Dr Stefan Schiller Zbsa

Keywords: biomaterials, biopolymer, biocompatibility, polyesters, tissue engineering, surfaces

Cell compatibility of biomaterials is one of the most important issues in Tissue Engineering. It is affected by the catalyst, the chemical composition and the surface topography of the material. Our aim was to study the influence of these factors for three standard FDA-approved biopolymers on the proliferation of preadipocytes. The analyzed materials were polyesters, which were synthesized by ring-opening polymerization. Tin(II)octanoate is the standard catalysts for the ring-opening polymerization (ROP); however, considered to be cytotoxic. Therefore, we compare its compatibility with iron catalysts. Our study suggests that neither polymers nor catalysts or mechanical properties have a significant impact on cell compatibility in short term 2D cell cultures in vitro. In contrast, nanoroughness can influence how cells attach and spread in a material, showing an influence on preadipocyte proliferation.

Higher nanoroughness leads to higher cell compatibility, which becomes more pronounced with time.

Introduction

Cell compatibility is one of the most important preconditions for the use of biopolymers in tissue engineering. Biopolymers can be divided into natural biopolymers like collagen and synthetic biopolymers. Among the synthetic biopolymers, poly(ε-caprolactone), poly(lactide) and poly(glycolide) are the most commonly used in tissue engineering since they are FDAapproved while poly(lactide) has also found wide application in daily products. In addition, they span a wide range of degradation times and elastic properties. Moreover, their different degradation times allow to adapt their use in many different applications: from fast degrading sutures made of poly(glycolide) to slow degrading contraceptive devices made of poly(εcaprolactone). [START_REF] Gunatillake | Biodegradable Synthetic Polymers for Tissue Engineering[END_REF][START_REF] Woodruff | The return of a forgotten polymer-Polycaprolactone in the 21st century[END_REF] Even though these biopolymers found widespread applications in the biomedical filed, their diversity in terms of their chemical nature (homo-or copolymers), chemicals used during synthesis (e.g. catalysts), as well as material properties resulting from polymer processing (e.g. surface topography/roughness) create a vast parameter range impacting cells, which has not yet been analyzed in concert to verify if critical differences occur and which limitations or freedom for material choices may exist (Scheme 1). Scheme 1: Biopolymers parameter space impacting cell compatibility: composition, catalyst used in ring-opening polymerization and surface topography can affect cells -in this study preadipocytes were used. On the lower left the three biopolymer constituting monomeric building blocks used for biopolymer synthesis are depicted: ε-caprolacton,, D,L-lactic acid and glycolide. The latter were used to prepare poly(DL-lactide) and poly(ε-caprolactone) homopolymers, as well as all copolymers combinations of the three monomers to binary copolymers and the ternary copolymer comprising all three monomers (see Table 1 &2).

Synthetic biopolymers are usually polymerized with tin(II)octanoate as a catalyst. However, this catalyst is considered to be cytotoxic. [START_REF] Tanzi | Cytotoxicity of some catalysts commonly used in the synthesis of copolymers for biomedical use[END_REF] As a consequence, other catalysts like rare earth metals, iron, zinc or organo-catalytic systems or enzymes were examined. [START_REF] Arbaoui | Metal catalysts for ε-caprolactone polymerisation[END_REF][START_REF] Hege | Non-toxic catalysts for ring-opening polymerizations of biodegradable polymers at room temperature for biohybrid materials[END_REF][START_REF] Gowda | Zinc acetate as a catalyst for the bulk ring opening polymerization of cyclic esters and lactide[END_REF][START_REF] Dove | Organic Catalysis for Ring-Opening Polymerization[END_REF][START_REF] Kobayashi | Enzymatic ring-opening polymerization and polycondensation for the green synthesis of polyesters[END_REF] Many studies compare the cell compatibility of different polymers , [START_REF] Kanitkar | Synthesis of novel polyesters for potential applications in skin tissue engineering[END_REF][START_REF] Pariente | In vitro biocompatibility evaluation of naturally derived and synthetic biomaterials using normal human bladder smooth muscle cells[END_REF][START_REF] Sliedregt | In vitro biocompatibility testing of polylactides Part I Proliferation of different cell types[END_REF][START_REF] Rogero | Biocompatibility study of polymeric biomaterials[END_REF] but to our knowledge there are only a few studies which examine the cell compatibility of the catalysts, too. [START_REF] Tanzi | Cytotoxicity of some catalysts commonly used in the synthesis of copolymers for biomedical use[END_REF][START_REF] Schappacher | Comparative in vitro cytotoxicity toward human osteoprogenitor cells of polycaprolactones synthesized from various metallic initiators[END_REF] Especially there are nearly no studies with a combined scope of the effect of surface topography and polymer identity/composition on cell proliferation. The choice for preadipocytes as cellular test system scan derives from their large potential in soft tissue engineering because of their potential to differentiate into different tissues. However, to the best of our knowledge there is no study analyzing cell compatibility of a combined approach including polymer identity/composition, catalyst and surface topography towards these cells.

Our main objective is to develop favourable scaffolds and growth chambers for soft tissue engineering composed of synthetic biopolymers. To achieve this goal, it is important to analyze the impact of polymers and catalysts towards cell compatibility. As it is hardly possible to completely remove tin(II)octanoate from the polymer, a potential influence towards cell compatibility has to be assumed. [START_REF] Stjerndahl | Industrial utilization of tin-initiated resorbable polymers: synthesis on a large scale with a low amount of initiator residue[END_REF] Schappacher et al. analysed the cell compatibility of several catalysts, by examining samples made with different catalysts with the finding that none of them were toxic. [START_REF] Schappacher | Comparative in vitro cytotoxicity toward human osteoprogenitor cells of polycaprolactones synthesized from various metallic initiators[END_REF] However Tanzi et al. analyzed four different catalyst including tin(II)octanoate and found that this is the second most toxic catalyst among them. [START_REF] Tanzi | Cytotoxicity of some catalysts commonly used in the synthesis of copolymers for biomedical use[END_REF] Until now there is no study, which compares tinand iron catalysts.

Concerning the cell compatibility of polymers, the current studies can be divided into in vitro and in vivo analyses or both simultaneously. In vitro studies do not show great differences between the cytocompatibility of different biopolymers. [START_REF] Pariente | In vitro biocompatibility evaluation of naturally derived and synthetic biomaterials using normal human bladder smooth muscle cells[END_REF][START_REF] Sliedregt | In vitro biocompatibility testing of polylactides Part I Proliferation of different cell types[END_REF][START_REF] Rogero | Biocompatibility study of polymeric biomaterials[END_REF] Brackett et al. analyzed endodontic materials for 6 weeks including poly(ε-caprolactone), indicating that poly(εcaprolactone) is non-cytotoxic, and was one of the best materials tested. [16] In the case of clinical application of poly(lactide), swellings could be observed three years after operation, induced by the degrading polymer. [17] Kobayashi et al. performed toxicity tests using implanted biodegradable polymers in rabbit cornea and found that biodegradable cyanoacrylates cause severe inflammation. [START_REF] Robert | Biocompatibility and resorbability of a polylactic acid membrane for periodontal guided tissue regeneration[END_REF] Other in vivo tests were conducted by implantation of biopolymers into animals: the implants were analyzed using different methods, most importantly histologically. Investigated biopolymers such as PLA mainly showed good biocompatibility. [START_REF] Gourlay | Biocompatibility testing of polymers: In vivo implantation studies[END_REF][START_REF] Kunzler | Systematic study of osteoblast and fibroblast response to roughness by means of surface-morphology gradients[END_REF] Until now there is no study, which analyzes the cell compatibility of different types of polymers with preadipocytes.

The effect of the substrate topography was investigated in cell-culture experiments. For example, it was shown that osteoblasts prefer surfaces with micro-roughness in contrast to fibroblasts. [START_REF] Richards | The effect of surface roughness on fibroblast adhesion in vitro[END_REF][START_REF] Chung | Enhancing growth and proliferation of human gingival fibroblasts on chitosan grafted poly (\epsilon-caprolactone) films is influenced by nano-roughness chitosan surfaces[END_REF] However, if surfaces with different nanoroughnesses are compared, the fibroblasts prefer rougher surfaces. [START_REF] Chung | Enhancing growth and proliferation of human gingival fibroblasts on chitosan grafted poly (\epsilon-caprolactone) films is influenced by nano-roughness chitosan surfaces[END_REF] For osteoblasts a combination of both nanoroughness and microroughness enhances differentiation. [START_REF] Gittens | The effects of combined micron-/submicron-scale surface roughness and nanoscale features on cell proliferation and differentiation[END_REF] Neither the effects of nanoroughness nor the effects of microroughness on preadipocytes are described in the literature yet.

Due to their importance in soft tissue engineering, preadipocytes were chosen for our study.

In this paper, we analyzed the influences of the following factors on preadipocyte compatibility: biopolymer type, catalyst, and (nano)roughness. Although several previous studies reported in the literature, examine one of the mentioned issues, as far as we know the present study is the first one that addresses all three aspects at once. We examined different polymers and copolymers made of ε-caprolactone, DL-lactide and glycolide. While their influence was already analyzed towards other cells, it was not yet analyzed towards preadipocytes.

1.

Experimental Section

Materials

Iron(II)bromide anhydrous +98% was purchased from Alfa Aesar and used as received. DLlactide and benzyl alcohol were purchased from Sigma-Aldrich and used as received. Iron(III) perchlorate from Aldrich was dried in vacuum prior to use and stored in a baked-out Schlenkflask under nitrogen. e-Caprolactone was dried over CaH2, distilled two-times and stored over molecular sieve and under nitrogen atmosphere in a baked-out Schlenk-flask. Chloroform from VWR was used as received. Methanol (VWR) was used as technical grade.

Preparation of Polymers

The polymers were synthetized by Ring-Opening-Polymerization (ROP).

A 25-50 ml flask was baked out three times and flushed with nitrogen. The solid chemicals were weighed on a weighing boat and added to the flask. The liquid monomers were continuously kept under nitrogen atmosphere. They were added with Eppendorf microliter pipettes. Last the initiator was added. Usually the ratio (2000:1:2, M:C: I) was used; however, if the molecular weight was not high enough with this ratio, a ratio of (2000:1:5) was used instead.

For example, to a solution of 6.6 ml (6.85 g, 0.06 mol) ε-caprolactone, 9.7 µl (12.15 mg, 30 µmol) Sn(octanoate)2 and 6.2 µl (6.5 mg, 60 µmol) benzyl alcohol were added. For other polymers or copolymers, the same polymerization scheme was used. The flask was sealed, evaporated and flushed with nitrogen three times. The reaction was stirred at140°C in an oil bath.

The raw product was dissolved in chloroform and poured into cold methanol in 50 ml falcon tubes. (3 ml of polymer solution to 25 ml of cold methanol). The falcon tube was centrifuged for 15 min at 12,000 rpm. Methanol was decanted and the polymer dissolved in chloroform.

The chloroform was evaporated and the polymer dried until a constant weight was reached.

Polymer chain lengths were characterized with PSS GPC SeCcurity and polystyrol standards.

NMR spectra of yielded polymers were analyzed using a Bruker AVII-300 MHz.

2D-Spin-Coats

Plasma-cleaned glass-slides

The glass slides (VWR, 3 cm diameter, ECN 631-1585) were cleaned with acetone and a lintfree paper (VWR, Kimberly Clark, Kimcare). Then the cleaning process was continued by exposure to oxygen plasma (Femto 100 W, Diener electronic, Germany) for 300 s.

A 2% polymer solution in chloroform was prepared. The glass slides were spin-coated (Spincoater Model P6700, Speciality Coating System Inc.) at 200 rpm for 5 s and then 12000 rpm for 20 s. A volume of 120 µl to 150 µl of polymer solution was used, adjusted to the viscosity of the polymer solution.

Functionalized glass slides

To verify, if the adhesion of the polymers to the glass slides can be enhanced, the glass slides were functionalized with silanes. The glass slides were cleaned with acetone and a lint-free paper (VWR, Kimberly Clark, Kimcare) and afterwards cleaned with Caro's acid (sulfuric acid with hydrogen peroxide) for 30 minutes. The glass slides were rinsed four times with distilled water and dried with compressed air followed by silanization for 2 hours with toluene:hexamethyldisilane (4:1). After this step, they were rinsed 4 times with distilled water and dried with compressed air. The spin-coating process was the same as described for plasma-cleaned glass-slides. The films were spincoated using the spin coater SCC 200 KLM with a volume of 120 µl to 150 µl of polymer solution, adjusted to the viscosity of the polymer solution.

Mechanical Measurements

Topography and film thickness were investigated by Atomic Force Microscopy (AFM) in intermittent contact mode in air with an AFM5500 (Agilent). Image processing and film thickness measurements were performed with Gwyddion software. All AFM force curves were taken in contact mode with a Z-closed loop system. Spring constants of silicon rectangular cantilevers (k = 2.1 N/m) used in this study were calibrated using thermaloscillation. [START_REF] Hutter | Calibration of atomic-force microscope tips[END_REF] The tip radii (R = 15-20 nm) were measured by blind reconstruction of the tip geometry. [START_REF] Villarrubia | Morphological estimation of tip geometry for scanned probe microscopy[END_REF] The characterization of hardness H and Young's modulus E were performed using an ultra nanoindenter from CSM-Instruments (Switzerland). The standard instrumented indentation tests were driven at constant load rate/load of 0.05 s-1 during both loading and unloading phases. Five different maximum loads were chosen: 0.1, 0.25, 0.5, 1.5 and 25 mN corresponding to a maximum penetration depth of about 2.5 µm while a principle limitation in our samples is based on the thin polymer thickness. A constant load step of 200 s was achieved before unloading. Each test was performed three times to ensure reproducibility.

Results reported here are averages and were calculated following the procedure described in the literature. [START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology[END_REF] 2.5 Cell culture

Human preadipocytes

Fat tissue was obtained during surgery of the abdomen and used with the informed consent of the patient. Subcutaneous human preadipocytes were isolated as described in the literature. [START_REF] Van | Cytological and enzymological characterization of adult human adipocyte precursors in culture[END_REF] The cells were immediately stored in 200 ml α-Minimal Essential Media (α-MEM, Lonza).

To the media was added 200 ml Hams F12 Nutrient Mixture (Lonza), 100 ml Fetal bovine serum (FBS, Biochrom), 5 ml chicken embryo extract (Seralab), Glutamine (Biochrom), 5 ml Penicillin Streptomycin (Pen-Strep, Biochrom) and 5 ml Amphotericin B (Ampho B, Biochrom). After 70-80% confluence the cells were frozen in liquid nitrogen. The cells were seeded at a concentration of 5*10 4 cell per well. They were seeded on the spin-coated 3 cm glass plates, placed in 6-well dishes (TPP).

Rat preadipocytes

Rat preadipocytes were isolated from of Sprague-Dawley-rats, which were killed before. For isolation of the cells the same procedure as for human preadipocytes were used. [START_REF] Van | Cytological and enzymological characterization of adult human adipocyte precursors in culture[END_REF] The cells were stored in medium 199 (Lonza) supplemented with 1%-Pen-Strep (Biochrom) and 10%-FBS. Media were changed the next day. After 70-80% confluency the cells were frozen in liquid nitrogen. The cells were seeded at a concentration of 5*10 4 cell per well. They were seeded on the spin-coated 3 cm glass plates, in 6-well dishes (TPP).

Cell counting

Dead cells swimming in the medium were extracted by suction. The living cells were trypsinized and suspended in the cell media mentioned above. The cell compatibility was analyzed by cell counting using a hemocytometer. human apotransferrin (Sigma-Aldrich) and 400µl Insulin Rapid (Sanofi). In the first week of differentiation the cells were cultured in a solution containing the media, 1.5 ml FBS (5%, Biochrom), 600 µl isobutyl-1-methylxanthine (IBMX, Serva), 30 µl cortisol (Sigma-Aldrich), 30 µl Troglitazon (trogli, Sigma-Aldrich) and 150 µl indomethacin (Sigma-Aldrich). In the second week only 1.5 ml FBS (5%, Biochrom) and 30 µl trogli (Sigma-Aldrich) were added to the media.

The cells were stained with oilred. The surfaces with differentiated adipocytes were photographed in 4x and 10x magnification. The pictures were analyzed using ImageJ and the cells counted with the cell counting plugin.

Statistic section

The results were analyzed with one-way ANOVA. The calculations were performed with

Rstudio. [START_REF]RStudio: Integrated Development for R[END_REF] The p-value for significance was defined as p = 0.05. To be able to speak of a tendency for one possibility, an additional p-value of p = 0.3 was defined. Additionally, when statistical significance or tendency is obtained, a Tukey HSD is performed.

Results and Discussion

The cell compatibility experiments, were composed of two sets of experiments, one with human preadipocytes (with the plasma-cleaned glass slides), the other with rat preadipocytes (with silanized glass slides). The second set of experiments was performed with rat preadipocytes due to limited access to human preadipocytes. The cells were isolated from five genetically identically Sprague-Dawley rats and not pooled. To isolate rat preadipocytes the rats were killed in accordance with the animal welfare officer of the University Tübingen and approval of the regional administrative authority Tübingen.

To verify if mechanical or topological qualities have an impact mechanical and topological measurements were made with AFM and AFM indentation. AFM indentation was used to measure the hardness and Young modulus (elasticity). In case of some polymer samples it was hard to measure the mechanical properties of the films, because of not sufficient height of the film or inhomogeneity.

Poly(DL-lactide) samples gave improper results probably because of the morphology of the film: it is highly inhomogeneous so that the mechanical property assessment could not be correctly achieved. As reported in the materials and methods section, the tests were performed up to approximately 2.5 µm and show for both investigated parameters (Young's modulus E and Hardness H) a unique asymptotic behavior for all the samples when the penetration depth is higher than 600 nm, i.e. when higher than the film thickness. At this penetration depth the mechanical property of the substrate assembly (mineral glass + cyanoacrylate glue layer), impact the measured film properties: the characteristics of the films seem to be far lower than the one of the substrate assembly. Nevertheless, the representative values of the films, i.e. for small indentation depth, do not show a fundamental difference between the samples: the elasticity of most of the measured polymers was around 4-6 GPa. The poly(DL-lactide) samples polymerized with iron(III)bromide were inhomogeneous and showed deviations. The polymers exhibit hardness around 0. 1 Polymers used for cell studies with human preadipocytes. PCL=Poly(ε-caprolactone), PLA=Poly(DL-lactide), PCLLA=Poly(ε-caprolactone-co-DL-lactide)

First set of experiments: human preadipocytes

In the first set of experiments the focus was on the influence of the catalysts. Each biopolymer sample was synthesized two times, one using tin(II)octanoate as a catalyst and the other with iron(II)bromide or iron(III)perchlorate according to reported literature. [START_REF] Hege | Non-toxic catalysts for ring-opening polymerizations of biodegradable polymers at room temperature for biohybrid materials[END_REF] Human primary cells from four different donors were used, four experiments were performed for each polymer type, and the cells were counted three times each time a measurement was made. Cells were counted on day 1 and after 2, 4, 7, 9 and 11 days. The results are plotted in figure 1. In two of four cases, the number of cells was not enough to perform the analysis on the last day, day 11. As the results differ substantially from cell to cell, only results up to day 9 were analyzed.

The cell compatibility of different polymers was examined regarding the influence of composition, catalyst, mechanical and topological properties. The tests were conducted with primary cells of rats and humans. The sample consisted in all cases of spin-coated glass slides.

The cell growth on different polymer films as well as on the control is similar. In order to validate the impact of the catalyst, the cell growth of human preadipocytes was analyzed disregarding the influence of the polymers, shown in figure 2. The preadipocytes seem to grow slightly better on the iron-catalyzed samples, but the difference is not significant. The noticeable small effect of the catalysts towards cell compatibility is supported by a study of Schappacher et al. They analyzed the cytotoxicity of poly(caprolactone) samples, polymerized with different catalysts, and the effects of the catalysts were found to be quite similar. [START_REF] Schappacher | Comparative in vitro cytotoxicity toward human osteoprogenitor cells of polycaprolactones synthesized from various metallic initiators[END_REF] It is important to state, that these short time experiments only show acute toxicity and surface effects. Deviations result from cell-based diversity, as there is a cell to cell difference for catalyst and polymer preference if differences are not critical/toxic.

The next step was to analyze the influence of the polymers while disregarding the influence of the different catalysts, see figure 3. We find no clear preference of the cells for one surface. The last aspect examined, was the surface roughness, measured by AFM and reported in table 1 andtable 2. In figure 4 the cell growth is presented in ascending order depending on the roughness of the polymer surfaces. In the case of the roughest surface with roughness 35 nm, the cell growth increases continuously. At day 9 one finds a preference for the roughest surface and a tendency of higher cell growth on higher roughness. The difference was not significant, but there was a tendency (ANOVA(Roughness): 0.3). The Tukey HSD test also showed that the p-value was smaller the bigger the roughness difference was. So nanoroughness seems to be a positive property human preadipocytes. The effect becomes more pronounced with time. 2 Polymers used for rat preadipocyte cell cultures. PCL = Poly(ε-caprolactone), PCLLA = Poly(ε-caprolactone-co-DL-lactide), PLGA: Poly(DL-lactide-co-glycolide), PCLGA = Poly(ε-caprolactone-co-glycolide), PCLLAGA = Poly(ε-caprolactone-co-DLlactide-co-glycolide).

For the second set of experiments, polymer samples were prepared with a difference in nanoroughness of only 10 nm. Nearly all polymers except one were synthesized with tin(II)octanoate. Primary cells from five rats were used. The cells were counted at day 0, d2, d4, d7, d9 and d11. The results are shown in figure 5. Poly(ε-caprolactone) is part of nearly all polymer compositions in the second part of the study. The polymer compositions contain poly(ε-caprolactone) in different ratios. To investigate if there is a monomer ratio of the copolymer, which is preferred by rat preadipocytes, the average cell growth was ordered in respect to an increasing ε-caprolactone ratio (see figure 6). In case of the PLGA samples, the average value was used. Until day 11 the cell density is lowest on the pure poly(ε-caprolactone) sample. At day 7 and 9 the cell density decreases with increasing ε-caprolactone-ratio. An explanation for this could be that with increasing ε-caprolactone ratio the hydrophobicity also increases. The hydrophobicity seems to slow down cell proliferation. This matches the study of Altankov et al. who compared the biocompatibility of hydrophobic and hydrophilic surfaces with the result that the hydrophilic surfaces are better for the cells than the hydrophobic. [START_REF] Altankov | Reorganization of substratum-bound fibronectin on hydrophilic and hydrophobic materials is related to biocompatibility[END_REF] At day 11 the cell density is on nearly the same level on all surfaces. On the PCLLA-surfaces the cell density is significantly higher than on poly(ε-caprolactone)-surfaces (ANOVA(PCLLA/PCL: 0.02). Another reason why the PCLLA-surface could be preferred, could be the better adhesion on PCLLA compared with the PCL-surface. Comparing the best adhesive surfaces (PCLLA and PLGA (80/2) with PCL leads to significant results (ANOVA(PCLLA-PLGA80/PCL: 0.024). However, no significant influence of the adhesion could be shown in the first set of experiments.

In case the of the types of polymers, the study showed that the cell compatibility of the polymers does not differ significantly. This is in line with the study of Pariente et al. who analyzed different natural and synthetic biomaterials concluding that all tested materials are biocompatible. [START_REF] Pariente | In vitro biocompatibility evaluation of naturally derived and synthetic biomaterials using normal human bladder smooth muscle cells[END_REF]

Comparisons

The studied biopolymer systems provide important insights for material design in tissue engineering and cell culture application. The use of biopolymers with distinct compositions and surface properties are exemplified and validated for 2D cell cultures.

Concerning the cell compatibility of the catalysts no statistical influence could be seen in time frame typical for 2D cultures. In the second set of experiments, iron-catalyzed poly(εcaprolactone-co-lactide) was significantly better then the tin-catalyzed poly(ε-caprolactone).

However, in the first set of experiments the tin-catalyzed poly(ε-caprolactone-co-lactide) showed better cell compatibility then the iron-catalyzed. Since the absolute differences are minor, one cannot infer a strict preference for one catalyst over another despite statistic significance.

In contrast nanoroughness does seems to have a pronounced influence on cell compatibility of the surfaces. The iron-catalyzed poly(ε-caprolactone) sample showed substantial growth by day 9, see figure 1. The nanoroughness of this polymer sample is with 35 nm the largest surface roughness of all polymer systems investigated, see figure 5. In the first set of experiments, one can also see that the cell density increases with increasing nanoroughness. This matches the findings of a study of Chung et al. who found that fibroblasts prefer nanorough surfaces in contrast to smooth surfaces. [START_REF] Chung | Enhancing growth and proliferation of human gingival fibroblasts on chitosan grafted poly (\epsilon-caprolactone) films is influenced by nano-roughness chitosan surfaces[END_REF] If the difference of the nanoroughness is only around 10 nm, as found in set 2, no influence of the roughness is found. This matches the study of Xie et al. which also showed no influence if the difference in nanoroughness was smaller as 12 nm. [START_REF] Xie | Nanoscale modifications of PET polymer surfaces via oxygen-plasma discharge yield minimal changes in attachment and growth of mammalian epithelial and mesenchymal cells in vitro[END_REF] 3.3 Differentiation experiments quickly eluted and distributed within the comparably large volume of the culture media, in addition biopolymer degradation is low within the time frames investigated.

Increased nanoroughness seems to be advantageous for preadipocytes, and should be subject of further studies. If the nanoroughness is comparable for all slides, for example in the second set of experiments, the material composition becomes more important.

Concerning the differentiation, no polymer-specific effect could be identified, which significantly stimulates differentiation. But the comparison of the polymer films and the pure glass slides showed big differences. On the polymer films many preadipocytes differentiated into adipocytes (see Supporting Information). In contrast to this finding, nearly no preadipocytes differentiated on pure glass samples despite the applied differentiation cocktail.

So as a major finding we can postulate that differentiation of preadipocytes favors polymer surfaces in contrast to glass surfaces.

In sum we can conclude that the type/chemical composition and catalyst as well as mechanical properties have limited influence on preadipocyte cell cultures. It could be shown, that a nanoroughness of 35 nm in contrast to 10 nm has a positive effect of preadopocytes.

Therefore the type of biopolymer to be chosen for certain biomedical applications may be attributed to specific needs of the application such as mechanical stiffness, time of degradation and processability. This is an important and promising finding fostering a multitude of applications such as current efforts towards porous scaffolds for 3D tissue replacement explored by us.

2. 5 . 4

 54 Differentiation Human preadipocytes were seeded on spin-coated glass slides at concentration of 5*10 4 cell per well. The cells were differentiated in media consisting of 50 ml Dulbeccos Modified Eagle Medium (DMEM) (Lonza) and 50 ml Hams F12 Nutrient Mixture (Lonza), supplemented with 1ml glutamine (Biochrom), 1 ml Pen-Strep (Biochrom), 200 µl Ampho B (Biochrom), 100 µl calcium D-pantothenate (Sigma-Aldrich), 1 µl Biotin (Serva), 200 µl

Figure 1

 1 Figure 1 Average cell growth of human preadipocyte cells on the polymer films and the glass

Figure 2

 2 Figure 2 Average cell growth (human preadipocytes) as function of time, disregarding the

Figure 3

 3 Figure 3 Average cell growth (human preadipocytes) as function of time, disregarding the

Figure 4 3 . 2

 432 Figure 4 Average cell growth (human preadipocytes) listed as a function of time with respect

Figure 5

 5 Figure 5 Average cell growth of rat preadipocytes on polymer films and glass control as

Figure 6

 6 Figure 6 Average cell growth of rat preadipocytes shown as a function of time in ascending

  ). The molecular weight and roughness of these polymers are given in table 2.

	Polymer/copolymer Catalyst	Average	Thickness	Roughness	Tack trend
			molecular	[nm]	[nm]	(adhesion)
			weight Mw
			[kg/mol]
	PLGA (75/25)	Sn(Oct)2	50	240	1	++
	PLGA (80/20)	Sn(Oct)2	39	190	3	--
	PCLLA (50/50)	Fe(ClO4)3 22	180	3	++
	PCL	Sn(Oct)2	80	650-700	11	+
	PCLGA (80/20)	Sn(Oct)2	66	360	14	--
	PCLLAGA	Sn(Oct)2	53		Could not be	Could not be
	(65/25/15)				measured	measured
	Control	Pure glass
	Table

Acknowledgements: We gratefully acknowledge the support of the BMBF (Research Prize Next Generation of Biotechnological Processes 2014 Biotechnology2020+, FK: 031A550), the Zentrum für Biosystem Analyse (ZBSA), the Freiburg Institute for Advanced Studies (FRIAS), the institute for macromolecular chemistry, the Institute for Micro System Engineering (IMTEK) & EXC 294 BIOSS Centre for Biological Signalling Studies and the Rectorate of the University of Freiburg for support. We want to thank Birgit Schreiner and Ulrike Schmidt for the cell experiments. And, especially Ulrike Schmidt for the valuable help by writing the article. We also would like to thank Marina Hagios for GPC measurements.

Differentiation experiments were performed to examine the impact of different surfaces on the differentiation of preadipocytes. In order to investigate this question, cells were differentiated according to the method mentioned in section 2.5.4.

Regarding differentiation there is no obvious difference between different polymer systems or catalysts investigated; the adipocyte density resides in the same range. However, there is a big disparity between the polymer surfaces and the glass control (see supporting information).

The adipocyte density is nearly 9 times lower than on the polymer surfaces. This is a clear indicator that polymer surfaces are better for adipocyte differentiation than glass.

Conclusions

The biopolymer systems studied represent a vast range of chemical entities and conditions covering important applicable combinations of the three major FDA-approved biopolmyers in use. Major factors for materials used in tissue engineering such as the type of biopolymer, combinations thereof, effect of catalysts used in biopolymer synthesis as well as mechanical properties and surface topological effects (roughness) could be validated and give important guidelines for the use of biopolymers and their effects exemplified and validated for 2D cell cultures.

Neither the polymer, nor the catalyst nor the mechanical properties have a critical influence on cell behavior. One exception concerning the polymer compatibility was found for rat preadipocytes which significantly preferred the PCLLA-surface over the PCL-surface.

Probably this is because of PCL's higher hydrophobicity compared to PCLLA. This is in line with a study of Altankov et al. who showed that cells prefer hydrophilic surfaces. [START_REF] Altankov | Reorganization of substratum-bound fibronectin on hydrophilic and hydrophobic materials is related to biocompatibility[END_REF] Also the better adhesion of cells on the PCLLA surface compared to the PCL surface can be important.

The role of the catalyst was shown to have no critical effect in the context of these studies if cell proliferation and differentiation is concerned. This may be attributed to a limited liberation of catalyst from the material: the catalyst able to diffuse out of the material will be


Bassem Boukhebouz
email: bassem.boukhebouz@cea.fr

Xavier Lamy
email: xavier.lamy@cea.fr

Guillaume Mercère
email: guillaume.mercere@univ-poitiers.fr

Édouard Laroche
email: laroche@unistra.fr

Mathieu Grossard
email: mathieu.grossard@cea.fr

Keywords: System identification, frequency domain, nonlinear distortions, flexible joint robots

In this paper, a nonparametric identification method is used to identify the dynamic model of a flexible joint robot (FJR) that includes friction and transmission nonlinearities. More precisely, the Best Linear Approximation (BLA) approach has been considered in a closed-loop setup to provide the frequency response of the system and an evaluation of the nonlinearities. Based on simulated data, the paper investigates the tuning of an adequate multisine excitation signal that enables good identification results. First, an algorithm is adapted in order to shape the spectrum of the control signal adequately. Second, based on an evaluation of the effects on the Coulomb friction and transmission nonlinearity, it is showed that an optimal amplitude can be found.

Identification of single flexible-joint robot dynamics: a nonparametric approach

I. INTRODUCTION
When manipulator robots are used on large workspace with fast dynamical changes, nonlinear behavior may occur [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. This is all the more true when these robotic systems have strong flexibilities (like structural [START_REF] Mercère | Identification of a flexible robot manipulator using a linear parameter-varying descriptor statespace structure[END_REF] or joint flexibilities [START_REF] De Luca | Robot with flexible elements[END_REF]) because of (i) the coupling between the rigid bodies and elastic deformations of joints, (ii) position dependent nonlinearities such as the well known Coulomb friction [START_REF] Hamon | New dry friction model with load-and velocity-dependence and dynamic identification of multidof robots[END_REF] or (iii) variable stiffness due to motor-to-joint transmission [START_REF] Fichera | On the modeling and identification of stiffness in cable-based mechanical transmissions for robot manipulators[END_REF] to name a few. White box models of such nonlinear systems can be generated based on mechanical modeling of rigid and/or flexible multibodies chains. Because such an approach requires an accurate knowledge of the robot under study as well as high level skills in robotics, system identification [START_REF] Ljung | System Identification: Theory for the User[END_REF] is often suggested as a valuable data based tool to bypass these modeling difficulties. In order to benefit from the white box analysis in the system identification procedure, parametric models are often used for modeling manipulator robots [START_REF] Khalil | Modeling, identification and control of robots[END_REF], [START_REF] Gautier | Dynamic identification of robots with power model[END_REF], [START_REF] Wernholt | Frequency-domain gray-box identification of industrial robots[END_REF], [START_REF] Östring | Closed-loop identification of an industrial robot containing flexibilities[END_REF]. This kind of gray box model often neglects strong nonlinearities (for example the stiffness of the transmission) in order to keep, in the end, a model structure (i) generic to mimic the system behavior, (ii) simple enough to ensure that the controller can be designed easily. Neglecting the effect of nonlinearities on resonance phenomena may unfortunately lead to conservative low-bandwidth controllers or simplified models which do not involve the resonance phenomena (this can damage the system if such phenomena occur within the control bandwidth). This is one of the main reasons why, at a first step, it is often suggested performing a nonparametric identification of the flexible joint robot [START_REF] Wernholt | Estimation of nonlinear effects in frequency domain identification of industrial robots[END_REF], mainly in the frequency domain, in order to provide knowledge of the input-output behavior of the system without requiring strong structural knowledge of the robotic system to control. By resorting to a divide and conquer strategy, nonparametric frequency response function (FRF) identification algorithms can be used locally to determine linear models of the robot behavior in various configurations around user defined working points. This paper is dedicated to the nonparametric estimation of the FRF of FJRs in the presence of friction and transmission nonlinearities using the Best Linear Approximation -BLAmethod described in details in [START_REF] Pintelon | Experimental characterization of operational amplifiers: a system identification approach-part i: theory and simulations[END_REF] and [START_REF] Pintelon | Experimental characterization of operational amplifiers: a system identification approach-part ii: calibration and measurements[END_REF]. Our contributions firstly consist in adapting the multisine amplitudes by restoring to an iterative procedure which shapes the multisine spectrum in order to optimize the quality of the identified model. Furthermore, by applying odd random phase multisine as excitation signal in a closed loop setup, we determine the BLA of the underlying linear transfer from the motor torque to the motor position, the variance of the stochastic nonlinear contributions as well as the noise variance. Secondly, the impact of the nonlinear nature of the transmissions is revealed by mapping the resonances and antiresonances frequencies of the FRF to the levels of excitation signals.

The paper is organized as follows. In Section II, the modeling of the flexible joint robot is introduced. Section III is dedicated to the basics of the frequency response function estimation with the BLA approach. In Section IV, we investigate the tuning of the multisine excitation signal and we show the simulation results of the identification of FJR. Concluding remarks are provided in section V.

II. MODELING OF FLEXIBLE JOINT ROBOT DYNAMICS

When it comes to the compliant manipulation in contact with human, flexible joint robots are more suitable and provide better performance than rigid robots. By FJRs, we refer to robots in which the flexibility is located at the articulation level [START_REF] De Luca | Robot with flexible elements[END_REF]. In this section, we first present the mathematical description of a FJR with n degrees of freedom. Then, the modeling of single FJR is discussed along with the description of nonlinearities considered in the current work.

The most commonly used model to describe the dynamics of FJRs is called the reduced model proposed by [START_REF] De Luca | Robot with flexible elements[END_REF]. In this model, the inertial coupling between the motors and the links of the robot is neglected. The dynamics of the robot typically consist of two coupled equations as follows M (q)q + C(q, q) q + τ G (q) + τ f l ( q) + τ e (q, θ) = 0,

J m θ + τ fm ( θ) -τ e (q, θ) = τ, (1)
with q is the joint position, while θ is the motor position. M (q) represents the inertia matrix of the rigid bodies of the poly-articulated chain. J m denotes the inertia matrix of the actuators, C(q, q) q is the Coriolis torques, τ G (q) is the gravity torque, τ f l ( q) and τ fm ( θ) are the friction torques on the link side and motor side, respectively, while τ e (q, θ) is the elastic torque.

The friction torques on the link and the motor side according to Coulomb friction law are given by

τ f l = f v q + f s sgn( q), τ fm = f vm θ + f sm sgn( θ), (2)
where f v and f s are the viscous and dry friction coefficients on the joint side, respectively, while f vm and f sm are the viscous and dry friction coefficients on the motor side, respectively. The flexibility at the joint level transforms the deflection q -θ into an elastic torque as follows

τ e = K(q, θ) (q -θ), (3)
where the stiffness K(q, θ) of the transmission is deflection dependent. This stiffness can be a nonlinear polynomial function of order n according to [START_REF] Fichera | On the modeling and identification of stiffness in cable-based mechanical transmissions for robot manipulators[END_REF]. In our work, we consider the case of a second order polynomial stiffness as given below, where k 0 and k 1 are stiffness coefficient, i.e.,

K(q, θ) = k 0 + k 1 (q -θ) 2 . ( 4
)
Consider a single FJR on a horizontal plane, hence, without gravity (see Fig. 1). When the friction torques as well as the elastic torque given by Eq. ( 2) and Eq. ( 3), respectively, are included in Eq. ( 1), the dynamical model is

J q + f v q + f s sgn( q) + k 0 (q -θ) + k 1 (q -θ) 3 = 0, J m θ + f vm θ + f sm sgn( θ) + k 0 (θ -q) + k 1 (θ -q) 3 = τ.
(5) Eq. ( 5) is nonlinear, coupled and characterized by elastic modes. We consider the case where only the motor position θ is available, and we locally calculate the underlying linear transfer function from the motor torque to the motor position (τ → θ). Such transfer is of prime importance for feedback control design and is known as the collocated transfer [START_REF] De Luca | Robot with flexible elements[END_REF]. By omitting the nonlinear terms in Eq. ( 5), we derive the transfer functions of interest, i.e., from the input τ to the output θ, we get

G 0 (s) = θ(s) τ (s) = Js 2 + (f v + f s )s + k 0 den(s) , (6)
with den(s) given by

den(s) = {JJ m s 3 + [J(f vm + f sm ) + J m (f v + f s )]s 2 +[(J + J m )k 0 + (f v + f s )(f vm + f sm )]s +[(f v + f s ) + (f vm + f sm )]k 0 }s. (7)
By neglecting the friction terms, G 0 (s) becomes

G 0 (s) = s 2 + ω 2 ar s 2 (s 2 + ω 2 a ) . ( 8
)
where the antiresonance ω ar and the resonance ω r are given by In practice, calculating G 0 (s) is not carried out by going through the linearization since the nonlinearities are usually unknown. For this reason, the FRF is a fundamental tool which allows the estimation of G 0 (s) from the measurement of the input-output signals of the system under study. Such a step is crucial and essential in the identification process because it gives the first insights about the input-output behavior of the system and allows the user to determine the frequency band of interest. In the next section, we present the BLA method [START_REF] Pintelon | System identification: a frequency domain approach[END_REF] to reach this goal. This method enables to accurately estimate the underlying linear system G 0 (s) in the presence of nonlinearities, to detect and to quantify nonlinear distortions and noise contributions [START_REF] Pintelon | System identification: a frequency domain approach[END_REF].

ω ar = k 0 J , ω r = (J + J m )k 0 JJ m . (9)

III. FREQUENCY RESPONSE FUNCTION MEASUREMENT

In the frequency domain, nonparametric identification consists in estimating the FRF of a system given the measurements of the input u(t) and output y(t) signals by calculating their discrete Fourier transforms -DFTs-. In this section, we introduce a method which allows accurate estimation of a linear model of a nonlinear system. Furthermore, the dedicated tools to minimize the impact of nonlinearities are presented along.

A. Best linear approximation

For systems that are subject to nonlinear distortions represented by a Volterra series [START_REF] Schetzen | The Volterra and Wiener Theories of Nonlinear Systems[END_REF] and for a class of Gaussianlike excitations [START_REF] Schoukens | Robustness issues of the equivalent linear representation of a nonlinear system[END_REF], the FRF can be written as [START_REF] Pintelon | System identification: a frequency domain approach[END_REF]

G(jω k ) = Ĝ0 (jω k ) + G s (jω k ) + N G (k), (10)
where ω k (k = 1, 2, . . . , F max ) stands for the excited frequencies, Ĝ0 is an estimate of the underlying linear model G 0 , G s is a zero mean stochastic nonlinear distortions and N G represents the noise contributions. It was shown in [START_REF] Pintelon | System identification: a frequency domain approach[END_REF], that a consistent estimation of G 0 , called the best linear approximation G BLA , can be obtained using Gaussian-like excitations.

Due to the instability of the model given by Eq. ( 5), the identification procedure is performed in a closed-loop setup [START_REF] Van Den Hof | Closed-loop issues in system identification[END_REF] as shown in Fig. 3. The difficulty is that the input u(t) depends on the nonlinear distortions because of the feedback loop, which introduces bias in the estimation of BLA. In [START_REF] Pintelon | Frf measurement of nonlinear systems operating in closed loop[END_REF], this problem was solved by presenting a generalized definition of the BLA. The idea consists in using the BLA via the indirect method [START_REF] Söderström | System Identification[END_REF], where we use the reference signal r(t) (typically the excitation signal), the input signal u(t) and the output signal y(t) to identify two systems, i.e, G ry (jω k ) and G ru (jω k ), respectively. The BLA is then given by

G BLA (jω k ) = S yr (jω) S ur (jω) , (11)
where S yr (jω k ) and S ur (jω k ) are the reference-output and the reference-input cross-power spectrum, respectively.

B. Best linear approximation approaches

Two estimation approaches were presented in [START_REF] Pintelon | Experimental characterization of operational amplifiers: a system identification approach-part i: theory and simulations[END_REF] and [START_REF] Pintelon | Experimental characterization of operational amplifiers: a system identification approach-part ii: calibration and measurements[END_REF] to calculate the BLA of a system in presence of nonlinear distortions:

• The fast approach is based on a single realization of odd random phase multisine excitation given by Eq. ( 13), i.e., with M = 1 realization and P ≥ 2 periods (see Fig. 2 (a)). From the excited harmonics of the odd multisine, we randomly eliminate some harmonics called the nonexcited lines (detection lines).

The elimination of harmonics consists in (i) gathering the excited odd harmonics into groups of F group consecutive harmonics (typical values for F group are two, three, or four [START_REF] Pintelon | System identification: a frequency domain approach[END_REF]) (ii) for each group, one randomly selected harmonic is omitted. For a nonlinear system, even harmonics in the input spectrum show up in the output spectrum at even nonexcited harmonics only, regardless of the parity of the nonlinearity, while the odd harmonics in the input spectrum appear at even and odd nonexcited harmonics in the output spectrum according to the parity of the nonlinearity. The level of the output spectrum at the detection lines corresponds to the stochastic nonlinear distortions at the adjacent excited lines. The detection lines enable the calculation of the variance of nonlinear distortions σ2 N L and the variance of noise contributions σ2 noise [START_REF] Pintelon | System identification: a frequency domain approach[END_REF].

• In the robust method, the information about the stochastic nonlinear distortions are obtained based on the analysis of the mean and variance of the input/output signals by performing several realizations of random phase multisines. For each new realization, a new odd random phase excitation is generated (see Fig. 2 (b)). It was shown in [START_REF] Pintelon | System identification: a frequency domain approach[END_REF] that the variance of the nonlinear distortions are inversely proportional to M and it is recommended to choose M ≥ 7 and P ≥ 2. The comparison between the variance of the nonlinear distortion and the noise variance allows the user to distinguish whether the system behaves nonlinearly or not. In order to reduce the impact of the nonlinear distortions, we increase the number of realizations M .

P periods

Transient 𝐺 [1,1] 𝐺 [1,2] . . . 𝐺 [1,𝑃] . . . . . . 𝐺 [2,1] M realizations 𝐺 [2,2] 𝐺 [2,𝑃] 𝐺 [𝑀,𝑃] 𝐺 [𝑀,1] 𝐺 [𝑀,2] . . . Fig. 2. BLA approaches [START_REF] Pintelon | System identification: a frequency domain approach[END_REF].

P periods

In this work, both approaches are used. First, the fast approach is considered for the classification of the nonlinear distortions in even and odd contributions and for the shaping of the excitation spectrum. In a second stage, the robust approach is used to reduce the distortions of the nonlinearities and noise contributions.

IV. BEST LINEAR APPROXIMATION OF FLEXIBLE JOINT

ROBOTS In this section, we apply the method described in section III to estimate the FRF of a single FJR from simulated data and to study the impact of the nonlinearities on the estimation. The model parameters are: motor inertia J m = 0.2 kg.m 2 , link inertia J = 0.3 kg.m 2 , parameters of the Coulomb friction f s = 5 N.m, f sm = 6 N.m, f v = 4 N.m.s.rad -1 , f vm = 2 N.m.s.rad -1 , deflection dependent stiffness parameters k 0 = 1500 N.m.rad -1 and k 1 = 3000 N.m.rad -2 . The plant controller C(s) is a parallel PD given by

C(s) = P + D N 1 + N 1 s , (12)
where P = 200 N.m.rad -1 , D = 60 N.m.s.rad -1 and the filter is N = 24 rad.s -1 .

A. Multisine excitation

From the class of Gaussian-like excitations, the multisine excitation is usually preferred for the estimation of the FRF [START_REF] Schoukens | Survey of excitation signals for fft based signal analyzers[END_REF]. The multisine enables (i) to reduce the measurement duration, (ii) to use its periodicity in order to avoid the leakage, (iii) to classify, detect and quantify the nonlinear distortions by a specific choice of the harmonics of the multisine [START_REF] Pintelon | System identification: a frequency domain approach[END_REF]. The multisine excitation is defined as follows

r(t) = Fmax k=1 A k cos(2πf k t + φ k ), (13)
where A k are the amplitudes, F max is the number of harmonics, f k are integer multiples of the fundamental frequency f 0 with k = 1, 2, . . . , F max and φ k are random phases with a uniform distribution in [0, 2π]. An integer number P of periods of r(t) are acquired after the transient has vanished, thus, providing N = P N period samples, where N period is the number of samples over one period.

B. Excitation signal shaping

In order to evaluate the quality of the estimated FRF and thus to tune the excitation signal accordingly, the following metric has been used that compares the estimated FRF with the actual frequency response of the system G 0 :

V = 1 F max Fmax k=1 |20 log(G 0 (jω k )) -20 log(G BLA (jω k ))| 2 . ( 14
)
In experimental conditions, G 0 is not available. The metric Eq. ( 14) might be replaced by the variance of the nonlinear distortions σ2

N L , to evaluate the quality of the identification. Fig. 4 represents the variations of the metric with respect to the excitation amplitude in four cases: (i) red line: when both nonlinearities given by Eq. ( 2) and Eq. ( 3) are included, (ii) blue line: when the transmission nonlinearity is active, (iii) black line: when the friction nonlinearity is used and (iiii) green line: when the data is noisy and both nonlinearities are included. It is clear from Fig. 4 that an optimal amplitude A opt exists.

Using a flat spectrum on the reference signal results in a non flat spectrum on the control signal, leading to possible saturation and low excitation at some frequencies (see both spectrum on sub-figures (a) and (b) in Fig. 5). In order to get a flat spectrum on the control signal, we have used Algorithm 1 that has been adapted from [START_REF] Saupe | Design of excitation signals for the closed loop identification of industrial robots[END_REF]. The results of Algorithm Algorithm 1: Multisine spectrum shaping Input: desired input spectrum U des (jω) Output: reference spectrum R i (jω) 1 condition← true; 2 generate multisine reference signal r i (t) defined by (13) of a flat spectrum R i (jω); 3 while condition do 4 excite the robot with r i (t) defined by (13); Whereas, only some odd harmonics are excited (in black in subfigures (a) and (c)), the energy is also observed on the control signal at the nonexcited odd harmonics, which confirms according to [START_REF] Pintelon | System identification: a frequency domain approach[END_REF] that the system is nonlinear and contains almost one odd nonlinearity. Moreover, the spectrum of the input becomes flat as desired (see Fig. 5 (d)). By using Algorithm 1, the metric Eq. ( 14) on the quality of the estimation at the optimal amplitude was reduced from 0.94dB to 0.78dB.

C. Nonlinearity impact on the FRF estimate

The impacts of the friction and nonlinear transmission on the FRF estimate are first evaluated separately and then together. The data is affected by a Gaussian zero mean white noise of SN R = 20dB. Friction nonlinearity impact: As can be seen on the black line in Fig. 4, the friction nonlinearity mostly introduces errors in the estimation for small amplitudes. Thus, to study the friction impacts, we choose a small excitation amplitude (A k = 10 -3 rad) and we consider that the stiffness is linear (i.e., k 1 = 0). As a result, the only acting nonlinearity is the Coulomb friction (see Fig. 6 (left)). In the first instance, the FRF is estimated by making a single realization (M = 1 and P = 7). The resulting FRF estimate is shown in Fig. 7. We observe that the Coulomb friction nonlinear distortions are indeed present and cause larger variability in the FRF (resonances and antiresonances are masked). This is sensible because the dry friction acts as a damping.

In order to reduce these nonlinear distortions and improve the quality of the estimation, 40 realizations (M = 40 and P = 4) are performed, because increasing the number of periods P reduces the noise contribution only on the FRF estimate [START_REF] Pintelon | System identification: a frequency domain approach[END_REF]. By averaging 40 realizations, these distortions are reduced. The plot in black in Fig. 9. reveals that the best linear approximation corresponds (almost) to the linearized model.

Stiffness nonlinearity impact

In this case, we increase the value of the excitation amplitude (A k = 9.10 -2 rad), such that the nonlinearity of the transmission is active (see Fig. 6). In addition, we omit the nonlinear terms in the friction torques, by setting the Coulomb friction coefficients to zero (f s = f sm = 0). As a result, the only source of nonlinearity is the deflection depending stiffness. First, the FRF is estimated by making a single realization (M = 1 and P = 7). The results are shown in Fig. 8. We observe that the nonlinear distortions are larger than the measurement noise contributions (variance of noise contributions σ2 noise is at least 100 dB lower than the variance of the nonlinear distortions σ2 N L . Thus, these fluctuations in the FRF are mostly due to the nonlinear distortions of the transmission. By performing 100 realizations, we reduce the transmission nonlinear distortions via the robust method (see the result in green in Fig. 9). The resonance and anti-resonance frequencies are stiffness dependent according to Eq. ( 9), thus, for high amplitudes they are shifted to the right.

Friction and stiffness nonlinearities impact:

The optimal amplitude (A opt = 8.10 -3 rad) from Fig. 4 is used to estimate the FRF in the presence of both nonlinearities. Fig. 9 in red, shows that, for the optimal amplitude and even in the presence of both nonlinearities, the FRF estimation via the robust method fits G 0 perfectly.

V. CONCLUSION AND PERSPECTIVES

In this paper, a nonparametric approach based on the BLA method has been investigated to estimate the FRF of a single FJR, in the presence of Coulomb friction and transmission nonlinearities, in a closed-loop setup. To get an accurate FRF estimate and to reduce the nonlinear distortions, several experiments were performed with different phase initializations of the multisine excitation signal. Our investigations have highlighted the dependence of the estimation on the signal amplitude. Moreover, an optimal amplitude could be found, that makes a trade-off in terms of quality of the identified model between the effects of the Coulomb friction and the nonlinearity of the transmission. Future work will consist in using the BLA method on an experimental data, in order to detect nonlinearities and accurately estimate the FRF of a flexible joint robot.

Fig. 1 .

 1 Fig. 1. Single flexible joint.

Fig. 3 .

 3 Fig. 3. Closed loop setup.

Fig. 4 .

 4 Fig. 4. Variation of the quality of the identification as a function of the excitation amplitude.

5 9 scale

 9 measure the input signal and calculate its spectrum U (jω); 6 estimate the FRF of G i,r→u (jω) according to Eq. (11); 7 calculate the inverse of G i,r→u (jω) (assumed to be invertible) and the new reference spectrum: R i+1 (jω)= G -1 i,r→u (jω)U des (jω); 8 calculate the new reference signal r i+1 (t) with the inverse Fourier transform i.e., ifft function of MATLAB; the time signal r rmi+1 (t) to avoid saturation;10 if U (jω) = U des (jω) return R i (jω); 1 are shown in Fig. 5, for an odd random phase multisine, within the band of excitation 0.1Hz-50Hz and a frequency resolution f res = 0.1Hz. The top of Fig. 5, i.e., (a) and (b), represents the case without the input shaping and the bottom, i.e., (c) and (d) with the use of the input shaping. Excited odd harmonics (in black), non-excited odd harmonics (in red) and non-excited even harmonics (in blue) are shown.

Fig. 5 .

 5 Fig. 5. Reference (multisine) and input (motor torque) spectrum.

Fig. 6 .

 6 Fig. 6. Friction and elastic torques characteristics.

Fig. 7 .

 7 Fig.7. Fast approach (M = 1 and P = 7) for estimating the gain of FRF between τ and θ in the presence of friction nonlinearity. The gain of G BLA (in black) is compared with the gain of the linearized model G 0 (in blue), variance of nonlinear distortions (in red) and variance of noise contributions (in green).

Fig. 8 .

 8 Fig.8. Fast approach (M = 1 and P = 7) for estimating the gain of FRF between τ and θ in the presence of transmission nonlinearity. The gain of G BLA (in black) is compared with the gain of the linearized model G 0 (in blue), variance of nonlinear distortions (in red) and variance of noise contributions (in green).

Fig. 9 .

 9 Fig.9. Robust approach for estimating the gain of FRF between τ and θ.