Dr Nina Yu Kostina

Dr Sebastien Blanquer

Dr Ognen Pop-Georgievski

Dr Khosrow Rahimi

Dr Barbara Dittrich

Anita Höcherl

Dr Jiří Michálek

Dirk W Grijpma

Dr Cesar Rodriguez-Emmenegger

Zwitterionic

Dr A Höcherl

Prof. D W Grijpma

Prof. D W W J Grijpma

Kolff

Zwitterionic Functionalizable Scaffolds with Gyroid Pore Architecture for Tissue Engineering

Keywords: gyroid pore structure, hydrogels, stereolithography, zwitterionic

Zwitterionic biodegradable hydrogels with gyroid pore architecture are fabricated by stereolithography. The addition of zwitterionic carboxybetaine to the polymerization resins endow hydrogels with improved swelling and functional groups. Hydrogels show suppression of fouling from protein solutions while could be selectively functionalized by preactivation of the carboxylate groups.

Introduction

The primary goal of tissue engineering is to provide a biological surrogate to regenerate damaged or malfunctioning tissues. [START_REF] Ratner | Biomaterials Science: An Introduction to Materials in Medicine[END_REF][START_REF] Langer | [END_REF][3][4] The reconstruction of most of the tissues requires a template guiding the organization of cells. Thus, scaffolding was introduced to tissue engineering to provide active support and DOI: 10.1002/mabi.201800403 signals for cells' growth, migration, and new tissue formation. [START_REF] Ratner | Biomaterials Science: An Introduction to Materials in Medicine[END_REF]5,6] Cell migration is controlled by a complex set of mechanisms that are affected not only by extraand intracellular signaling but also by surrounding extracellular environment. [7] Scaffold microstructure, such as porosity, pore size and shape, specific surface area, interconnectivity, has been shown to significantly affect cell adhesion, growth, and differentiation. Such scaffolds are used for in vivo regeneration of various tissues, such as skin, cartilage, bone, and peripheral nerves. [7][8][9][10][START_REF] Madaghiele | Biomedical Foams for Tissue Engineering Applications[END_REF] Access to nutrients is a cornerstone for cell growth and survival. Thus, interconnectivity of pores are essential for unrestricted fluid flow through the pore space and to ensure diffusion of nutrients, metabolic waste products, cytokines, and paracrine factors through the scaffold. [START_REF] Melchels | [END_REF] Conventional strategies for scaffold preparation rely on the covalent or physical cross-linking of polymers to create a matrix. [13,14] However, the size of such mesh (nanometer scale) is well below cell size (larger than several µm) which makes it impossible for cells to crawl and colonize the whole matrix. To circumvent this, several strategies to create macroporosity (micrometer scale) have been introduced, including solvent casting/particulate leaching, fibril networking, and phase separation. [15][16][17][18][19][20] However, the use Q4 xxxx Stereolithography-assisted fabrication of hydrogels of carboxybetaine methacrylamide and a α,ω-methacrylate poly(d,l-lactide-block-ethylene glycolblock-d,l-lactide) telechelic triblock macromer is presented. This technique allows printing complex structures with gyroid interconnected porosity possessing extremely high specific area. Hydrogels are characterized by infrared spectroscopy, X-ray photoelectron spectroscopy, and confocal laser scanning microscopy. The copolymerization with zwitterionic comonomer leads hydrogels with high equilibrium water content, up to 700% while maintaining mechanical robustness. The introduction of carboxybetaine yields excellent resistance to nonspecific protein adsorption while providing a facile way for specific biofunctionalization with a model protein, fluorescein isothiocyanate labeled bovine serum albumin. The homogeneous protein immobilization across the hydrogel pores prove the accessibility to the innermost pore volumes. The remarkably low protein adsorption combined with the interconnected nature of the porosity allowing fast diffusion of nutrient and waste product and the mimicry of bone trabecular, makes the hydrogels presented here highly attractive for tissue engineering. of these techniques is limited due to the difficulty of having a perfect control over the internal pore architecture resulting in structures with irregular pore sizes and often poor pore interconnectivity. [21] Alternatively, the advances in additive manufacturing significantly improved the production of scaffolds with precise geometry and internal morphology. [22][23][24] Among them, stereolithography is one of the most developed computer-aided rapid prototyping technique. [START_REF] Melchels | [END_REF]23,24] It exploits the 3D-controlled solidification of liquid-based resins by photopolymerization. The structure is polymerized in a layer-by-layer manner which allows the fabrication of complex 3D scaffolds with precise external and internal architecture with the highest accuracy. [START_REF] Melchels | [END_REF][24][25][26] In particular gyroid-type structures can be written with very high precision. Gyroid structures of various length scales are ubiquitous in nature from membranes of organelles to milk particles. [27] The triply periodic gyroid-structure has unique properties such as the highest surface area available and perfectly interconnected porosity. [28] For tissue engineering not only is the architecture of the pores important but also the chemistry of the exposed surface. The polymers used for stereolithography play an important role in the scaffold fabrication. Photo-cross-linkable polymer solutions-resins-are often based on the mixture of (meth) acrylate-functionalized oligomers, photoinitiators, and a nonvolatile solvent and the viscosity of the mixture is carefully optimized. [23,29] However, the surfaces generated from such polymers are rapidly fouled by proteins from the tissue culture media causing an abundance of negative side effects.

To induce specific cell attachment, the surface of the scaffold should be decorated with biological cues, such as peptides, or proteins interacting with integrin receptors of the cell. [30,31] However, it becomes very challenging for real application when the decorated scaffold comes in contact with biological fluids, such as blood plasma or other bodily fluids. This induces nonspecific protein adsorption, that is, protein fouling, leading to irreversible changes in the properties of the material. [32,33] This protein fouling interferes with the signaling mechanisms by masking the biological elements immobilized at the surface and by creating a film of proteins which supports the nonspecific adhesion of cells. [32][33][34][35][36][37][38][39][40][41][START_REF] Thompson | Biological Fluid-Surface Interactions in Detection and Medical Devices[END_REF] Thus the fabrication of materials able to resist nonspecific protein adsorption is essential. [START_REF] Thompson | Biological Fluid-Surface Interactions in Detection and Medical Devices[END_REF] Although several strategies to modify the surface of 2D materials with coatings resistant to fouling have been achieved, the problem of protein fouling for 3D scaffolds used in tissue engineering is generally neglected. [33,35,37,[START_REF] Rodriguez-Emmenegger | [END_REF] Only a few attempts were made in the preparation of 3D scaffolds able to resist protein fouling and most of them were based on poly(ethylene glycol) (PEG). [34,38,39,44,45] However, recently Riedel et al. proved that their contact with real biological media led to fouling regardless of the type of polymer architecture. [33,37,46,47] None of the PEG-based surface modifications could resist the fouling from human blood plasma. The deposits on the best antifouling brushes consisted of apolipoprotein A-I, apolipoprotein B-100, complement C3, complement C4-A, complement C4-B, Fbg, histidine-rich glycoprotein, the Ig mu chain C region, and HSA. These proteins were ubiquitous in all of the deposits on PEG-based surfaces suggesting some degree of biospecificity, thus alternative to PEG should be sought. [46] Previously, it has been shown that polymers based on zwitterionic carboxybetaines are able to completely suppress fouling from complex biological media, such as saliva, urine, cerebrospinal liquid, blood plasma, and whole blood. [35,37,48,49] Recently, our group has introduced antifouling hydrogels consisted of zwitterionic carboxybetaines. [38,39] These hydrogels were prepared by copolymerization of 2-hydroxyethyl methacrylate with carboxybetaine acrylamide or methacrylamide and showed a dramatic decrease of protein fouling from complex biological fluids, such as fetal bovine serum and undiluted human blood plasma. [38,39] Additionally, the carboxybetaine moieties endowed hydrogels with available functional groups which can be utilized for the immobilization of specific biological molecules. [38,39,50] Herein we introduce a novel type of zwitterionic antifouling hydrogels with precise gyroid pore architecture by employing the unique properties of nonfouling zwitterionic poly(carboxybetaine)s and a technique for fabrication of scaffolds with designed architecture, such as stereolithography. The hydrogels were prepared by copolymerization of zwitterionic carboxybetaine methacrylamide (CBMAA) and a macromer α,ω-methacrylate poly(d,l-lactide-block-ethylene glycol-block-d,llactide) telechelic triblock copolymer (MA-PDLLA-PEG-PDLLA-MA) as cross-linker. The poly(d,l-lactide) (PDLLA) block is introduced for biodegradability, while the poly(ethylene glycol) (PEG) block confers hydrophilicity. Both ends of the macromer are functionalized with methacrylate groups to allow the copolymerization with the zwitterionic comonomer. The successful copolymerization was confirmed by infrared spectroscopy with attenuated total reflectance (FTIR-ATR) and X-ray photoelectron spectroscopy (XPS). The copolymerization of MA-PDLLA-PEG-PDLLA-MA macromer with CBMAA monomer led to hydrogels with unique properties. On one hand, these hydrogels showed antifouling character-the absence of nonspecific interaction with proteins, but on the other hand, their available carboxyl groups can be utilized for specific immobilization of bioactive compounds. In order to demonstrate this, the hydrogels were functionalized with fluorescently labeled bovine serum albumin (BSA), which was selected as a model protein.

Experimental Section

2.1. Materials 3,6-Dimethyl-1,4-dioxane-2,5-dione (d,l-lactide), α,ω-dihydroxy poly(ethylene glycol) (α,ω-dihydroxy PEG) ( M n = 4000 g mol -1 ), tin(II) 2-ethylhexanoate (Sn(Oct) 2 ) (92.5-100%), methacrylic anhydride (MAAH) (94%), triethylamine (TEA) (≥99%), N-[3- (dimethylamino)propyl]methacrylamide (99%), N-hydrox- ysulfosuccinimide sodium salt (sulfoNHS) (≥98%), N-(3-dimethylaminopropyl)-N′-ethylcarbodiimide hydrochloride (EDC), BSA-fluorescein isothiocyanate conjugate (BSA-FITC)
were purchased from Sigma-Aldrich. β-Propiolactone was purchased from Serva electrophoresis GmbH. Lucirin TPO-L (ethyl-2,4,6-trimethylbenzoyl phenylphosphinate) was acquired from BASF (Germany). Phenol red was supplied by Riedel de Haën.

Isopropanol, dichloromethane, diethyl ether, chloroform, tetrahydrofuran, and methanol were purchased from Lachner. Tetrahydrofuran (anhydrous THF, ≥99.9%), diethyl ether (anhydrous, ≥99.7), and phosphate-buffered saline (PBS, pH 7.4) were purchased from Sigma-Aldrich.

Macromol. Biosci. 2019, 1800403 U N C O R R E C T E D P R O O F

Synthesis of MA-PDLLA-PEG-PDLLA-MA Macromer

The synthesis of MA-PDLLA-PEG-PDLLA-MA macromer (Scheme 1) was performed in a two-step reaction. In the first step d,l-lactide was polymerized from the hydroxyl end groups of PEG. This was followed by conjugation with methacrylate groups to generate a telechelic macromer.

Synthesis of PDLLA-PEG-PDLLA

PDLLA-PEG-PDLLA was synthesized by a ring-opening polymerization of d,l-lactide initiated from the terminal hydroxyl groups of α,ω-dihydroxy PEG catalyzed by Sn(Oct) 2 (Scheme 1). [51] The targeted molecular weight of the oligomers was between 4500 and 5000 g mol -1 . Firstly, 300 g (75 mmol) of α,ω-dihydroxy PEG was molten and dried in a three-necked flask, under vacuum and magnetic stirring, at 120 °C for 10 h. Then 86.5 g (0.6 mol) of d,l-lactide and 0.6 g (1.5 mmol) Sn(Oct) 2 was added under argon atmosphere. The molar ratio of d,l-lactide/Sn(Oct) 2 was 400/1. The reaction was carried out for 3.5 h at 140 °C. Afterwards, isopropanol was added to the reaction mixture. This alcohol is a good solvent for the monomer and catalyst while the polymer precipitates and can be recovered by filtration with copious washing.

Synthesis of MA-PDLLA-PEG-PDLLA-MA Macromer

The oligomer was functionalized with methacrylate groups (MA) by acylation of the terminal hydroxyl groups of the PDLLA-PEG-PDLLA oligomer with MAAH (Scheme 1). Solutions of PDLLA-PEG-PDLLA (C = 0.25 g mL -1 ) and MAAH (C = 0.2 g mL -1 ) in dry DCM were prepared in separate flasks and purged with argon for 20 min. Then TEA (18 mL) was injected into the solution of oligomer under argon. The solution of MAAH was added to the solution of oligomer and TEA by dropping funnel. The reaction was allowed to proceed for 10 days with continuous stirring. The formed MA-PDLLA-PEG-PDLLA-MA was precipitated into dry diethyl ether and purified by dissolving in chloroform and reprecipitating into diethyl ether. [52]

Synthesis of Carboxybetaine Methacrylamide (3-Methacryloylamino-propyl)-(2-carboxy-ethyl)-dimethyl-ammonium)

CBMAA was synthesized by a modified procedure reported earlier (Scheme 2). [53,54] (The synthetic procedure is described in the ESI).

Optimization of the Stereolithography Resin Composition and Building of Hydrogel Structures

Resins of different compositions were prepared (Table 1). The MA-PDLLA-PEG-PDLLA-MA macromer, CBMMA monomer (Table 1) and phenol red dye were dissolved in Milli-Q water to give solution 1. with a digital micro-mirror device that enables projections of 1280 × 1024 pixels, each measuring 32 × 32 µm 2 . Hydrogels with gyroid-type porosity were prepared by the sequential illumination, polymerization, and cross-linking of layers of the resin. For this, the resin was exposed to a pixel pattern of blue light (wavelength 400-550 nm, peak intensity at 440 nm) while the z-position of the platform was moved step-wise (25 µm) to allow the formation of a 3D structure. The gyroid pore structures of hydrogels were designed using Rhinoceros 3D (McNeel Europe) and K3DSurf computer software as described elsewhere. [55] The fabrication of the hydrogel structures was carried out by exposing the photopolymerizable resin to light intensities of 10 mW cm -2 . The curing time for a 25-µm-thick layer was evaluated from the working curves obtained for each type of resin formulation-40 s for 0%CBMAA, 54 s for 10%CBMAA and 57 s for 20%CBMAA. After photo-polymerization, the obtained structures were washed with acetone (1 day), water (2 days), acetone (1 day) to remove any non-cross-linked compounds. The samples were dried at room temperature for 1 day under atmospheric pressure and subsequently in vacuum until constant weight. Flat nonporous samples were also produced for XPS and FTIR analysis.

U N C O R R E C T E D P R O O F

Functionalization of Hydrogels with BSA-FITC

BSA-FITC was covalently immobilized on the pores of the gyroids structures using EDC/sulfoNHS coupling. Hydrogels of each formulation (0% CBMAA, 10% CBMAA, and 20% CBMAA) were swollen in water and then placed in individual vials containing a solution of EDC (C = 0.522 m) and sul-foNHS (C = 0.115 m) in water. This solution creates active esters of the CBMAA monomers which can be attacked by amine groups of proteins for their covalent immobilization. The samples were shaken for 2 h at room temperature. Subsequently, hydrogels with activated carboxyl groups were washed three times with water, allowing at least 20 min for each wash. Then the activated hydrogels were placed to PBS for 1 h. After incubation, each sample was placed into 4 mL of fresh PBS solution and 2 mL of stock-solution of BSA-FITC (C = 3.33 mg mL -1 ) was added. The reaction was allowed to proceed in the dark for 2 h at room temperature and then for 12 h at 4 °C. Finally, the hydrogels were thoroughly washed with copious amounts of PBS and stored in this buffer overnight. Confocal imaging was performed after an incubation period of at least one day.

In order to prove the absence of nonspecific adsorption of BSA-FITC on the hydrogels, blank experiments were performed. Hydrogels of each formulation were swollen in PBS solution. Then the swollen hydrogels were transferred to vials containing 4 mL of fresh PBS to which 2 mL of stock-solution of BSA-FITC (C = 3.33 mg mL -1 ) was added. The incubation time, washing procedures, and imaging were performed as above.

Characterization of Oligomer, Macromer, and Hydrogels

Proton Nuclear Magnetic Resonance

1 H NMR spectra of reaction products were recorded on Bruker DPX 300 and utilized to determine conversion of d,l-lactide, chemical structure, molecular weight of PDLLA-PEG-PDLLA and degree of functionalization with MAAH. CDCl 3 was used as a solvent.

Fourier Transform Infrared Spectroscopy with Attenuated Total Reflectance

The chemical structure of synthesized PDLL-PEG-PDLLA oligomer and MA-PDLLA-PEG-PDLLA-MA-co-CBMAA hydrogels were characterized by FTIR-ATR. Spectra were recorded using an FTIR Thermo Nicolet Nexus 670 spectrometer equipped with a Specac Golden Gate attachment and a diamond reflection prism.

X-Ray Photoelectron Spectroscopy

The chemical composition of the hydrogels was investigated by XPS. The measurements were carried out with a K-Alpha+ spectrometer (Thermo Fisher Scientific). The samples were analyzed using a micro-focused, monochromated Al Kα X-ray source (400 µm spot size). The kinetic energy of the electrons was measured using a 180° hemispherical energy analyzer operated in the constant analyzer energy mode (CAE) at 200 eV and 50 eV pass energy for survey and high resolution spectra. Data acquisition and processing were performed using Thermo Advantage software. The XPS spectra were fitted with one or more Voigt profiles (binding energy (BE) uncertainty: ± 0.2 eV). The analyzer transmission function, Scofield sensitivity factors, [56] and effective attenuation lengths (EALs) for photoelectrons were applied for quantification. EALs were calculated using the standard TPP-2M formalism. [57] All spectra were referenced to the C 1s peak of hydrocarbons at 285.0 eV BE controlled by means of the well-known photoelectron peaks of PET and metallic Cu, Ag, and Au.

Swelling Properties of Hydrogels

The equilibrium water content (EWC) of porous hydrogels was determined gravimetrically. Five samples of each composition of hydrogels were immersed in deionized water and allowed to swell until constant mass (5 days) and the mass of the swollen samples was determined (m s ). Subsequently, the samples were dried under vacuum at room temperature to constant weight and the mass of dry hydrogels (m d ) was determined. The EWC was calculated as:

) ( = - % * 100% EWC m m m s d d (1)

Laser Scanning Confocal Microscopy

Hydrogels with immobilized BSA-FITC were visualized using the Olympus multiphoton LSCM with the FV10-ASW viewer software (Olympus, Japan). Images were recorded using a 4×-air objective and a 488 nm laser for excitation of FITC; emission was detected at 500-530 nm. 3D reconstructed images were obtained by stacking 20 LSCM xy-scans at 0.5 µm intervals using the ImageJ software.

Results and Discussion

In this study, we report the preparation of a novel type of zwitterionic functionalizable hydrogels with gyroid pore architecture. These hydrogels were prepared by copolymerization of macromer (MA-PDLLA-PEG-PDLLA-MA) with the zwitterionic monomer (CBMAA) utilizing stereolithography.

Synthesis of PDLLA-PEG-PDLLA Oligomer and MA-PDLLA-PEG-PDLLA-MA Macromer

The PDLLA-PEG-PDLLA oligomer was synthesized by ringopening polymerization of d,l-lactide, initiated by the hydroxyl terminal groups of the PEG chain (Scheme 1). The conversion of d,l-lactide was determined from 1 HNMR spectra of the reaction mixture (Figure S1, Supporting Information). When the lactide cycle is opened, protons of the methyl groups present in d,l-lactide are shifted from δ 1.64-1.61 to δ 1.60-1.37 ppm. The conversion was calculated as the ratio of the integral of the shifted peaks (δ 1.60-1.37 ppm) to the sum of integrals of shifted peaks and peaks of residual d,l-lactide (δ 1.64-1.61 ppm) (Equation S1, Supporting Information) and was equal to 93%.

A prominent band at 1750 cm -1 in the FTIR-ATR spectrum of PDLLA-PEG-PDLLA assigned to ester carbonyl confirms the successful polymerization of d,l-lactide (Figure S2, Supporting Information). On the 1 HNMR spectrum a peak at δ 4.13-4.42 ppm, corresponding to methylene protons of the PEG connected to PDLLA could be discerned proving the structure of PDLLA-PEG-PDLLA (Figures S1 and Figure S3, Supporting Information). The degree of polymerization of d,l-lactide (m) was calculated by comparing the integrals corresponding to protons of methylene groups of PEG (δ 3.20-3.90 ppm in Figure S3, Supporting Information) and protons of methyl lactyl units (δ 1.37-1.60 ppm in Figure S3), considering the average degree of polymerization of PEG is DP n = 91 (Equation S2, Supporting Information). The NMRdegrees of polymerization were used to calculated the number average molecular weight of PDLLA-PEG-PDLLA oligomer M n = 5250 g mol -1 .

The hydroxyl end groups of the synthesized PDLLA-PEG-PDLLA oligomer were acylated with MAAH to obtain the MA-PDLLA-PEG-PDLLA-MA macromer (Scheme 1). The incorporation of methacrylate groups to PDLLA-PEG-PDLLA was confirmed by the appearance of three signals in the 1 HNMR spectrum of the macromer: at δ 6.14, 5.58, and 1.90 ppm, corresponding to the = CH 2 and -CH 3 of the methacrylate groups (Figure S4, Supporting Information). The complete disappearance of the signals of terminal hydroxyls in PDLLA-PEG-PDLLA (δ 2.83) indicates that all end-groups contain a methacrylate. This was further confirmed by comparing the integrals of protons of the methacrylate group (δ 6.14 and 5.57 ppm) with those of PEG (δ 3.40-3.80 ppm) considering the numberaverage degree of polymerization (DP n = 91). [52]

Preparation and Characterization of Hydrogels

A precise control over the cure depth of resin is necessary for fabrication of structures by stereolithography. In order to achieve well defined structures it is necessary to tune the cross-linking depth. The polymerization is initiated from an inverted platform that moves upward to generate each layer. Upon irradiation, only part of the resin, which is at focal plane of the laser beam, is cross-linked. The cross-linked volume must be thick enough so that when the platform moves upwards (away from the focal plane) the next cross-linked volume will be grown from the previously formed structure. On the other hand if the depth of cross-linking is too deep then the structure blurs and the pores close. [24,58,[START_REF] Srivatsan | Additive Manufacturing: Innovations, Advances, and Applications[END_REF] The cure depth is determined by the light energy to which the resin is exposed. This energy can be adjusted by the illumination time or by regulating the light intensity. The relationship between the effective cure depth of the resin and the light irradiation dose can be expressed using Equation (2) derived from the Beer-Lambert law:

⋅       = ln C D E E d p c (2)
where C d is cure depth, E is the energy of the light to which the resin has been exposed, E c is the critical energy for curing, which represents the minimum energy level required for crosslinking of the resin, and D p is the penetration depth. Experimentally, an optimal exposure time is determined by constructing the working curve, a plot of the C d versus the natural logarithm of the energy ln(E). From the intercept and slope of the working curve, the critical energy E c and penetration depth of light into resin D p can be determined. The critical exposure and penetration depth are very important parameters and must be evaluated for each resin formulation before the scaffold fabrication to prevent the failure of interlayer bonding or inaccuracies in the architecture. [52,55] Resins based on MA-PDLLA-PEG-PDLLA-MA with different weight ratio of CBMAA were prepared (Table 1). A 0.05 wt% of phenol red dye was added to each formulation to restrict the penetration depth of the light and gain better control over the layer thickness. [START_REF] Ronca | [END_REF] The working curves for all resin compositions were obtained by varying the energy of the light exposed Q5 at the resin surface (E) and measuring the resulting cure depths (C d ) of the resins (Figure 1). The required curing times for a 25-µm-thick layer at a light intensity equal to 10 mW cm -2 were calculated to be 40 s, 54 s, and 57 s for 0% , 10% , and 20% CBMAA resins respectively.

Macromol. Biosci. 2019, 1800403 U N C O R R E C T E D P R O O F
Nonporous films and hydrogels with gyroid pore architecture were fabricated by stereolithogaphy utilizing resins containing different amount of zwitterionic comonomer CBMAA. All hydrogel structures were soft and highly flexible, their shape and structure precisely matched their design. The chemical structure of the hydrogels was confirmed by FTIR-ATR (Figure 2). The increase in the amount of zwitterionic comonomer was evidenced by the concomitant increase in the intensity of amide bands at 1650 cm -1 and 1535 cm -1 , and the carboxyl band at 1600 cm -1 arising from the CBMAA added to the resin formulation.

In addition, the chemical composition of hydrogels was further evidenced using XPS. Figure 3 reports the high resolution core level C 1s and N 1s regions of spectra of the bulk poly(CBMAA) material and 0%CBMAA, 10%CBMAA, and 20%CBMAA hydrogels. The C 1s spectrum of poly(CBMAA) bulk material is characterized by a CC contribution at 285.0 eV, which is accompanied by contributions at 285.6 ± 0.1 eV, 286.2 ± 0.2 eV, 287.9 ± 0.1 eV, and 288.3 ± 0.1 eV arising from the secondary chemical shift,, that is, the effect of the amide group on the tertiary carbon atom in the CH 3 C*C(O)NH structure, the CN + moieties, the amide group C(O)NH and the carboxylic (OCO) group. The N 1s spectrum of poly(CBMAA) is characterized by contributions at about 399.5 ± 0.1 eV and 402.5 ± 0.1 eV of amides and charged quaternary ammonium groups, respectively. The 0%CBMAA hydrogel shows the characteristic contributions for the lactide and ethylene oxide monomers as well as the methacrylate groups that did not react during the photo-curing reaction. The contributions centered at 284. The swelling behavior and water uptake capability are essential parameters of scaffolds for tissue engineering. Highly wetted scaffolds provide a good environment for cell growth and proliferation. However, water content of all natural tissues differs depending on the tissue function. Thus, it is of high importance to be able to tune the water content in order to match the properties of targeted tissue. The copolymerization of macromer MA-PDLLA-PEG-PDLLA-MA with zwitterionic comonomer CBMAA can enable tuning the swelling properties of the hydrogels. [38,39] As expected, an increase in the amount of CBMAA in the polymerization resin lead to an increase of EWC of hydrogels (Figure 4). A 50% increase in swelling was observed with the addition of only 10% of CBMAA to the polymerization feed. A further increment in the CBMAA concentration to 20% led to hydrogel with remarkable water content reaching the value of 700%.

Protein Immobilization on Hydrogel Pores

A widely used approach to induce specific cell adhesion is to immobilize biomolecules, such as peptides or proteins, on the surface of the scaffold to interact with integrin receptors of the cell. To immobilize the bioactive compounds, the material of the scaffold must possess available functional groups while preventing any nonspecific interactions that can occur at the interface between the biological medium and the surface of the scaffold. To achieve this, CBMAA was introduced as a comonomer for the preparation of hydrogels. It was hypothesized that the copolymerization of the macromer with CBMAA will provide not only available carboxylic groups for the biofunctionalization but also will ensure the resistance to nonspecific interaction with proteins.

The pores of the hydrogels were functionalized with a fluorescently-labeled model protein, BSA-FITC. For this, carboxylate groups of the carboxybetaine monomers were activated with EDC/sulfoNHS in water, followed by incubation with a protein solution in PBS buffer (Figure 5D-F). To prove the specificity of the binding we performed control experiments by incubating with BSA-FITC for 14 h without any previous activation with EDC/sulfoNHS. No fluorescence was observed on nonactivated hydrogels during the scanning of all hydrogels (0%, 10%, and 20%CBMAA) with a 488 nm laser for excitation of FITC proving the absence of nonspecific albumin adsorption on the hydrogels (Figure 5A-C).

As expected, no BSA immobilization was observed on the hydrogel without any addition of CBMAA (0%CBMAA) (Figure 5D), due to the lack of functional groups on hydrogels. On the other hand, hydrogels containing 10% and 20% of CBMAA were successfully biofunctionalized with BSA-FITC as presented in Figure 5E,F. These hydrogels show strong fluorescence emission stemming from BSA-FITC. ImageJ software was used to measure the mean average intensity of the pixels in the fluorescent area in order to analyze the difference in the fluorescence intensity in hydrogels containing 10 and 20%CBMAA. The fluorescence intensity of hydrogels containing 10%CBMAA was 22.8 ± 4.3 a.u., while that of hydrogels with 20%CBMAA was 33.4 ± 5.0 a.u.; emitting almost 32% more light. The larger fluorescent signal is well in line with the higher amount of functionalizable carboxylate groups. Figure 5G depicts the 3D reconstruction of 20 confocal scans of 10%CBMAA hydrogel after biofunctionalization showing that the pores were homogeneously modified all across the structure.

Conclusions

Stereolithography was utilized to fabricate non-fouling hydrogels with a precisely designed interconnected porosity. The hydrogels were prepared by copolymerization of macromer MA-PDLLA-PEG-PDLLA-MA and CBMAA. The addition of CBMAA comonomer endowed the hydrogels with an increase in EWC, reaching an unprecedented level of 700% for 20%CBMAA. The CBMAA monomers endowed the hydrogels with antifouling properties while biofunctionalizable. The open interconnected porosity allowed the modification of the surface of the pores with proteins as evidenced by LSCM.

The hydrogels described here represent a novel and highly promising scaffold for tissue engineering. The shape and pore architecture of hydrogels can be precisely designed and implemented by stereolithography. The combination of oligomer based on hydrophilic (PEG) and degradable (PDLLA) segments and a zwitterionic comonomer endows these hydrogels the tunable swelling properties, antifouling character and available functional groups for the immobilization of bioactive compound.

Scheme 1 .

 1 Scheme 1. Synthesis of PDLLA-PEG-PDLLA oligomer and MA-PDLLA-PEG-PDLLA-MA macromer.

  5 ± 0.1 eV, 285.0 eV, 286.3 ± 0.2 eV, 287.0 ± 0.2 eV, and 289.0 ± 0.5 eV arise from the CC, CC, COC, OC*C(O)O, and ester OCO group, respectively. The N 1s spectrum of 0%CBMAA completely lacks nitrogen (amides and charged quaternary ammonium groups) contributions. The incorporation of CBMAA monomer units into the MA-PDLLA-PEG-PDLLA-MA structure is clearly observed in the high resolution C 1s and N 1s spectra of the 10%CBMAA and 20%CBMAA hydrogels. The presence of CBMAA can be verified by the rise of the contributions of CC at 285.0 eV, CH 3 C*C(O)NH at 285.6 ± 0.1 eV, the C-N + moieties Macromol. Biosci. 2019, 1800403

Figure 1 .

 1 Figure 1. Stereolithography working curves for A) 0% CBMAA, B) 10% CBMAA, and C) 20% CBMAA resins.

Figure 2 .

 2 Figure 2. FTIR-ATR spectra of hydrogels prepared from different formulations of resin: A) 0% CBMAA, B) 10% CBMAA, and C) 20% CBMAA.

Figure 3 .

 3 Figure 3. High resolution C 1s (left) and N 1s (right) XPS spectra of A) pure poly(CBMAA), B) 0%CBMAA, C) 10%CBMAA, and D) 20%CBMAA dry hydrogels.

  www.advancedsciencenews.com www.mbs-journal.de (which strongly overlap with the OC*C(O)O contributions of PDLLA at 286.3 ± 0.2 eV) and the C(O)NH group at 288.1 ± 0.2 eV within the C 1s spectra, as well as the occurrence of contributions arising from amide at 399.7 ± 0.1 eV and charged quaternary ammonium groups at 402.5 ± 0.1 eV in the high resolution N 1s spectra. The increase of CBMAA in the polymerization feed from 10 to 20% lead to an increase of the nitrogen content from 1.3 at.% to 2.8 at.%.

  Macromol. Biosci. 2019, 1800403

Figure 4 .

 4 Figure 4. The equilibrium water content (EWC) of hydrogels A) 0% CBMAA, B) 10% CBMAA, and C) 20% CBMAA, defined as the ratio between the mass of absorbed water after reaching the equilibrium and the mass of dried hydrogels (left). The photograph shows dry and swollen hydrogel of 10% CBMAA (right).

Figure 5 .

 5 Figure 5. Biofunctionalization of gyroid hydrogels with BSA-FITC observed by confocal laser microscopy. The contact of BSA-FITC with non-activated hydrogels shows no fluorescence (A-C for 0, 10 and 20% CBMAA hydrogels respectively). D-F are confocal images of hydrogels with 0, 10 and 20% of CBMAA preactivated with EDC/sulfoNHS. The strong fluorescence in E and F proves the successful biofunctionalization. Image G represents the 3D reconstruction of 20 confocal xy scans of 10% CBMAA hydrogels with immobilized BSA-FITC.

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© 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1800403 (3

Table 1 .

 1 Composition of formulations of hydrogels with different weight ratio of macromer to monomer.

	Hydrogels abbreviation	MA-PDLLA-PEG-PDLLA-MA / CBMAA
		[wt%]
	0% CBMAA	100 / 0
	10% CBMAA	90 / 10
	20% CBMAA	80 / 20

Macromol. Biosci. 2019, 1800403 © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1800403 (5 of 9) www.advancedsciencenews.com www.mbs-journal.de

Acknowledgements N.Y.K. acknowledges the Alexander von Humboldt Foundation, O.P.G. acknowledges the support from the Grant Agency of the Czech Republic (GACR) under Contract No. 16-02702S, J.M. acknowledges the support from the Grant Agency of the Czech Republic (GACR) under Contract No. 16-04863S, A.H. acknowledges the Ministry of Education, Youth and Sports of the Czech Republic (grant POLYMAT #LO1507).

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Conflict of Interest

The authors declare no conflict of interest.

Macromolecular Bioscience

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Guillaume Caron
email: guillaume.caron@u-picardie.fr

Yusuke Yoshiyasu
email: yusuke-yoshiyasu@aist.go.jp

Direct visual servoing in the non-linear scale space of camera pose

This paper proposes to consider direct visual servoing (DVS) for object manipulation by a robot arm. The convergence domain limits of DVS are overcome by introducing the non-linear scale space related to camera pose. Its use in a new, yet little complex, direct cost can enlarge twice the convergence domain of one state-of-the-art DVS, as many experiments of symmetric object orientation control assess.

I. INTRODUCTION

Visual Servoing (VS) controls a robot to actively align its camera stream to a desired view [6]. Image alignment is also performed by warping in Simultaneous Localization And Mapping (SLAM) [4] and Visual Odometry (VO) [START_REF] Chaumette | Visual servo control. i. basic approaches[END_REF]. Usually, the alignment minimizes a cost function defined from the desired image (or geometric data) and the current acquired image. They either rely on matched features [START_REF] Krombach | Feature-based visual odometry prior for real-time semi-dense stereo slam[END_REF], [START_REF] Puang | KOVIS: Keypoint-based visual servoing with zero-shot sim-to-real transfer for robotics manipulation[END_REF], direct minimization of image brightness differences, e.g. Photometric VS (PVS) [7], or color-depth (RGBD) data [START_REF] Steinbrücker | Real-time visual odometry from dense rgb-d images[END_REF], [START_REF] Argus | FlowControl: Optical flow based visual servoing[END_REF].

Most image alignment techniques have trouble balancing precision and width of convergence domain. Feature-based alignment can find its bound when matching features is impossible or imprecise [14]. On the other hand, direct alignment is very precise but within a tight convergence domain, as the seminal PVS work [7] showed (Fig. 1b). This fact limits direct alignment approaches to frame-by-frame tracking [10].

Recent progresses have made both Direct VS (DVS) [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF], [START_REF] Duflot | Wavelet and Shearlet-based Image Representations for Visual Servoing[END_REF], [12], [START_REF] Caron | Defocus-based direct visual servoing[END_REF] and image alignment [START_REF] Ahmine | Adaptive lucaskanade tracking[END_REF], [21] achieve large convergence domains. One of the key ideas was to control the Gaussian smoothing of images together with camera pose Degrees of Freedom (DoF) [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF], [START_REF] Ahmine | Adaptive lucaskanade tracking[END_REF]. Image smoothing removes local discontinuities of the image, hence those of the direct cost function [15], enlarging its convergence basin (Fig. 1c).

This work extends these lines of work by introducing a new Non-Linear Scale Space (NLSS) approach to DVS. To the best of our knowledge, NLSS has only been developed in the image denoising field for preserving natural edges [START_REF] Collewet | Photometric visual servoing[END_REF], [START_REF] Boscaini | Anisotropic diffusion descriptors[END_REF] or used as a consequence of optical properties, e.g. defocus [START_REF] Caron | Defocus-based direct visual servoing[END_REF]. The new NLSS considers anisotropic Gaussian filtering depending on camera pose DoFs. It provides larger convergence domains (Fig. 1d) than DVS with isotropic Gaussian filtering [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF].

The main contributions of this work are summarized as:

• New NLSS model related to camera pose DoF, which shows generality over isotropic Gaussian smoothing • A DVS control law based on the new NLSS capable of very large convergence domain (Fig. 1d) • A method to build the non-linear scale space of the desired image offline, even with unknown depth The contributions are evaluated for the automatic storage of goods on supermarket shelves, focusing on the difficult case of axially-symmetric objects: the robot grasps an object in unknown orientation around its axis of symmetry and must align the object to the desired orientation showing its main label (Fig. 1a). To our knowledge, this is the first time DVS is applied to object manipulation. Such a setup requires at least 3 camera DoFs that NLSS can formally deal with (see Sec. III-B). Experimental results with various types of objects in two setups show that NLSS can double the width of the convergence domain of [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF].

In the rest of the paper, Section II recalls the state-of-the-art Photometric Gaussian Mixtures-based VS relying on isotropic Gaussians. Section III introduces the new non-linear scale space related to camera pose. It is then considered for DVS (Sec. IV). After that, the new NLSS alignment cost is evaluated with respect to state-of-the-art costs on a dozen of objects in Section V. Finally, DVS experiments with a robot arm assess the contributions (Sec. VI) before conclusion (Sec. VII).

II. PHOTOMETRIC GAUSSIAN MIXTURES-BASED DVS A. PGM model and control law

We first review the Photometric Gaussian Mixtures-based DVS (PGM VS) [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF], which relies on isotropic Gaussians.

Instead of considering image I with N × M pixels as input to the control law, PGM VS takes a Photometric Gaussian Mixture of the input image, G. Note that G mixes Photometric Gaussians (PG) of every pixel I(u), i.e. I :

u ∈ U -→ I(u) ∈ [[0, 255]], with U = [[0, N -1]] × [[0, M -1]].
A single PG is characterized by its center u and its spread λ ∈ R * + , i.e. mean and standard deviation (R * + = R + \ {0}). Hence a PG is defined as g : u g ∈ U g -→ g(u g , I, u, λ):

g(u g , I, u, λ) = I(u) exp -||u g -u|| 2 /(2λ 2 ) . (1)
Here g models the strength of attraction at pixel location u in image I [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF]. If the definition domain of the PG is the same as the one of the image, then U g = U.

Then, the PGM G is defined with a single λ as:

G(u g , I, λ) = u∈U g(u g , I, u, λ). ( 2
)
As U is a discrete set, the exponential in the expression (1) of g is evaluated at discrete locations u ∈ U, thus it is sampled. Hence, in short, if λ reaches 0, then G(u g ) shrinks to I(u) and PGM VS to PVS [7] (not recalled here for conciseness).

The control law of PGM VS is designed to minimize the Sum of Squared Differences (SSD) cost between current G, i.e. the PGM of I with λ, and G * , i.e. the PGM of the desired image I * with λ * ∈ R * + . PGM VS considers λ as an additional DoF [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF] to those of camera pose p = [t , θ, w ] , representing the rigid transformation from the world frame F w to the camera frame

F c . In p, t = [t X , t Y , t Z ] ∈ R 3 is the 3D translation, and w = [w X , w Y , w Z ] ∈ R 3 : ||w|| = 1
with angle θ ∈ R represent the 3D rotation as axis-angle.

To express the PGM SSD cost, elements of G are stacked as vector G ∈ R |Ug|×1 and those of G * as G * ∈ R |Ug|×1 :

C P GM (p, λ) = 1 2 ||G(p, λ) -G * || 2 . ( 3
)
The control law minimizing C P GM is then expressed as:

v , λ = -µJ G + (G(p, λ) -G * ), (4)
where v ≈ ṗ, µ ∈ R * + is a gain to tune the convergence pace, J G ∈ R |Ug|×D+1 juxtaposes L G ∈ R |Ug|×D , i.e. the interaction matrix related to a PGM sample (natural integer D = 6 is all 6 DoF are controlled, least otherwise), and J λ ∈ R |Ug|×1 , i.e. the Jacobian of the PGM sample with respect to λ. [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF] details their expressions but one may note that L G is function of geometric interaction matrices involving the pinhole camera model. It transforms the 3D point expressed in the camera frame X = [X, Y, Z] ∈ R 3 into 2D image space. Let x ∈ R 2 be the projection of X into the intermediate normalized image plane:

x = x, y = X/Z, Y /Z , (5)
and x ∈ P 2 its homogeneous representation. Then, the 2D point ũ ∈ P 2 represented as the homogeneous coordinates of pixel location u is obtained by:

ũ = Kx, (6)
where K is a camera matrix containing intrinsic parameters

α u ∈ R + , α v ∈ R + , u 0 = (u 0 , v 0 ) ∈ R 2 + : K =   α u 0 u 0 0 α v v 0 0 0 1   . (7)
The evolution of λ (λ * is constant) while the camera is moving is one of the PGM VS's key to both wide convergence domain (with large λ) and precision (with small λ). [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF] reports a sequence of PGM VS: Step 1 with λ * large (exact value depends on experiments) and λ = 2λ * at initialization; Step 2 with constant λ = λ * = 1.

B. Analysis on PGM VS

The contributions of PGM VS actually comes with two issues. The first one is theoretical, related to isotropic Gaussian smoothing. Indeed, as the expression of PGM [START_REF] Argus | FlowControl: Optical flow based visual servoing[END_REF] shows, the image is smoothed uniformly. This makes distant objects smoothed as much as near objects, despite their scale differences in the image. Hence, the information brought by distant objects, which is usually relevant to constrain rotations [START_REF] Tardif | Monocular visual odometry in urban environments using an omnidirectional camera[END_REF], can simply disappear. Another nested issue regards the expression of PG [START_REF] Ahmine | Adaptive lucaskanade tracking[END_REF]. Its Gaussian spread λ is expressed in the image domain. Hence, λ depends on image resolution [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF] with currently no way to relate it to camera motion.

The second issue is practical. PGM VS smooths each acquired image for the spread value of current iteration. Although it may provide both large convergence domain and precision, its algorithmic complexity needs dedicated hardware and implementation to run about online (10 Hz with images of 100 × 100 pixels on the Graphics Processing Unit [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF]).

NLSS (Sec. III) is designed to address the theoretical aspect while NLSS VS (Sec. IV) targets the practical issue.

III. NON-LINEAR SCALE-SPACE RELATED TO POSE

This section introduces a new type of NLSS based on anisotropic filtering with Gaussian kernels depending on the camera pose DoF, the camera projection model and the pixel position in the image plane. NLSS considers the relative pose δp ∈ R 6 (defined similarly to p, Sec. II-A) between two camera frames F c * and F c . Rodrigues' formulae express c M c * ∈ SE(3) from δp, transforming coordinates from frame F c * to F c . Thus, by writing the homogeneous representation of 3D point X as X ∈ P 3 and expressing it in frames F c and

F c * , we have c X = c M c * c * X.
In the rest of this section, we express monodimensional Gaussians in the image plane along the curve defined by projections u of X, function of each element of δp.

A. Single DoF anisotropic kernels

For each pixel I(u), X projects as u for δp = 0. Then, we define u n = [u n , v n ] ∈ R 2 , the coordinates of filtered image I n , and ũn ∈ P 2 their homogeneous representation.

1) Anisotropic kernel related to t X : To express the monodimensional Gaussian curve of the anisotropic kernel, we write

u n function of X and δp = [t X , 0 1×5 ] , t X = 0: ũn = K c * X + t X / c * Z, c * Y / c * Z, 1 , (8)
leading to:

u n = u + α u t X / c * Z, 0 = u + ∆ u (t X ). (9)
u n hence belongs to the line of equation v n = v.

The anisotropic kernel related to t X is then expressed as:

f n (u n , u, t X ) = 1 √ 2πλ n (t X ) exp - (u n -u) 2 2λ n (t X ) 2 , ( 10
)
∀u n : v n = v, with: λ n (t X ) = ||∆ u ( tX )||, (11)
such that tX is the magnitude of camera horizontal translation parameterizing the scale (∼spread in Sec. II).

The above defined kernel is anisotropic since it depends on the Z coordinate of the 3D point projected as u (Fig. 2). Considering the same Z for every pixel leads back to the isotropic Gaussian kernel, but for the scale parameter tX .

2) Anisotropic kernel related to t Z : Generally speaking, t Z leads to a Gaussian kernel of 2 dimensions. Indeed, from δp = [0 1×2 , t Z , 0 1×3 ] , t Z = 0, we get:

ũn = K    c * X c * Z+t Z c * Y c * Z+t Z 1    =    αux c * Z c * Z+t Z + u 0 αvy c * Z c * Z+t Z + v 0 1    , (12)
rewritten for u n and v n as:

u n = u - t Z ( c * Z + t Z ) (u -u 0 ) (v -v 0 ) = u + ∆ u (t Z ). (13)
Hence, u n belongs to the half-line from u 0 toward ∆ u (t Z ). Then, let us express the kernel function f n (u n , u, t Z ), such that it represents a Gaussian centered at u, of spread:

λ n (t Z ) = ||∆ u ( tZ )||, (14)
similar to the anisotropic kernel of t X (Sec. III-A1), but only defined for points u n verifying (13):

f n (u n , u, t Z ) = exp -1 2 (u n -u) Σ -1 n (u n -u) √ 2πλ n (t Z ) , (15)
after posing:

Σ n = ∆ u ( tZ )∆ u ( tZ ) . (16)
The anisotropic kernel of (15) is even more anisotropic than the one of t X of (10), as its spread depends on u, in addition to Z, as Figures 3a and3b show.

3) Anisotropic kernel related to θ Y : First, express the curve in the image along which the monodimensional Gaussian related to θ Y (shorten "θw with w = [0, 1, 0] ") is defined.

Considering δp = [0 1×4 , θ Y , 0] , θ Y = 0, we get: ũn = K c * X cos θ Y + c * Z sin θ Y , c * Y -c * X sin θ Y + c * Z cos θ Y , 1 . ( 17
)
Multiplying the right vector of ( 17) by c * Z/ c * Z leads to:

ũn = K (x cos θ Y + sin θ Y ) , y -x sin θ Y +cos θ Y , 1 . (18)
The above vector defines a curve in the image plane (Fig. 3d), function of θ Y , ∀x, being the perspective projection of a circle perpendicular to the image plane, i.e. a hyperbola. To get a simpler expression, we express the same curve with respect to x = 0. To do so, we write y from x (right vector of ( 18)), seen as an orientation vector colinear to the line of sight of x, and x ∈ P 2 , an orientation vector of (x = 0, y ):

x = [x, y, 1] and x ∝ [0, y,

x 2 + 1] , (19)
since x and x are the same, up to a pure rotation around camera axis Y c . We then express:

y = y/ x 2 + 1. (20)
Substituting x and y with x and y in ( 18) and ( 17) we get:

u n = α u tan θ Y + u 0 (v -v 0 ) / cos θ Y √ x 2 + 1 + v 0 , (21)
with x = (u -u 0 ) /α u .

-1 -0.5 0 0.5 1 We then express the anisotropic kernel f n (u n , u, θ Y ) of hyperbolic support, for u n of (21). Posing:

d θ Y = [u n , 1][u 0 , 1] / (||[u n , 1]|| ||[u 0 , 1]||) , (22)
we get:

f n (u n , u, θ Y ) = 1 √ 2πλ n (θ Y ) exp - arccos (d θ Y ) 2 2λ n (θ Y ) 2 ,
(23) where:

λ n (θ Y ) = | θY |. ( 24
)
Figures 3c and3d show examples of kernels of θ Y . 4) Anisotropic kernels related to t Y , θ X and θ Z : These kernels are not detailed for conciseness but the anisotropic kernel related to t Y , resp. θ X , is very similar to the one related to t X (Sec. III-A1), resp. θ Y (Sec. III-A3), up to a transpose.

The anisotropic kernel related to θ Z is expressed following the same method than the other DoFs. Its shape is circular.

B. Combining anisotropic kernels

The most obvious combination of monodimensional kernels considers those related to t X and t Y (resp. Sec. III-A1 and III-A4) giving the bidimensional kernel F n ():

F n (u n , u, t X , t Y ) = f n (u n , u, t X )f n (u n , u, t Y ). ( 25
)
Assuming a fronto-parallel planar scene (Z constant for all u) and t X = t Y = t XY , F n of (25) becomes the conventional expression of the isotropic 2D Gaussian (homeoscedastic):

F n (u n , u, t X , t Y ) = exp - (un -u) 2 2λn (t X ) 2 √ 2πλn(t X ) exp - (vn -v) 2 2λn(t Y ) 2 √ 2πλn(t Y ) = 1 2πλ n (t XY ) 2 exp - (u n -u) 2 + (v n -v) 2
2λ n (t XY ) 2 .

(26) F n of (26) is a particular case of (25) equivalent to the Gaussian kernel used in every work of image alignment exploiting, explicitly or not, the linear scale space [START_REF] Steinbrücker | Real-time visual odometry from dense rgb-d images[END_REF], [10], [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF], [START_REF] Ahmine | Adaptive lucaskanade tracking[END_REF]. Ignoring the factor in front of the exponential, (26) is the same exponential as for a Photometric Gaussian (1) of PGM VS. Figures 4a to 4d show several examples of 2D kernels of (26), for various u and Z. By extension, mixtures of above mentioned 2D kernels are isotropic iff Z is constant (Fig. 4e and4f) but anisotropic in general (Fig. 4g and4h).

IV. NLSS VISUAL SERVOING

This Section describes the new VS control law exploiting the NLSS filtered desired image (Sec. IV-A) and the computation of the desired image for NLSS-VS (Sec. IV-B).

A. Control law

The new NLSS VS control law is similar to (4) but only filters the desired image I * anisotropically to obtain I * n . By stacking pixels of the current and desired images as vectors I(p) and I n * , the new NLSS SSD cost is defined as: Thus, as I * n is constant over time, the same interaction matrix L I as for PVS [7] is used in the NLSS VS control law:

C N LSS (p) = 1 2 ||I(p) -I n * || 2 . ( 27
)
v = -µL I + (I(p) -I n * ) . (28)
In (28), L I + is the pseudo-inverse of the photometric interaction matrix L I and µ is a gain (see (4)). L I involves image gradients, camera intrinsic parameters and scene depth at each pixel (see [7] for the details). When the scene depth is unknown, setting the same constant depth for every pixel has shown the robustness of (4) to such approximation [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF], [7]. Thus, the same behavior is expected for (28).

B. Generating the desired image for the control law

In our context, a static camera looks at a symmetric object rotated by a robot arm around its axis of symmetry. This is dual to the camera rotating around the object at constant depth. Hence, three camera degrees of freedom are sufficient: one rotation around the object axis of symmetry and two translations in the plane orthogonal to that axis. We consider the object axis Z o fits the axis of symmetry. Assuming Z o is perpendicular to the plane formed by camera axes X c and Z c , the three camera DoFs are: t X , t Z , θ Y . These DoFs are those to consider when generating the NLSS desired image I * n . Thus, the NLSS combines kernels (dropping u n and u for compactness) f n (t X ), f n (θ Y ) and f n (t Z ) (resp. ( 10), ( 23), ( 15)) as

F n (t X , θ Y , t Z ) in a same fashion as F n (t X , t Y ) does (25).
Ideally, the desired image I * n in (28) would be the result of the NLSS filtering with kernel F n (t X , θ Y , t Z ) of a single acquired image of the object at desired orientation o θ * Z (here expressed in the object frame F o for brevity). As the 2D anisotropic kernel F n (t X , θ Y , t Z ) involves translations, it depends on scene depth, thus object shape. We propose a procedure to compute I * n , even when the depth is not available. Thus, only a passive monocular camera is required and 3D models or depth measurements are not necessary.

To compute I * n , equivalently to the explicit filtering of I * with kernels of the NLSS (Sec. III), we acquire a set of images with several object orientations around o θ * Z and compute once the Gaussian weighted sum of this set of images. The interval of orientations and the Gaussian weights depend on the NLSS scale parameter o θZ , equivalent to three scale parameters in the camera frame. So, for a given o θZ , we define the range of orientations Ω around the object desired orientation o θ * Z as:

Ω = [ o θ * Z -N λ o θZ , o θ * Z + N λ o θZ ], (29)
where N λ is nothing but a well-known integer factor leading Ω to cover 68.3%, 95.5% or 99.7% of the Gaussian curve integral for N λ equal, respectively, to 1, 2 or 3. Then, each pixel value I * n (u) of the desired image is computed as the bounded integral of pixel values acquired at the same location u but spanning the whole range Ω, i.e.:

I * n (u) = Ω I(u, o θ Z ) √ 2π| o θZ | exp - ( o θ Z -o θ * Z ) 2 2| o θZ | 2 d o θ Z .
(30) Although a continuous acquisition procedure exposing the camera while rotating the object at a speed varying accordingly to the inverse of the Gaussian of (30) could be imagined, it would be very complex to implement. To make the computation of I * n (u) easy, Ω is sampled with step δ θ ∈ R * + , and a discrete set of images is acquired for each element of Ω. These images are summed, each weighted with the Gaussian in (30). The cardinality of Ω gives a normalizing factor to ensure pixel values of the desired image I * n remain in the [0, 255] interval.

V. COST FUNCTIONS EXPERIMENTAL COMPARISON

In order to compare cost functions of NLSS VS (27) with respect to the state-of-the-art PGM VS (3) and the classical PVS [7], we consider 13 objects of various sizes (bottles and food boxes, Fig. 5a,5b). Each object is axial-symmetric and its frame is set so that axis Z o fits the axis of symmetry. Hence, we can simply note o θ Z as the object angle around its axis of symmetry. It should be noted that due to this symmetry, only the object appearance, not its geometry, allows us to detect orientation changes around Z o . 6850 images (http://mis.u-picardie.fr/~g-caron/pub/data/ConveJRL.zip, 4 GB) of 720×960 pixels were acquired by a webcam, while each object slowly rotated for 360 degrees around Z o on the center of a turn table. The constant rotation speed was set so that the angle between neighboring acquired images is δ θ = 0.8 degrees (Sec. IV-B) on average. Then, to compare the NLSS cost (27) (N-cost) with baseline PGM (4) (G-cost) and photometric (P-cost: SSD between I and I * ) costs, one image I * per object is selected as the one facing the most the main label to the camera (Fig. 5a,5b). After that, I * n is computed from images before and after I * , in the acquired image sequence, among the range of angles spanning Ω (Sec. IV-B). As Ω, hence I * n , depends on the scale parameter The convergence domain width is experimentally evaluated. For an object, every acquired image while the object was rotating is used to evaluate the cost functions from -180 to +180 degrees (see Fig. 1b, 1c and 1d for the ayataka object). Then, the cost function is numerically differentiated to find left o θ (l) Z and right o θ (r) Z borders of the convergence domain of the global minimum. As these borders might not be symmetric, the convergence domain width is computed as twice the minimum absolute value among o θ (l)

Z and o θ (r) Z .
Table I shows the width of convergence domain of all three costs for various values of spread (Sec. II) and scale (Sec. III), for every object. The N-cost always outperforms the P-cost, sometimes with very large ratios, e.g. ×10 for noodles. Surprisingly, the G-cost does not outperform the P-cost for one object. For three objects, N-cost is on par with G-cost. Out of them, N-cost shows a larger domain than G-cost in 6 cases. Conversely, G-cost shows a larger domain than N-cost in 4 cases. Actually, local minima appear in the N-cost very close to the global optimum with large o θZ . A slight modification of the N-cost, filtering isotropically the current image with a little spread (see the 8 columns on the right of Tab. I), overcomes the issue and makes the N-cost largely outperforming the Gcost for 11 over 13 objects.

VI. VISUAL SERVOING EXPERIMENTS

We use a Universal Robot 10 (UR10) with a Robotiq 3-finger gripper (Fig. 1a). They are connected via a wired network to a laptop (Intel Core i7, Ubuntu 16.04) that runs VS control laws. Images with 512 × 640 pixels are acquired by a Flir 30 FPS camera (Computar lens, focal length 8 mm). Each object area in the image is manually selected to avoid perturbations and ensure a fair comparison of VS methods.

The camera intrinsic parameters (7) were computed from device datasheets. The relative transformation between the end-effector to the object frame was set to the identity matrix as the gripper could grasp objects from their cap or top cover (its Z-axis is aligned with each object axis of symmetry). The camera to robot calibration was done from a single pose computation of a 4 dots marker held by the gripper using the ViSP library [13] and the kinematics obtained from UR10.

PVS, PGM VS and the new NLSS VS were applied on the objects of Section V. Initial poses o θ (0) Z were set by incrementally increasing the angular error with respect to the desired orientation o θ * Z with a step of 10 degrees, reduced to 5 to help ranking methods. Gains of the three control laws were manually set to allow the maximum speed without I. On its side, PGM VS runs with two steps but λ = λ * constant as Table I shows different values can change the global optimum. It also saves time at runtime, even if images had to be resized to 105 × 86 pixels to meet the same 10 Hz control loop rate as [START_REF] Crombez | Visual servoing with photometric gaussian mixtures as dense features[END_REF]. Then, PGM VS runs with λ = 1. If it fails to converge, then λ is successively set to 2, then 5, then 10.

NLSS VS clearly outperforms PVS, always extending the convergence domain, up to 22 times. For instance, NLSS VS converges for -40 degrees

≤ o θ (0) Z -o θ * Z ≤ 180 degrees with cheese whereas PVS diverges if | o θ (0) Z -o θ * Z | > 5
degrees. PGM VS outperforms PVS as well but not always. For instance, PVS converges between -35 and 30 degrees with corn but PGM VS only converges between -10 and 5 degrees (-35 and 35 degrees for NLSS VS). Hence, on this criterion, NLSS VS performs better than PGM VS too.

When PGM VS performs better than PVS, NLSS VS can also largely outperform PGM VS, for example with jagabee: NLSS VS could converge for -30 ≤ o θ (0) Z -o θ * Z ≤ 180 degrees, whereas for PGM VS -50 ≤ o θ (0) Z -o θ * Z ≤ 20 degrees. However, there are objects, such as suntory, for which NLSS VS' converge domain is only 20 % of PGM VS' (25 versus 120 degrees). Actually, on the 13 considered objects, the average convergence domain width is almost the same for NLSS VS (132.7 degrees) than for PGM VS (131.9 degrees), while PVS' being 34.2 degrees. But as VS results with corn, jagabee and suntory let think, the deviation of the convergence domain width is larger for PGM VS (75.5 degrees) than for NLSS VS (69.6 degrees), making PGM VS' performances harder to predict than NLSS VS ones.

To finalize the comparison, we compute the balance of the convergence domain around o θ * Z by averaging the absolute difference of absolute values of the domain limits (the lower, the better balanced): PGM VS reaches 84.7 degrees, whereas NLSS VS stands at 50.4 degrees, i.e. 1.7 times better balanced.

The experiments are gathered in the accompanying video.

Figure 6 shows VS results for the noodles box. NLSS VS converges smoothly from the farthest initial angle 30 times the one of PGM VS, 18 times the one of PVS. The error at convergence (Fig. 6d) cannot be zero as I * n is filtered but I not (recall the N-cost at the optimum, Fig. 1d). Despite that fact, precision at convergence is 1.2 degrees (0.2 degrees for PVS and 2.3 degrees for PGM VS, both for much smaller maximum initial error than for NLSS VS, see Fig. 6).

Execution times are exactly the same for PVS and NLSS VS: 5 ms per image on average. PGM VS takes 83 ms per image when λ = 10 and 36 ms when λ = 1, despite a careful implementation on precomputations and processing images with 36 times less pixels than PVS and NLSS VS.

VII. CONCLUSION

This paper introduced for the first time the non-linear scale space related to all camera DoF. It enlarges significantly the convergence domain of direct visual servoing for a much lower computational complexity than previous reference works on photometric Gaussian mixtures (similar to a linear scale space). Future works will adapt online the non-linear scale space parameters to improve the convergence pace.

Fig. 1 :

 1 Fig. 1: (a) Axially-symmetric object showing its main label. (b-d) Direct cost functions (abscissa: object angle in degrees) for (b,c) baseline and (d) new costs; dashed lines are convergence borders of width (degrees): (a) 25, (b) 54, (c) 107.

  x = [-0.6; -0.4] x = [-0.6; -0.4] x = [-0.6; -0.4]

Fig. 2 :

 2 Fig. 2: Anisotropic kernel function of t X on two x (black circles) for tX = 0.2 m: (a-d) Z = 1.0 m; (e-f) Z = 0.5 m.

Fig. 3 :

 3 Fig. 3: Kernels of: t Z on (circles) (a) x = [0.1; 0], (b) x = [0.6; -0.4], for tZ = 0.2 m, Z = 0.5 m; θ Y on (c) x = [0; 0.01], (d) x = -[0.4; 0.6] for θY = π/14, i.e. 13 degrees.

Fig. 4 :

 4 Fig. 4: (a-d) isotropic kernels for tX = tY = 0.2 m on two x: (a,c) x = 0.0, y = 0.0; (b,d) x = 0.6, y = -0.4. (e-f) mixtures of isotropic kernels (a, b), resp. (c, d). (g-h) mixtures of anisotropic kernels (b, c), resp. (a, d).

o

  θZ of the NLSS related to the rotation around object axis of symmetry, several values of o θZ are considered for comparison but a single N λ = 3 (examples of I * n filtered with o θZ = 6 degrees, Fig. 5c, 5d).

Fig. 5 :

 5 Fig. 5: (a-b) Axial-symmetric objects (2 among 13). Ayataka is a tea bottle. (c-d) Corresponding desired NLSS images I * n .

  -7 -3 -1 19 -6 -3 -1 22 -0 -3 -0 127 § -1 -1 -0 17 17 20 60 89 100 120 6 1719 20 87 90 130 133 † 133 † corn 34 34 7 -1 -0 100 6 -1 -0 99 -1 -0 -0 3 -0 -0 -0 34 100 100 100 100 130 § 130 § 1 34 100 100 100 100 169 † 169 † 168 noodles 6 10 10 10 9 10 10 13 -3 32 0 0 -3 0 0 -3 -3 6 12 61 § 61 § 61 § 61 § 1 0 10 12 87 † 69 69 71 71 71 cheese 6 13 13 13 22 13 13 16 3 77 † 3 3 16 47 § 47 § 14 14 14 11 11 16 56 † 56 † 56 † 14 14 gum 10 10 10 0 -2 13 -0 -2 -0 86 § -2 32 84 84 86 87 † 87 † 2 jagabee 7 7 7 7 -10 9 10 -10 -0 34 § -0 -0 -0 0 -0 -0 -0 7 10 34 § 34 § 34 § -0 1 1 7 23 42 44 47 49 † 47 47 koala 33 34 33 31 -1 37 33 34 -1 48 § 37 -1 -1 45 -1 -1 -3 33 35 44 44 44 48 § 21 21 34 37 44 44 48 48 62 65 † oolong 88 114 48 -4 -2 114 48 -6 -2 118 -13 -2 -2 120 -11 -2 -0 88 114 114 122 122 123 † 123 † 0 114 114 118 120 122 122 123 † 123 † suntory 7 19 †2 3 -0 -0 -0 -0 7 19 † 2 3 -0 -0 -0 -0 oscillations at the optimum. NLSS VS runs with the tradeoff value o θZ = 6 degrees suggested by Table

Fig. 6 :

 6 Fig. 6: Noodles box servoing on main label. (a-c) Cropped initial errors (images or PGMs) at the maximum angular error (degrees) allowing to converge. (d) NLSS VS final error.

TABLE I :

 I Convergence domain width [degrees] comparison between P-cost, G-cost and N-cost, for various spreads (sp) λ and λ * (pixel unit), and scales (sc) o θZ (degrees). Bold † shows the largest domain per object. If no tie, Bold § shows the second largest domain with another method. A negative sign indicates that the method changes the global optimum. -2 25 61 61 61 106 106 107 † 107 † 55 66 69 104 104 106 § 106 § 106 § chips 17 17

	P	G-cost	N-cost		N-cost with filtered I
	sc/sp *	1 1 1 1 2 2 2 2 5 5 5 5 10 10 10 10 0.2 1.7 3.2 4.7 6.2 7.7 9.2 10.7 0.2 1.7 3.2 4.7 6.2 7.7 9.2 10.7
	sp	1 1.2 1.3 2 2 2.3 2.6 4 5 5.8 6.5 10 10 11.5 13 20		1 1 1 1 1	1	1	1
	kitsune 17 19 19 7 -1 44 22 7 -1 62 § 10 0 -0 58 0 -6 -1 17 62 § 58 55 55 3	3	3 19 62 62 65 55 70 † 2	2
	ayataka 25 56 9 -2 -2 43 9 -2 -2 43 -2 -2 -2 8	-2 -2