, no significant differences in imbibition was observed between seed samples. This observation suggested that seed moisture was quite comparable between seed samples and that the resulting differences in germination kinetics are related to later phases. Since the environmental factor (year) more strongly shapes the bacterial and fungal communities, we assessed whether the change in microbiota composition could explain variation in germination rates. Authors demonstrated that the environmental conditions, age of seed and genotype [START_REF] He | Interaction between parental environment and genotype affects plant and seed performance in Arabidopsis[END_REF][START_REF] Nagel | Genome-wide association mapping and biochemical markers reveal that seed ageing and longevity are intricately affected by genetic background and developmental and environmental conditions in barley: Genetic and biochemical markers of seed ageing[END_REF][START_REF] Leprince | Late seed maturation: drying without dying[END_REF] affect different seed performances like secondary dormancy and seed vigor. The requirement to perform germination assays for all seed samples simultaneously ultimately results in differences in longevity of seed collected in Y1 and Y2. However, long-term storage (i.e. several years) of B. napus seeds samples does not impact germination time but rather germination percentage [START_REF] Nagel | Seed longevity in oilseed rape (Brassica napus L.) -genetic variation and QTL mapping[END_REF], which was not different between Y1 and Y2. Moreover, Astrid and Tenor genotypes did not support the ageing effect hypothesis as their germination rates were either similar or inverted respectively between Y1 and Y2 compared to the other genotypes. In addition, for Tenor different bacterial and fungal ASVs were effectively described in seed assemblages. Seed moisture cannot be involved as there was no significant differences during imbibition phase.

Therefore, because of the most probable assumption of microbiota involvement, further experiments are required to investigate further the implication of microbial assemblages in germination or extended seed phenotypes variations.

In conclusion, the microbial assemblages of B. napus seeds are mainly influenced by the environment but to a lesser extent by the plant genotype. The inheritance of some abundant bacterial and fungal ASVs also differs between genotypes. Further studies will explore more extensively the host genetic determinants driving the microbial assemblages and the impact of such assemblages on seed vigor. Observed richness (number of detected ASVs) and (E) diversity (Inverse Simpson's index)

were assessed with ITS1 ASVs for fungi. ASVs was performed with an in-house gyrB database [START_REF] Bartoli | In situ relationships between microbiota and potential pathobiota in Arabidopsis thaliana[END_REF] and the UNITE v7.1 database [START_REF] Abarenkov | The UNITE database for molecular identification of fungi -recent updates and future perspectives[END_REF], respectively. Unknown taxa represented ASVs that are not affiliated at the order level. et al., 2004, 2005;[START_REF] Wagner | From knowledge-based research towards accurate and rapid testing of seed quality in winter rape[END_REF]. Four subsamples of 25 seeds by genotype and year (Y1 and Y2) were incubated at 20°C on germination paper (GE Healthcare, type 3644) continuously moistened for 5 days and image acquisition was performed every two hours. Europe, among the 5 WOSR genetic groups. Cultivars were genotyped using 628 SNP [START_REF] Clarke | A high-density SNP genotyping array for Brassica napus and its ancestral diploid species based on optimised selection of single-locus markers in the allotetraploid genome[END_REF]. The structure of the collection was studied using ADMIXTURE [START_REF] Alexander | Fast model-based estimation of ancestry in unrelated individuals[END_REF]. Each genetic group is represented with a colour. A cultivar belongs to the group of which the colour is majority. The 9 selected genotypes of the study are marked with a yellow star. S1).

Table S2 Cropping conditions for the B. napus seed production.

Table S3 Genotype-specific bacterial and fungal ASVs for Y1.

Table S4 Genotype-specific bacterial and fungal ASVs for Y2. [START_REF] Bartoli | In situ relationships between microbiota and potential pathobiota in Arabidopsis thaliana[END_REF] and the UNITE v7.1 database [START_REF] Abarenkov | The UNITE database for molecular identification of fungi -recent updates and future perspectives[END_REF], respectively.

Unknown taxa represented ASVs that are not affiliated at the order level. et al., 2004, 2005;[START_REF] Wagner | From knowledge-based research towards accurate and rapid testing of seed quality in winter rape[END_REF]. Four subsamples of 25 seeds by genotype and year (Y1 and Y2) were incubated at 20°C on germination paper (GE Healthcare, type 3644) continuously moistened for 5 days and image acquisition was performed every two hours. [START_REF] Clarke | A high-density SNP genotyping array for Brassica napus and its ancestral diploid species based on optimised selection of single-locus markers in the allotetraploid genome[END_REF]. The structure of the collection was studied using ADMIXTURE [START_REF] Alexander | Fast model-based estimation of ancestry in unrelated individuals[END_REF]. Each genetic group is represented with a colour. A cultivar belongs to the group of which the colour is majority. The 9 selected genotypes of the study are marked with a yellow star.

Fig. 1

 1 Fig. 1 Diversity of seed-associated microbial assemblages of B. napus. Seed samples of nine

Fig. 2 Fig. 3

 23 Fig. 2 Similarities in microbial composition between B. napus genotypes seed-associated

Fig. 4

 4 Fig. 4 Generalist and specific bacterial ASVs. Prevalence of bacterial ASVs across plant

Fig. 5 Fig. 6

 56 Fig. 5 Generalist and specific fungal ASVs. Prevalence of fungal ASVs across plant genotypes

Fig. S1

 S1 Fig. S1 Distribution of the 116 genotypes representing the diversity of WOSR in Western

Fig. S2

 S2 Fig. S2 Climatic data for the cropping periods Y1 and Y2. (A) Rainfall level in mm (RR-DC) and

Fig. S4

 S4 Fig. S4 Number of bacterial (A) and fungal (B) ASVs shared between years or specific of one

Fig. S5

 S5 Fig. S5 Relationships between B. napus genetic distances and similarities of microbial

Fig. S6

 S6 Fig. S6 Different phenotyping parameters calculated for seeds of each year and genotype.

Fig. 1

 1 Fig. 1 Diversity of seed-associated microbial assemblages of B. napus. Seed samples of nine genotypes were collected during two consecutive years (Y1 and Y2). Community profiling of the seed microbiota was performed on five seed subsamples per genotype. (A) Estimated richness (Chao1 index), (B) diversity (Inverse Simpson's index) and (C) phylogenetic diversity (Faith's phylogenetic diversity index) were monitored with gyrB ASVs for bacteria. (D) Observed richness (number of detected ASVs) and (E) diversity (Inverse Simpson's index) were assessed with ITS1 ASVs for fungi. 940x529mm (72 x 72 DPI)

Fig. 4

 4 Fig. 4 Generalist and specific bacterial ASVs. Prevalence of bacterial ASVs across plant genotypes were investigated with UpSetR (Lex et al., 2014). Upper histograms represented the number of ASVs associated to one specific intersection in Y1 (A) or Y2 (B). Plant genotype associated with each particular intersection is highlighted with a dot. Intersection associated with all genotypes is displayed in red. The histogram located on the left of each UpSetR representation indicated the number of ASVs per genotype. Finally, boxplot represented the median relative abundance (log10) of bacterial ASVs associated with each intersection. 680x940mm (72 x 72 DPI)

  Fig.S1Distribution of the 116 genotypes representing the diversity of WOSR in Western Europe, among the 5 WOSR genetic groups. Cultivars were genotyped using 628 SNP[START_REF] Clarke | A high-density SNP genotyping array for Brassica napus and its ancestral diploid species based on optimised selection of single-locus markers in the allotetraploid genome[END_REF]. The structure of the collection was studied using ADMIXTURE[START_REF] Alexander | Fast model-based estimation of ancestry in unrelated individuals[END_REF]. Each genetic group is represented with a colour. A cultivar belongs to the group of which the colour is majority. The 9 selected genotypes of the study are marked with a yellow star.

  Fig. S2 Climatic data for the cropping periods Y1 and Y2. (A) Rainfall level in mm (RR-DC) and mean humidity in % (UM-MD). (B) Mean calculated temperature (TMC), minimal (TN) and maximal (TX) temperatures in °C. All data are reported for 10 days (decade) during the years 2015-2016 (Y1) and 2016-2017 (Y2) from records at Le Rheu station 35240002 -La Grande-Verrière -(48,113°N, -1,781°E) (INRA Agroclim network) 694x740mm (72 x 72 DPI)

Table 1

 1 Percentage of explained variance for beta-diversity estimators. Percentage of explained variance estimated with PERMANOVA analysis (P < 0.01) for β-diversity of bacterial (gyrB) and fungal (ITS1) assemblages.

		Year	Genotype	Year:Genotype	Genotype (Y1)	Genotype (Y2)
	gyrB	16.4	19.3	16.6	44	40.8
	ITS1	65.5	11.5	13.3	66.2	90.7

Table 2

 2 Core members of Brassica napus seed microbiota. Prevalence (expressed as a percentage) of each amplicon sequence variants (ASV) in B. napus seed samples collected in year 1 (Y1), year 2 (Y2) and in seed samples (Others) of Brassica oleracea, Brassica rapa and Raphanus sativus reported by[START_REF] Barret | Emergence Shapes the Structure of the Seed Microbiota[END_REF][START_REF] Rezki | Differences in stability of seed-associated microbial assemblages in response to invasion by phytopathogenic microorganisms[END_REF][START_REF] Rezki | Assembly of seedassociated microbial communities within and across successive plant generations[END_REF]

	ASV	Taxonomy	Y1 (%)	Y2 (%)	Others (%)
	Bacteria
	ASV00001	Pantoea agglomerans	0.0	100.0	90.6
	ASV00003	Ralstonia pickettii	0.0	100.0	0.6
	ASV00028	Pseudomonas	20.0	100.0	34.9
	ASV00038	Sphingomonas	0.0	100.0	10.6
	ASV00046	Frigoribacterium	10.0	100.0	11.1
	ASV00124	Burkholderiales	0.0	100.0	0.0
	Fungi
	ASV0001	Alternaria infectoria	100.0	100.0	72.6
	ASV0002	Cladosporium	100.0	100.0	97.6
	ASV0003	Cladosporium delicatulum	100.0	100.0	100.0
	ASV0004	Botrytis fabae	100.0	100.0	47.9
	ASV0005	Sporidiobolales sp.	100.0	100.0	60.1
	ASV0006	Alternaria infectoria	100.0	100.0	66.2
	ASV0007	Alternaria brassicae	100.0	90.0	95.4
	ASV0008	Cryptococcus victoriae	100.0	0.0	34.8
	ASV0009	Cryptococcus victoriae	100.0	0.0	37.5
	ASV0010	Cryptococcus oeirensis	100.0	0.0	37.8
	ASV0011	Filobasidium stepposum	100.0	100.0	95.4
	ASV0012	Alternaria infectoria	100.0	100.0	62.8
	ASV0013	Alternaria infectoria	0.0	100.0	47.0
	ASV0014	Bulleromyces sp.	100.0	0.0	0.3
	ASV0015	Bensingtonia sp.	100.0	90.0	58.5
	ASV0017	Erysiphe cruciferarum	100.0	20.0	12.8
	ASV0018	Cryptococcus oeirensis	0.0	100.0	0.0
	ASV0019	Udeniomyces pannonicus	60.0	100.0	14.0
	ASV0021	Cladosporium aggregatocicatricatum	100.0	100.0	47.3
	ASV0022	Alternaria	90.0	100.0	0.3
	ASV0025	Cryptococcus victoriae	0.0	100.0	0.0
	ASV0028	Cryptococcus tephrensis	100.0	90.0	32.9
	ASV0029	Cryptococcus carnescens	90.0	100.0	32.9
	ASV0030	Cladosporium perangustum	0.0	100.0	4.6
	ASV0038	Phaeosphaeria caricicola	100.0	0.0	11.9
	ASV0041	Cladosporium ramotenellum	0.0	100.0	5.2
	ASV0044	Cryptococcus victoriae	0.0	100.0	0.0
	ASV0054	Ramularia vizellae	0.0	100.0	0.9

Tables and Supporting information captions.

Table S1

 S1

Characteristics of the nine B. napus genotypes. Quality: 00 or ++ is for absence or presence of Glucosinolate-Erucic acid. WOSR groups are defined from the genetic diversity study of 116 WOSR accessions with 628 SNPs (Fig.

Table S2

 S2 Cropping conditions for the B. napus seed production.

		Distance between fields in meters
		Y0 1	Y1	Y2
	Y0	0
	Y1	170	0
	Y2	966	860	0

1 Self-pollination with cages of 1.80 m height, 0.65 m diameter, covered with a polyester veil, Lenglart, France 2 30x70 mm bags ref SM570Y, Sealed air Cryovac, France Location

of INRA fields in Le Rheu, France

		Years		Year code	GPS coordinates
		2012-2013	Y0	48°06'36.29''N 1°46'22.04''0
		2015-2016	Y1	48°06'40.86''N 1°46'26.08''0
		2016-2017	Y2	48°06'37.21''N 1°47'07.98''0
				Cropping history
	Years	Year code	WOSR Genotypes	Sowing date	Date of self-attach 2 pollination bag	Harvest date

The datasets supporting the conclusions of this article are available in the SRA database

under the accession number PRJEB31617.

ACKNOWLEDGEMENTS

We want to apologize to authors whose relevant work was not included in this article due to space constraints. The authors wish to thank Muriel Bahut from the ANAN platform (SFR Quasav) for amplicon sequencing,the Phenotic Platform for performing seed germination experiments and Thomas Baldwin for English proofreading.

Table S3 Genotype-specific bacterial and fungal ASVs for Y1.

Bacterial ASVs Fungal ASVs

Genotype ASV Taxonomy Ab (log10) ASV Taxonomy Ab (log10)


Keywords: Set-membership estimation, Kalman filter, linear systems, ellipsoidal set

This paper presents a comparison in terms of accuracy and complexity between two approaches used for state estimation of linear systems: a classic Kalman filter and a guaranteed set-membership state estimation technique. The main goal of this paper is to analyze the advantages of these techniques and to combine them in the future in a new accurate and simple extension that handles system uncertainties and chance constraints. Two academic examples illustrate the main differences between the compared techniques.

INTRODUCTION

Generally, process control requires accurate information about the plant. However, the measured variables do not totally describe the behavior of the system. Particularly, the entire system state is not always accessible. This is why it is important to get access to the unknown information using available data/knowledge. Various methods for state estimation are suggested in the literature and they can be divided into two categories. Stochastic approaches such as the Kalman Filter (see [START_REF] Kalman | A new approach to linear filtering and prediction problems[END_REF]) assume the prior knowledge of the distribution of the perturbations and the measurement noises (in general Gaussian distribution) taking into account certain characteristics like the mean and the covariance. This assumption can be sometimes unrealistic. Thus deterministic approaches [START_REF] Bertsekas | Recursive state estimation for a set-membership description of uncertainty[END_REF], [START_REF] Fogel | On the value of information in system identification-bounded noise case[END_REF]) that considers unknown but bounded perturbations and bounded noises have been elaborated. There are several deterministic approaches used for state estimation, like set-membership state estimation [START_REF] Schweppe | Recursive state estimation: Unknown but bounded errors and system inputs[END_REF]), interval observers [START_REF] Pourasghar | Comparison of set-membership and interval observer approaches for state estimation of uncertain systems[END_REF]) or robust filtering methods [START_REF] El Ghaoui | Robust filtering for discrete time systems with bounded noise and parametric unicertainity[END_REF]). In the implementation of set-based deterministic estimation methods, various sets are used: polytopes [START_REF] Walter | Exact recursive polyhedral description of the feasible parameter set for bounded-error models[END_REF]), zonotopes [START_REF] Combastel | A state bounding observer based on zonotopes[END_REF], [START_REF] Alamo | Guaranteed state estimation by zonotopes[END_REF], [START_REF] Le | Zonotopic guaranteed state estimation for uncertain systems[END_REF]), ellipsoids [START_REF] Kurzhanski | Ellipsoidal calculus for estimation and control[END_REF], [START_REF] Durieu | Multi-Input Multi-Output ellipsoidal state bounding[END_REF], [START_REF] Polyak | Ellipsoidal parameter or state estimation under model uncertainty[END_REF], [START_REF] Daryin | Reachability approaches and ellipsoidal techniques for closed-loop control of oscillating systems under uncertainty[END_REF], [START_REF] Daryin | Estimation of reachability sets for large-scale uncertain systems: from theory to computation[END_REF], [START_REF] Chernousko | State estimation for dynamic systems[END_REF]). The low complexity of ellipsoids makes them widely used compared to polytopes which offer better accuracy of the estimation. [START_REF] Combastel | Zonotopes and Kalman observers:gain optimality under distinct uncertainty paradigms and robust convergence[END_REF] recently proposed a combi-

The first author is a first year PhD student.

nation between stochastic and deterministic approaches, more exactly a zonotopic Kalman filter.

In the present paper, a comparison in terms of accuracy and computation complexity is made between two estimation techniques studied in the litterature:

PRELIMINARIES AND SETUP

Consider the following discrete-time Linear Time Invariant (LTI) system

x k+1 = Ax k + Bu k + E w w k y k = Cx k + Du k + F v v k (1)
where x k ∈ R nx is the state vector of the system, u k ∈ R nu is the input vector, and y k ∈ R ny is the measured output vector at sample time k. The matrices A, B, C, D, E w and F v have the appropriate dimensions. Here, w k ∈ R nx is a vector containing the state perturbations, while v k ∈ R ny contains the measurement noises.

Combining the state perturbations and the measurement noises in one vector ω k = [w k v k ] ∈ R nx+ny , the system (1) can be rewritten in an equivalent form

x k+1 = Ax k + Bu k + Eω k y k = Cx k + Du k + F ω k (2)
with the matrices E = E w 0 nx,ny and F = 0 ny,nx F v .

In this work, we aim to compare an estimate of the state of the system (1) provided by two approaches that are further detailed in Sections 3 and 4.

GUARANTEED ELLIPSOIDAL SET-MEMBERSHIP STATE ESTIMATION

This section briefly describes the guaranteed ellipsoidal set-membership state estimation proposed by Ben Chabane et al. (2014a) for the system (2).

In this context, consider that the initial state x 0 belongs to the ellipsoid:

E(P 0 , x 0 , ρ 0 ) = {x ∈ R nx : (x -x 0 ) P 0 (x -x 0 ) ≤ ρ 0 }
with the shape matrix P 0 = P 0 0, the center x 0 and the so called radius ρ 0 . Given an ellipsoidal estimation set for x k , with xk the nominal estimated state, the objective of this technique is to obtain an ellipsoidal set estimation for x k+1 . Figure 1 illustrates the 2-step procedure (prediction and correction) to calculate the estimation set. At each sample time k, the green set represents the predicted state set. The yellow strip represents the set of states compatible with the measurements y k+1 . The blue ellipsoid (which contains the state estimation set) overapproximates the intersection of the predicted state set and the measurement strip. Repeating the procedure at each time k leads to a guaranteed estimation set that contains the state of the system.

β,Y k ,ρ k+1 ρ k+1 subject to          βP * * 0 ρ k+1 -βρ k * P A -Y k C (P E -Y k F )ω k P 0 ρ k+1 ≤ βρ k + σ 0 < β < 1 (3) for all ω k ∈ B nx+ny , with Y k = P L k and the nominal estimated state xk+1 = Ax k + Bu k + L k (y k -C xk -Du k ).
The symbol "*" denotes symmetrical terms.

In fact, expression (3) guarantees that the system state x k+1 belongs to the ellipsoid E(P, xk+1 , ρ k+1 ).

An improvement of this method in terms of accuracy is proposed in Ben Chabane et al. ( 2014b), with the advantage that it can deal with interval uncertainties on the evolution matrix A. The main difference with respect to Ben Chabane et al. ( 2014a) is the use of the measurement y k+1 together with additional quadratic constraints on the perturbations ω k . This improvement allows us to reduce even more the size of the ellipsoidal estimated state set by solving an additional optimization problem min ρ k+1 ,P ,x k+1 ,H,τ,µi Supposing that x k+1 ∈ E(P, xk+1 , ρ k+1 ), the expression (4) offers an improved ellipsoidal state estimation set E (P , x k+1 , ρ k+1 ).

ρ k+1 subject to                                           τ P + C HC * * η 1 -τ x k+1 P η 2 - nx+ny i=1 µ i * P -P x k+1 P      0, P 0, F HF < nx+ny i=1 µ i T i , τ ≥ 0, τ < 1, ρ k+1 > τ ρ k+1 , µ i ≥ 0, i = 1, . . . , n x + n y (

KALMAN FILTER

Recall the LTI system (1) taking into consideration that w k and v k are random, independent white Gaussian noises, with the covariance matrices denoted by G w and G v , respectively. Notice that the state is a random Gaussian vector denoted by x ∼ N (x, σ) and particularly the initial state is represented by

x 0 ∼ N x 0|-1 , G 0|-1 .
The Kalman filter design is divided into two steps:

• Prediction. A previously estimated state xk-1|k-1 and the linear nominal model (without any perturbation) are used to predict the value of the next estimated state xk|k-1 as well as the state estimate covariance

G k|k-1 xk|k-1 = Ax k-1|k-1 + Bu k-1
(5)

G k|k-1 = AG k-1|k-1 A + E w G w E w (6) • Correction.
The current output measurements and the statistical properties of the model are used to correct the state estimation, leading to compute the state estimate covariance

S k = CG k|k-1 C + F v G v F v (7) K k = G k|k-1 C S -1 k (8) xk|k = xk|k-1 + K k (y k -C xk|k-1 ) (9) G k|k = (I -K k C)G k|k-1 (10)
with K k the Kalman gain and S k the innovation covariance at the sample time k.

COMPARISON

The main difference between the approaches presented in Sections 3 and 4 can be mainly spotted in terms of system modeling. 2014b) offers guaranteed bounds for the state estimation despite the presence of possible interval uncertainties on the evolution matrix A of the system (1) or (2). However, the Kalman filter offers a reduced computation complexity with respect to the considered set-membership estimation method. In fact, the Kalman equations are based on basic matrix operations and the computational complexity can be approximated by the number of multiplications per loop. Using the expressions ( 5)-( 10) and considering the worst case scenario (i.e. full matrices) we can approximate the filter computational complexity to O(N 3 ), with N = max(n x , n y ).

The computational complexity of the ellipsoidal state estimation method relies on solving a LMI optimization problem. The mincx solver of the Matlab Robust Control Toolbox is based on the interior point method [START_REF] Nesterov | Interior point polynomial methods in convex programming: Theory and applications[END_REF] which is an iterative technique solving a least square problem at each iteration. The complexity of the method in the worst case scenario can be approximated to O(m 2.75 l 1.5 ) with m the number of decision variables and l the number of constraints [START_REF] Vandenberghe | Positive definite programming[END_REF]. Notice that m = (n x + n y ) 2 + n x n y + 2 and l = 2 nx+ny + 3 for the optimization problem (3) and m = 0.5(n 2 x + n 2 y ) + 2.5n x + 1.5n y + 2 and l = n x + n y + 6 for the optimization problem (4).

The comparison allows us to conclude that the Kalman filter offers us a better result in terms of complexity, thus faster computations. In terms of accuracy, and for each iteration, the ellipsoidal method computes an ellipsoidal set to which the real state is guaranteed to belong.

The set-membership estimation setup (Section 3) offers the possibility to use correlated/uncorrelated perturbations and measurement noises, however the choice of the perturbation bounds needs good knowledge of the plant. The Kalman filter uses the assumption of Gaussian noises, which can be difficult to verify for some real plants.

Starting from this results, the aim of our future research work is to combine the advantages of the two presented techniques in order to propose an extented method that handles systems uncertainties (i.e. interval uncertainties in the system matrices) and chance constraints.

ILLUSTRATIVE EXAMPLES

Two numerical examples are considered in this section to illustrate the comparison of the presented state estimation techniques.

Example 1. Consider the following stable LTI system   

x k+1 = -0.8 0.2 -0.3 0.1 x k + -0.12 0.02 w k y k = [-2 1] x k + 0.2v k (11)
In this example, we present the results obtained by the improved guaranteeed elipsoidal set-membership state estimation (4) compared to the results obtained by Kalman filter.

In order to make a valid comparison between these two techniques, appropriate assumptions should be taken regarding the initial state, and noises. In fact, we consider that x 0 ∼ N x 0|-1 , G 0|-1 and w k ∼ N (0, 1), v k ∼ N (0, 1) for the Kalman filter. For the ellipsoidal set-membership approach, the initial state x 0 belongs to E(P 0 , x 0|-1 , ρ 0 ), and the perturbations and measurement noises are bounded, i.e. |w k | ≤ 1 and |v k | ≤ 1. Notice that x 0|-1 = [5 5] , G 0|-1 = I 2 , P 0 = 10 -9 I 2 and ρ 0 = 2 • 10 -8 .

Figure 2 and 3 show the bounds of x 1 and x 2 respectively after 10 iterations obtained by the ellipsoidal setmembership state estimation method (4) and the Kalman filter. The real state x (red asterix) is always inside the guaranteed bounds (in dashed blue) calculated by the ellipsoidal set-membership method (4). It can be noticed that, in this example, the state estimated with the Kalman filter (black asterix) has a slower convergence and it is not always inside the guaranteed bounds obtained with the improved set-membership method Ben Chabane et al.

(2014b).

Concerning the computational complexity, the classic Kalman filter takes around 0.21ms per iteration, while the set-membership estimation technique (LMIs (3) and ( 4)) spends around 9ms to determine the estimation bounds.

Example 2. The aim of using an unstable system, in which the states do not converge to 0 is to prove the efficiency of the ellipsoidal method. In this context, we consider the previous example with a different evolution matrix A, while keeping the same perturbations, noises and initial conditions Figure 4 and its zoom (Fig. 6) show the bounds of x 1 (in dashed blue) obtained with the ellipsoidal estimation method. The bounds of x 2 are illustrated in Fig. 5 and its zoom (Fig. 7). It can be noticed that the real state (red asterix) is guaranteed to be inside these bounds, while it is not the case for the classical Kalman filter (black asterix).

   x k+1 = -1.5 0.2 -0.3 0.1 x k + -0.12 0.02 ω k y k = [-2 1] x k + 0.2v k (

CONCLUSION

In this paper, a brief comparison has been made between two methodogies used for the state estimation of discretetime linear time invariant systems, subject to perturbations and measurement noises. The guaranteed ellipsoidal set-membership estimation method Ben Chabane et al.

(2014b) is compared to the classic Kalman Filter, in terms of accuracy and complexity. The best estimation results (i.e. guaranteed bounds) are obtained with the improved estimation method Ben [START_REF] Ben Chabane | Ellipsoidal state estimation for systems with interval uncertainties[END_REF]. The main advantage of the Kalman filter is its lower computational complexity. In order to take advantage of the benefits of the two proposed methodologies, future work will consist on finding a new estimation technique that guarantees high accuracy, with a small computational charge. Additionally, parametric uncertainties and chance constraints will be considered in future developments.

Fig. 1 .

 1 Fig. 1. State estimation using ellipsoids More precisely, at each time k, the radius of the ellipsoidal set is minimized by solving a Linear Matrix Inequality (LMI) problem (see Ben Chabane et al. (2014a) for more details)

  4) with η 1 = -(y k+1 +Du k+1 ) HC and η 2 = ρ k+1 -τ ρ k+1 + τ x k+1 P xk+1 + (y k+1 + Du k+1 ) H(y k+1 + Du k+1 ).

Fig. 3 .

 3 Fig. 2. Bounds of x 1

Fig. 6 .

 6 Fig. 4. Bounds of x 1

  Comparison between two state estimation techniques for linear systems D. Merhy * C. Stoica Maniu * T. Alamo * * E.F. Camacho * * S. Ben Chabane * * *

  Fig. 7. Zoom on the bounds of x 2

							Ellipsoidal method
		15					Real state
							Kalman filter
		10
	2
	Bounds of x	0 5
		-5
		-10
		7.2	7.4	7.6	7.8	8	8.2	8.4	8.6
				Sample time k


Gauthier Rousseau
email: gauthier.rousseau@supelec.fr

Cristina Stoica Maniu
email: cristina.stoica@supelec.fr

Sihem Tebbani
email: sihem.tebbani@supelec.fr

Mathieu Babel
email: mathieu.babel@parrot.com

Impact of propellers inertia and asymmetries on a V-shaped quadrotor

Keywords:

  L'archive ouverte pluridisciplinaire

INTRODUCTION

The use of quadrotors has dramaticaly increased during the last decade, with the rise of consumer quadrotors. These drones notably have applications in video making, but also in industries, such as construction sites management, agriculture monitoring or infrastructure inspection.

The most common quadrotor configuration is composed of four fan propellers, symmetrically disposed around the center of mass of the drone, alternating clockwise and anticlockwise directions of rotation, and all pointing toward the vertical axis of the drone. This configuration is often refered to as an 'X4' configuration. The modelling and control of X4 quadrotors has been intensively studied, in such works as [START_REF] Shapovalov | Research of the controlled flight dynamics based on the full and simplified quadrotor models[END_REF], [START_REF] Hamel | Dynamic modelling and configuration stabilization for an X4-flyer[END_REF], [START_REF] Bangura | Real-time model predictive control for quadrotors[END_REF], Pounds et al. (2006), etc. With four propellers to control their six degrees of freedom (DOF) of motion, quadrotors constitute underactuated systems. Consequently, these 6DOF cannot be controled independently. Nevertheless, the actions of the propellers on the drone can be intuitively decoupled on an X4 quadrotor, which constitutes one advantage of this configuration.

However, with such a configuration, the dynamics on the yaw axis is usually inferior to those on the roll and pitch axes. This is due to the rotation around the yaw axis being mostly controlled using the propellers drag torques, while their lift is used for pitch and roll control, through the leverage it induces. Depending on the configuration, the latter tends to be 10 to 100 times stronger than the drag torques, resulting in weaker dynamics on the yaw axis.

A solution to balance these dynamics is to tilt the propellers axes in order to have a leverage induced by the lift on the yaw axis, too. This can be achieved by actuating the orientation of the propellers, as in Moutinho et al. The first author is a first year PhD student. This work was supported by Parrot Drones.

(2015), or by using non-X4 configurations. In [START_REF] Hossain | Design and development of an y4 copter control system[END_REF], an Y4 ('V-tail') configuration is studied, with the rear propellers tilted in a 'V' shape. Studies on a Lynxmotion Hunter V400 frame, a popular V-tail frame among hobbyists, has also been lead recently in [START_REF] Bellocchio | Modelling and simulation of a quadrotor in V-tail configuration[END_REF].

Furthermore, when adding payloads on the drone, the center of mass could be moved and the action of the different propellers could become asymmetric. Such an asymmetry can also be a matter of design, for instance, in order to achieve a better aerodynamic profile.

The main contribution of this paper consists in analyzing the effects of the propellers orientation and asymmetries on a V-shaped quadrotor. More specifically, we highlight a nonminimum phase behaviour on the pitch dynamics due to the action of propellers inertia, appearing when the V-angle increases. In addition, for asymmetrical drones, coupling terms can arise between the different axes.

Notation. In the sequel, s x , c x , and t x are used to denote sin x, cos x, and tan x, respectively.

QUADROTOR MODELLING

We consider the movement of a quadrotor equipped with four propellers in the inertial reference frame R W = (O, x W , y W , z W ). We choose R B = (G, x B , y B , z B ) as the body-fixed frame, with G the center of mass of the quadrotor (Fig. 1). We assume z W = z B in case of hovering without perturbation. The quadrotor and its propellers are supposed to be rigid. The position of the center of mass in the inertial frame is given by r = x x W + y y W + z z W , and its speed in the inertial frame is denoted by ṙ

= ẋ x W + ẏ y W + ż z W = u x B + v y B + w z B .
We use Euler angles (ϕ, θ, ψ) -convention ZY X -to describe the attitude of the drone, and thus we define the rotation matrix R, between R W and R B

ẋ ẏ ż = c θ c ψ s ϕ s θ c ψ -c ϕ s ψ c ϕ s θ c ψ + s ϕ s ψ c θ s ψ s ϕ s θ s ψ + c ϕ c ψ c ϕ s θ s ψ -s ϕ c ψ -s θ s ϕ c θ c ϕ c θ R u v w
Denoting by Ω B/W = p x B +q y B +r z B the rotation speed vector of the drone in the inertial frame, the following expression holds

φ θ ψ = 1 sϕt θ cϕt θ 0 cϕ -sϕ 0 sϕ/c θ cϕ/c θ p q r (1)
For small ϕ and θ angles, (1) is equivalent to φ θ ψ ≈ (p q r). Finally, let m be the total mass of the quadrotor and J = diag(J x , J y , J z ) its inertia tensor in R B .

We will next consider the propellers, the gravity and the body drag influence in order to derive a linear model for the system dynamics near the hovering equilibrium.

Propellers. The drone is equipped with four propellers, (p 1 , p 2 , p 3 , p 4 ). Denoting by P i the center of mass of the propeller p i , its position in the body-fixed frame R B is --→ GP i = l xi x B +l y i y B +l z i z B . The rotation speed vector of the propeller p i in R B is Ω pi/B = ω i z pi , with ω i > 0. The orientation of z pi in R B is given by the angles ψ i ∈ [-π, π] and θ i ∈ [0, π], as illustrated in Fig. 2,

z pi = s θi c ψi x W + s θi s ψi y W + c θi z W .
A quadratic model is chosen for the lift force T i and drag torque Γ i applied on the i-th propeller

T i = α i ω 2 i z pi , Γ i = β i ω 2 i z pi (2
) with α i > 0 or α i < 0 depending on the orientation of the propeller in R B , and β i < 0. We can linearize (2) near the hovering equilibrium state by defining ω i = ω hi +δω i , with ω hi the rotation speed of the i-th propeller when hovering (without any perturbation such as wind or ground effect). This allows approaching

T i • z pi ≈ T 0i + a i δω i , Γ i • z pi ≈ Γ 0i + b i δω i with T 0i = α i ω h 2 i , a i = 2α i ω hi , Γ 0i = β i ω h 2 i , b i = 2β i ω hi .
The leverage induced by propellers lift is then

Γ li = --→ GP i × T i
Let us denote by J p i the inertia of the i-th propeller with respect to its rotation axis z pi . The reaction torque applied on the quadrotor via the i-th propeller is given by Γ ri = -J p i ωi z pi We neglect the gyroscopic torques generated by the propellers, as they are much smaller than those induced by the propellers drag and lift for small rotation speeds of the drone. Since the quadrotor has four propellers, the total force applied by the propellers on the drone is 4 i=1 T i , and the total torque is

4 i=1 (Γ i + Γ li + Γ ri ).
Gravity. The gravity action on the drone is modeled by a constant force f g = mg z W . For small angles, this expression becomes

f g ≈ -mg θ x B + mg ϕ y B + mg z B ( 3
)
Body drag. We can add a simplified linear model of the aerodynamic drag of the quadrotor body around the hovering equilibrium state, as the combination of a force f d and a torque τ d , given by

f d = -C x u x B -C y v y B -C z w z B τ d = -C p p x B -C q q y B -C r
r z B Thus, with small but constant angles, the drone model reaches a terminal velocity instead of accelerating indefinitely. Close to the hovering equilibrium, the action of body drag torques is reduced, since the rotation speed of the drone is supposed to be small anyway.

Linear Model. Applying the fundamental principle of dynamics leads to

mr = f g + f d + 4 i=1 T i J ΩB/W = τ d + 4 i=1 (Γ i + Γ li + Γ ri ) -Ω B/W ×JΩ B/W
(4) At the hovering equilibrium, with (3), the following expression holds

0 = mg z B + 4 i=1 T 0i z pi 0 = 4 i=1 Γ 0i z pi (5)
For small angular speeds of the quadrotor, the expression (4) can be linearized by neglecting the term

Ω B/W × JΩ B/W . Using (5), it comes m u = -mg θ -C x u + u u J x ṗ = -C p p + u p m v = mg ϕ -C y v + u v J y q = -C q q + u q m ẇ = -C z w + u w J z ṙ = -C r r + u r (6)
where

(u u u v u w u p u q u r ) = B ω→u • (δω 1 δω 2 δω 3 δω 4 ) + B ω→u •(δ ω1 δ ω2 δ ω3 δ ω4 ) , with B ω→u = (B 1 B 2 B 3 B 4 ), B ω→u = (B •1 B •2 B •3 B •4
), and

B i =        a i s θ i c ψ i a i s θ i s ψ i a i c θ i b i s θ i c ψ i + a i ly i c θ i -lz i s θ i s ψ i b i s θ i s ψ i + a i lz i s θ i c ψ i -lx i c θ i b i c θ i + a i s θ i lx i s ψ i -ly i c ψ i        , B •i =       0 0 0 -Jp i s θ i c ψ i -Jp i s θ i s ψ i -Jp i c θ i       (7)
Ideally, each of the 6DOF of the quadrotor would be decoupled and controlled independently. However, with only 4 propellers, the drone is underactuated and this is not possible. Nevertheless, it is usually possible to decouple the action of the propellers on the three rotation axes and the z translation axis. In this context, let us define the 4 × 4 mixing matrix B ω→v = (0 4,2 I 4 ) B ω→u and its inverse, the unmixing matrix

B v→ω = (B ω→v ) -1
A way to decouple w, p, q, and r is to introduce the decoupled control signals (v w v p v q v r ) such as

(u u u v u w u p u q u r ) = B ω→u • B v→ω • (v w v p v q v r ) + B ω→u • B v→ω • ( vw
  ψ 1 θ 1 lx 1 ly 1 lz 1 a 1 b 1 Jp 1 ψ 2 θ 2 lx 2 ly 2 lz 2 a 2 b 2 Jp 2 ψ 3 θ 3 lx 3 ly 3 lz 3 a 3 b 3 Jp 3 ψ 4 θ 4 lx 4 ly 4 lz 4 a 4 b 4 Jp 4   =   0 π lx -ly 0 a -b Jp 0 0 lx ly 0 -a -b Jp 0 π -lx ly 0 a -b Jp 0 0 -lx -ly 0 -a -b Jp   (8)
with l x > 0, l y > 0, a > 0, and b > 0. We then find the usual expressions

Bω→u =       0 0 0 0 0 0 0 0 -a -a -a -a lya -lya -lya lya lxa lxa -lxa -lxa b -b b -b       , B ω→u =       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Jp -Jp Jp -Jp      
The propellers action on the x and y translation axes are null and their action on the z translation axis and the three rotation axes can be fully decoupled:

B ω→u • B v→ω = 0 2,4 I 4 , B ω→u • B v→ω = 0 5,3 0 5,1 0 1,3 Jp b
. Then, we can write

u u (s) = 0, u v (s) = 0, u w (s) = v w (s), u p (s) = v p (s), u q (s) = v q (s), u r (s) = (1 + Jp b s )v r (s),
(9) with s the Laplace variable. To simplify the notation, the dependence on the s variable of the input/output signals is further omitted. Finally, using (6) and ( 9), we can write the following transfer functions

u = -mg s (Cq+Jy s )(Cx+m s ) vq, w = 1 Cz +m s vw, v = mg s (Cp+Jx s )(Cy+m s ) vp, r = 1+ Jp b s
Cr +Jz s vr

Propellers inertia induces a zero in the yaw dynamics. The latter is usually significant, and allows us to reduce and to smooth the control signals on this axis during dynamic flight phases.

V4 Quadrotor. On an X4 configuration, the yaw axis is controlled using the propellers drag and the reaction torques. These tend to be one or more orders of magnitude inferior to the torques induced by the propellers lift on the roll and pitch axes. One way to improve a quadrotor dynamics on its yaw axis is then to tilt its propellers in a V-shape, as illustrated on Fig. 3. Sometimes only the rear propellers are tilted (called a V-Tail or Y4 configuration). In the present paper, we suppose that all the four propellers are tilted the same way (V4 configuration). In this context, we choose the same parameters as (8) with the difference (ψ

1 ψ 2 ψ 3 ψ 4 ) = π 2 π 2 -π 2 -π 2 , (θ 1 θ 2 θ 3 θ 4 ) = (π -ϑ ϑ π -ϑ ϑ) and (l z 1 l z 2 l z 3 l z 4 ) = (l z l z l z l z ), with ϑ ∈ 0, π
2 . Such a configuration leads to A part of the propellers lift is now added to their drag torques on the yaw axis, improving the dynamics of the drone on this axis. However, these drag torques also appear on the pitch axis now, and counter a part of the propellers lift, degrading the dynamics on this axis.

Inversing the rotation direction of each propeller (by changing the angles ψ i and θ i ) leads to

Bω→u =     0 0 0 0 as ϑ -as ϑ -as ϑ as ϑ -ac ϑ -ac ϑ -ac ϑ -ac ϑ a ly c ϑ -lz s ϑ -a ly c ϑ -lz s ϑ -a ly c ϑ -lz s ϑ a ly c ϑ -lz s ϑ bs ϑ + alxc ϑ bs ϑ + alxc ϑ -bs ϑ -alxc ϑ -bs ϑ -alxc ϑ -bc ϑ + alxs ϑ bc ϑ -alxs ϑ -bc ϑ + alxs ϑ bc ϑ -alxs ϑ    
The propellers lift now counters the propellers drag on the yaw axis, degrading the yaw dynamics. Since the propellers have been tilted in order to increase the yaw dynamics, this solution is not pertinent here. For this reason, hereafter, we will keep the directions of rotation as shown on Fig. 1 for the propellers.

Using the expressions

B ω→u • B v→ω =   0 0 0 0 0 K pv 0 0 I 4   B ω→u • B v→ω =   0 4,4 0 0 τ q 0 0 0 0 τ r   (10) with K pv = s ϑ lyc ϑ -lzs ϑ , τ q = Jps ϑ bs ϑ -alxc ϑ , τ r = Jpc ϑ
bc ϑ +alxs ϑ , the following transfer functions are obtained

u u = 0, u v = K pv v p , u w = v w , u p = v p , u q = (1 + τ q s )v q , u r = (1 + τ r s )v r
The differences with respect to (9) are the following: u v and u p are now coupled and a zero appears on the pitch axis, as a part of the torques induced by the propeller inertia. For realistic ϑ angles and a, b, and l x coefficients, the propellers lift action on the pitch axis is still much stronger than the action of the propellers drag torques, and bs ϑ < al x c ϑ . This zero on the pitch axis is usually a positive real scalar, inducing a nonminimum phase behaviour on the pitch axis. However, this is usually a high frequency zero that only affects the very beginning of the transient response of the pitch axis (see Section. 4).

Finally, using (6), we can write the next transfer functions It can be noticed that zeros are introduced in the transfer functions of the speeds u and v, in comparison to the X4 configuration.

Asymmetrical Quadrotor. Due to payloads, aerodynamic design or other reasons, industrial quadrotors can have asymmetric configurations. For instance, the distance || --→ GP i || can differ from one propeller to another, or the propellers themselves can be different. We study here a drone which has a front/back asymmetry, but the same reasoning can be applied to other configurations. The difference with ( 8) is (upperscript f for front, b for back)

  lx 1 ly 1 a 1 b 1 Jp 1 lx 2 ly 2 a 2 b 2 Jp 2 lx 3 ly 3 a 3 b 3 Jp 3 lx 4 ly 4 a 4 b 4 Jp 4   =    l f x -l f y a f -b f J f p l f x l f y -a f -b f J f p -l b x l b y a b -b b J b p -l b x -l b y -a b -b b J b p    (11)
Such a configuration leads to

B ω→u • B v→ω = 0 2,4 I 4 B ω→u • B v→ω = 0 5,4 0 τ pr 0 τ rr with τ pr = b b J f p -b f J b p a f b b l f y +a b b f l b y , τ rr = a f l f y J b p +a b l b y J f p a f b b l f y +a b b f l b y
. Depending on the configuration, propellers inertia can now couple the roll and yaw axes during dynamic flight phases, i.e.

u w = v w , u p = v p , u q = v q , u r = (1 + τ rr s )v r + τ pr s v p
For the same set of parameters as in ( 11), with the difference (ψ

1 ψ 2 ψ 3 ψ 4 ) = π 2 π 2 -π 2 -π 2 , (θ 1 θ 2 θ 3 θ 4 ) = π -ϑ f ϑ f π -ϑ b ϑ b , (l z 1 l z 2 l z 3 l z 4 ) = l f z l f z l b z l b z , i.
e. a V-shaped quadrotor with front/back asymmetries, we can show that

B ω→u • B v→ω =   0 0 0 0 0 K pv 0 K rv I 4   B ω→u • B v→ω =   0 4,4 τ wq 0 τ qq 0 0 τ pr 0 τ rr   (12)
with K pv , K pv , τ wq , τ pr having a null/positive/negative value, depending on the configuration, τ qq usually negative as previously seen and inducing a positive zero in the pitch dynamics, and τ rr usually positive. This leads to

u v = K pv v p + K rv v r , u w = v w , u p = v p , u q = (1 + τ qq s )v q + τ wq s v w , u r = (1 + τ rr s )v r + τ pr s v p
The transfer functions of the linearized model ( 6) become

u = -mg 1 (Cq + Jy s )(Cx + m s ) (1 + τqq s ) s vq -τwqvw v = mg + KpvCp s + KpvJx s 2 s (Cp + Jx s )(Cy + m s ) vp + Krv Cy + m s vr w = 1 Cz + m s vw, r = 1 + τrr s Cr + Jz s vr + τpr s Cr + Jz s vp
The front/back asymmetry can then couple the roll and yaw dynamics, and the pitch and vertical acceleration dynamics. Up to two additional zeros can be introduced in the pitch and yaw dynamics depending on the configuration. These real zeros can be positive or negative, depending on the configuration.

ω h |δω i | -v w step |δω i | -v q step |δω i | -v r step
Fig. 4. Propellers speed at the hovering equilibrium state ω h , and steady state propellers speed control signals δω i in response to v w , v q , and v r steps, for different V angles of a given symmetric V4 quadrotor.

SIMULATION RESULTS

Motor model. For the following simulation study, each brushless motor is controlled by a rotation speed controller which is assumed to confer the motors a second order lowpass filter behaviour near the hovering equilibrium state

δ ωi + 2ξ i ω 0i δ ωi + ω 0 2 i δω i = ω 0 2 i u ωi
We also assume that the control law is designed to give the same dynamics to each motor, regardless the asymmetry of the drone. For the following simulations, these second order filters have a damping factor ξ i close to 1.

Case 1: V4 quadrotor simulation. A simulation of the linear model was performed for a symmetric quadrotor in V4 configuration, for different values of the V-angle ϑ, and with a realistic set of parameters (mass, inertia, thrust and drag coefficients, || --→ GP i || distances). For a V-angle of 10 • on each propeller, the required propellers rotation speeds δω i to achieve a given torque on the yaw axis was divided by two in steady state (see Fig. 4). In the mean time, those required to obtain a given torque on the pitch axis and a given force on the z axis were nearly unchanged (less than 5% higher), up to a ϑ angle of 20 • ( ∼ 10% higher). Fig. 5 shows the evolution of the total propellers torque on the pitch axis u q in response to a step of decoupled control torque v q . The impact of the V-angle on a PIDcontrolled pitch angle closed-loop was also simulated. The PID controller was tuned for a 0 • V-angle, and its robustness regarding the positive real zero introduced by the V-angle is presented Fig. 6. As expected, the bigger the V-angle ϑ, the stronger the nonminimum phase behaviour in the pitch dynamics. However, the step response of u q /v q typically reaches its minimum in less than 10ms (Fig. 5), and the overshoot increases by 20% for a 40 • V-angle on Fig. 6, and could be reduced by retuning the PID controller. Hence, the nonminimum phase behavior on the pitch axis is hardly visible for an angle ϑ that could reasonably be encountered on a potential industrial quadrotor.

In the mean time, Fig. 7 illustrates that as ϑ increases, the impact of propellers inertia dramatically decreases on the yaw axis (∼ halved for ϑ = 20 • ). As a consequence, if the V4 configuration improves the static gain between propellers speeds and propellers torques on z B , it could degrade the bandwidth on the yaw axis.

The impact of the coupling gain K pv in (10) is illustrated by Fig. 8. The force generated by the term mg ϕ (in blue solid line) behaves more or less as a double integrator regarding v p (and does not depend on ϑ). The dashed red line stands for the evolution of the term K pv v p , consisting in a simple static gain. Only the motor dynamics remains (also present in the term mg ϕ), giving here a second order low-pass filter behavior. The red line represents the sum of these two terms, and can be compared to the blue line to emphasize the extra dynamics added by the term K pv v p on the V4 configuration.

Consequently, for a quadrotor designed to achieve high velocities along x B , it could be more interesting to tilt the propellers around y B , instead of x B . This way, the roll dynamics would be degraded, leading to a nonminimum phase behaviour, in favour of a better yaw dynamics, while the pitch dynamics would have a slightly higher bandwidth due to a coupling term between v q and u u . Case 2: Asymmetrical V-shaped quadrotor simulation. The model was also simulated in the case of V-shaped quadrotors containing front/back asymmetries. Several 'realistic' configurations were simulated, including a 3:1 ratio in terms of || --→ GP i || distances or propellers properties between the front and the back of the quadrotor, as well as V-tail configurations. The simulations confirmed the expected additional coupling terms brought by the asymmetries (see Section 5). However, their impact on the closed loop system was negligible most of the time. Only the term K rv in (12) appeared significant for some V-tail configurations.

CONCLUSION

An X4 quadrotor can profit from propellers inertia to reduce and smooth yaw control signals during dynamic flight phases. However, using a simplified linear model of the quadrotor, we highlighted that, for a V-shape configuration, these inertia confer a nonminimum phase behaviour to the pitch dynamics. This phenomenon may require caution for controllers stability on this axis. Nevertheless, simulations showed that for reasonable V angles, this behaviour can be neglected. Coupling terms were also highlighted when asymmetries arise between the propellers, especially in the case of V-tail configurations, for which yaw and lateral translation can be highly coupled.

In future work, a more profound study will be conducted on the nonlinear MIMO dynamics of the quadrotor, in order to better measure the impact of these coupling terms and additional zeros. Tests on a real system will be carried out in order to experimentally validate the results obtained in the simulation study.

In particular, far from the hovering equilibrium, not only the linearizations conducted Section 2 are not valid anymore, but much more complicated phenomena become significant. As described in [START_REF] Bristeau | The role of propeller aerodynamics in the model of a quadrotor UAV[END_REF] or [START_REF] Huang | Aerodynamics and control of autonomous quadrotor helicopters in aggressive maneuvering[END_REF], the propellers thrust can significantly deviate from their rotation axis due to aerodynamic effects. The drone body itself can also deform when experiencing high mechanical stress, modifying the orientation of the propellers.

In this context, there may be situations where, while the propellers are indeed oriented in a V-shape, their thrusts are not, or the effects of the V-shape become negligible regarding, for instance, other phenomena.

Fig

  Fig. 1. Inertial & bodyfixed frames Fig. 2. Position & orientation of propeller i

  vp vq vr ) 3. X4 AND V4 CONFIGURATIONS X4 Quadrotor. A quadrotor is in X4 configuration when its two plans (G, x B , z B ) and (G, y B , z B ) are symmetry plans and the 4 propellers are oriented toward ±z B . For an X4 configuration as shown on Fig. 1, we typically have the following values for the system parameters

Fig

  Fig. 3. A V-shaped quadrotor

  u = -mg 1+τq s s (Cq+Jy s )(Cx+m s ) vq, w = 1 Cz +m s vw, v = mg+Kpv Cp s +Kpv Jx s 2 s (Cp+Jx s )(Cy+m s ) vp, r = 1+τr s Cr +Jz s vr

  Fig. 5. uq vq step response