Shazia Ilyas

Nithya Joseph

Anthony Szymczyk

Alexander Volodin

Kitty Nijmeijer

Wiebe M De Vos

Ivo F J Vankelecom

Weak polyelectrolyte multilayers as tunable membranes for solvent resistant nanofiltration

Keywords: Weak polyelectrolytes, SRNF, Layer by layer assembly, Polyelectrolyte multilayers, Nanofiltration membranes

come

challenging polar aprotic solvent, N,N-dimethylformamide (DMF). Results of this study clearly demonstrate the potential of using pH as tuning parameter for weak PEMs to prepare SRNF-membranes optimized for specific applications.

Introduction

Nanofiltration (NF) is among the pressure-driven membrane processes which have gained a lot of attention recently, with separation characteristics falling in the range between RO and UF membranes. This makes the technology especially promising for separations on a molecular level. One of the key current challenges for NF is to broaden the range of applications from aqueous to organic feeds in the form of solvent-resistant nanofiltration (SRNF) membranes [START_REF] Vandezande | Solvent resistant nanofiltration: Separating on a molecular level[END_REF]. As SRNF separation does not involve an energy demanding phase transition, it has drawn considerable attention as an alternative separation technique to alleviate the energy costs and related environmental problems caused by conventional separation methods such as distillation [START_REF] Volkov | New membrane material for SRNF applications[END_REF][START_REF] Marchetti | Molecular Separation with Organic Solvent Nanofiltration: A Critical Review[END_REF]. SRNF has potential applications in strategic fields, such as the petrochemical industry, pharmaceutical industry and food industry to separate desired molecules from solvents and/or to recover solvents and solutes from waste streams [START_REF] Bhanushali | Advances in Solvent-Resistant Nanofiltration Membranes[END_REF][START_REF] Hendrix | SRNF membranes for edible oil purification: Introducing free amines in crosslinked PEEK to increase membrane hydrophilicity[END_REF][START_REF] Aerts | The use of solvent resistant nanofiltration in the recycling of the Co-Jacobsen catalyst in the hydrolytic kinetic resolution (HKR) of epoxides[END_REF][START_REF] Peeva | Continuous purification of active pharmaceutical ingredients using multistage organic solvent nanofiltration membrane cascade[END_REF][START_REF] Zwijnenberg | Important factors influencing molecular weight cut-off determination of membranes in organic solvents[END_REF]. However, there is lack of membranes that are thermally and chemically stable, that have a high permeability and that provide effective separation of molecules in the range of 200-1000 g·mol -1 in various organic solvents over long time-scales [START_REF] Vandezande | Solvent resistant nanofiltration: Separating on a molecular level[END_REF][START_REF] Marchetti | Molecular Separation with Organic Solvent Nanofiltration: A Critical Review[END_REF][START_REF] Mohammad | Nanofiltration membranes review: Recent advances and future prospects[END_REF][START_REF] Hermans | Recent developments in thin film (nano)composite membranes for solvent resistant nanofiltration[END_REF][START_REF] Silva | Solvent transport in organic solvent nanofiltration membranes[END_REF]. Polymers are an interesting material for SRNF related applications but very few polymeric membranes are stable: swelling and/or dissolution of the polymeric matrix often results in a loss of membrane selectivity.

In membrane technology, layer by layer (LbL) based self-assembly of charged polyelectrolytes (PEs) on an oppositely charged porous support has recently emerged as one of the simplest, most versatile, and most environmentally benign techniques to fabricate membranes with selective top-layers [START_REF] Joseph | Layer-by-layer preparation of polyelectrolyte multilayer membranes for separation[END_REF]. Fig. 1 shows schematically the LbL process to create membranes with a polyelectrolyte multilayer (PEM) top layer. A key strength of the PEM membranes is their versatility as the layer properties, and thus the separation properties, can be controlled via the types of PEs used, the number of deposited layers and the pH and ionic strength during coating [START_REF] Krasemann | Selective Ion Transport across Self-Assembled Alternating Multilayers of Cationic and Anionic Polyelectrolytes[END_REF]. PEMs have already been used to make different kinds of membranes such as reverse osmosis [START_REF] Jin | Use of Polyelectrolyte Layer-by-Layer Assemblies as Nanofiltration and Reverse Osmosis Membranes[END_REF][START_REF] Park | Desalination membranes from pH-controlled and thermally-crosslinked layer-by-layer assembled multilayers[END_REF], ion selective [START_REF] Krasemann | Selective Ion Transport across Self-Assembled Alternating Multilayers of Cationic and Anionic Polyelectrolytes[END_REF][START_REF] Hong | Recovery of phosphate using multilayer polyelectrolyte nanofiltration membranes[END_REF][START_REF] Liu | Ion-Exchange Membranes Prepared Using Layer-by-Layer Polyelectrolyte Deposition[END_REF][START_REF] Magnenet | Functionalization of organic membranes by polyelectrolyte multilayer assemblies: application to the removal of copper ions from aqueous solutions[END_REF][START_REF] Magnenet | Polyelectrolyte modification of ultrafiltration membrane for removal of copper ions[END_REF], nanofiltration [START_REF] Jin | Use of Polyelectrolyte Layer-by-Layer Assemblies as Nanofiltration and Reverse Osmosis Membranes[END_REF][START_REF] Stanton | Ultrathin, Multilayered Polyelectrolyte Films as Nanofiltration Membranes[END_REF][START_REF] Miller | Controlling the Nanofiltration Properties of Multilayer Polyelectrolyte Membranes through Variation of Film Composition[END_REF][START_REF] Malaisamy | High-Flux Nanofiltration Membranes Prepared by Adsorption of Multilayer Polyelectrolyte Membranes on Polymeric Supports[END_REF][START_REF] Liu | Size-Selective Transport of Uncharged Solutes through Multilayer Polyelectrolyte Membranes[END_REF][START_REF] Hong | Removal of Dyes, Sugars, and Amino Acids from NaCl Solutions Using Multilayer Polyelectrolyte Nanofiltration Membranes[END_REF][START_REF] Ouyang | Multilayer polyelectrolyte films as nanofiltration membranes for separating monovalent and divalent cations[END_REF][START_REF] Ng | A review on nanofiltration membrane fabrication and modification using polyelectrolytes: Effective ways to develop membrane selective barriers and rejection capability[END_REF] and SRNF membranes [START_REF] Ahmadiannamini | Multilayered polyelectrolyte complex based solvent resistant nanofiltration membranes prepared from weak polyacids[END_REF][START_REF] Ahmadiannamini | Influence of polyanion type and cationic counter ion on the SRNF performance of polyelectrolyte membranes[END_REF][START_REF] Ahmadiannamini | Multilayered PEC nanofiltration membranes based on SPEEK/PDDA for anion separation[END_REF][START_REF] Li | Solventresistant nanofiltration membranes based on multilayered polyelectrolyte complexes[END_REF][START_REF] Li | Morphology and performance of solvent-resistant nanofiltration membranes based on multilayered polyelectrolytes: Study of preparation conditions[END_REF][START_REF] Joseph | Up-scaling' potential for polyelectrolyte multilayer membranes[END_REF].

Tetrahydrofuran (THF), N,N-dimethylformamide (DMF), acetonitrile (ACN), dimethyl sulfoxide (DMSO), N-methyl-2-pyrrolidone (NMP) and dichloromethane (DCM) are some of the important industrial solvents that are still problematic for most of the currently available SRNF membranes [START_REF] Vandezande | Solvent resistant nanofiltration: Separating on a molecular level[END_REF].

These aprotic solvents demand an extremely high chemical stability from any membrane they come into contact with. Recently, PEM-based SRNF-membranes were reported to have promising and stable performance for some of these solvents [START_REF] Ahmadiannamini | Multilayered polyelectrolyte complex based solvent resistant nanofiltration membranes prepared from weak polyacids[END_REF][START_REF] Ahmadiannamini | Influence of polyanion type and cationic counter ion on the SRNF performance of polyelectrolyte membranes[END_REF][START_REF] Ahmadiannamini | Multilayered PEC nanofiltration membranes based on SPEEK/PDDA for anion separation[END_REF][START_REF] Li | Solventresistant nanofiltration membranes based on multilayered polyelectrolyte complexes[END_REF][START_REF] Li | Morphology and performance of solvent-resistant nanofiltration membranes based on multilayered polyelectrolytes: Study of preparation conditions[END_REF]. Furthermore, PEM-based membranes typically possess a highly charged top layer allowing the possibility for Donnan based exclusion of similarly charged molecules [START_REF] Krasemann | Selective Ion Transport across Self-Assembled Alternating Multilayers of Cationic and Anionic Polyelectrolytes[END_REF]. However, all previous studies utilized at least one strong PE, making their systems relatively insensitive to the pH at which the layers are formed. With such a combination of PEs, salt addition to the PE dipping solutions best controls the thickness and morphology of the adsorbed layer.

With the addition of salt the polymer chains become more coiled due to the screening of the charges along the polyelectrolyte chain. Furthermore, a the ionic strength can determine the structure of the formed polyelectrolyte complex. To describe this, Schlenoff et al., [START_REF] Schlenoff | Charge and Mass Balance in Polyelectrolyte Multilayers[END_REF] defined two types of charge compensations in polyelectrolyte complexation, intrinsic and extrinsic. In the former, a polymeric charge is balanced by an opposite charge of the other polyelectrolyte, while in the later most the polymeric charge is balanced by a counter ion. Increasing the ionic strength shifts the equilibrium from intrinsic to extrinsic charge compensation which leads to thicker, more mobile and more swollen layers [START_REF] Schlenoff | Charge and Mass Balance in Polyelectrolyte Multilayers[END_REF]. Thicker layers on a membrane lead to lower membrane permeabilities, whereas more open layers can increase the permeability [START_REF] De Grooth | The role of ionic strength and oddeven effects on the properties of polyelectrolyte multilayer nanofiltration membranes[END_REF]. However, the effectiveness of this parameter is often limited to a small range of salt concentrations because of either solubility problems or decomposition of the multilayer films when increasing the ionic strength [START_REF] Dubas | Polyelectrolyte Multilayers Containing a Weak Polyacid: Construction and Deconstruction[END_REF][START_REF] Kovacevic | Kinetics of Formation and Dissolution of Weak Polyelectrolyte Multilayers: Role of Salt and Free Polyions[END_REF].

The use of two weak PEs (both the polycation and the polyanion) can provide the opportunity to use the pH during coating to allow a large degree of control over membrane performance [START_REF] Hong | Optimization of flux and selectivity in Cl-/SO42separations with multilayer polyelectrolyte membranes[END_REF]. Unlike strong PEs, which remains charged over the entire pH range, the degree of ionization of weak PEs depends greatly on the deposition solution pH. So the resulting charge density of PEs and the thickness of deposited PEMs can be affected by both the pH and the salt concentration of the PE solutions [START_REF] Yoo | Controlling Bilayer Composition and Surface Wettability of Sequentially Adsorbed Multilayers of Weak Polyelectrolytes[END_REF][START_REF] Shiratori | pH-Dependent Thickness Behavior of Sequentially Adsorbed Layers of Weak polyelectrolytes[END_REF][START_REF] Izumrudov | Salt-Induced Multilayer Growth: Correlation with Phase Separation in Solution[END_REF][START_REF] Izumrudov | Ionization-Controlled Stability of Polyelectrolyte Multilayers in Salt Solutions[END_REF]. SRNF-membranes fabricated by the LbL deposition of weak PEs thus have an obvious advantage: the molecular structure inside of the PEM and the charge on the outside of the PEM can be controlled by an additional parameter, the pH. An added advantage of using weak PEs based membranes is that after fouling their cleaning can be made very easy using a sacrificial layer approach [START_REF] Ilyas | Multifunctional polyelectrolyte multilayers as nanofiltration membranes and as sacrificial layers for easy membrane cleaning[END_REF][START_REF] Ahmadiannamini | Sacrificial polyelectrolyte multilayer coatings as an approach to membrane fouling control: Disassembly and regeneration mechanisms[END_REF].

In this work, the first example of a PEM-based SRNF-membrane is presented prepared from weak PEs, a system where variation in pH can be used to tailor the membrane performance for specific solutes. More specifically, poly(acrylic acid) (PAA) and poly(allylamine hydrochloride) (PAH) based PEMs were used, as this system has already shown itself promising in traditional NF-applications [START_REF] Hong | Optimization of flux and selectivity in Cl-/SO42separations with multilayer polyelectrolyte membranes[END_REF][START_REF] Ilyas | Multifunctional polyelectrolyte multilayers as nanofiltration membranes and as sacrificial layers for easy membrane cleaning[END_REF]. The pH of the coating solution was used as a parameter to control the layer thickness and the molecular organization of the multilayers, which in turn leads to different separation behavior for different solutes. PEMs were prepared under three pH combinations (PAH/PAA: 7.5/7.5, 7.5/3.5 and 3.5/3.5) and were characterized on model surfaces in order to understand the effect of coating pH on their resulting multilayer composition.

The PEMs were then prepared on a hydrolyzed PAN support membrane under the same coating conditions and were characterized in terms of surface roughness, contact angle and IPA permeance. The performance of these multilayered membranes was characterized in terms of their ability to retain different organic solutes of varying size and charge from IPA in order to understand the effect of coating pH on membrane performance. Finally, the multilayered membranes were tested for their stability towards aprotic solvents such as acetonitrile (ACN), dimethylformamide (DMF) and tetrahydrofuran (THF) for long periods of time (> 50 h). Ontario, New York. Isopropyl alcohol (IPA), acetonitrile (ACN), tetrahydrofuran (THF), N,Ndimethylformamide (DMF) and dimethyl sulfoxide (DMSO) were obtained from VWR, Leuven, Belgium, and were used as solvents. Rose bengal (RB), acid fuchsin (AF), bromothymol blue (BTB), methylene blue (MB) and methyl orange (MO) were purchased from Sigma-Aldrich and used as solutes (see Table 1 for more details). The PEs used in this study were poly(allyl amine) hydrochloride (PAH; M w = 15,000 g•mol -1 ) and poly(acrylic acid) (PAA; M w = 15,000 g•mol -1 ). Sodium nitrate (NaNO 3 ) was used to adjust the ionic strength in all solutions. Sodium hydroxide (NaOH) and hydrochloric acid (HCl) were used to adjust the pH. PE solutions always contained 0.1 g•l -1 of polymer at various pH conditions and always with 50 mM of background electrolyte. Deionized water (Milli Q, 18.2 MΩ•cm) was used to rinse the membranes and to prepare polyelectrolyte solutions. All solutions were used within eight days after preparation.

Experimental

Membrane support preparation and LbL coating

The charged PAN-H support was prepared using a procedure reported earlier [START_REF] Ahmadiannamini | Multilayered polyelectrolyte complex based solvent resistant nanofiltration membranes prepared from weak polyacids[END_REF], but a short summary is given here. 15 wt. % PAN in DMSO was cast on a polypropylene/polyethylene support (FO 2471, The rinsing step removes any loosely bound or excess polyelectrolyte from the membrane surface and prevents complex formation in bulk solution. The described procedure was repeated to deposit the desired number of bilayers. The final membranes were then stored in Milli Q water for further use.

Preparation of PEMs on glass slides and silicon wafers

For ellipsometry studies, the PEMs were assembled on silicon wafers using a similar procedure as for the membrane support. For MB adsorption studies, PEMs were assembled on glass slides. Before assembly, the silicon wafers and glass slides were pre-treated with piranha solution (3:1 mixture of 98% H 2 SO 4 and 30% H 2 O 2 ; caution: piranha is a strong oxidizer and should not be stored in closed containers) for 2 h and rinsed in de-ionized water. Wafers and glass slides were used within 1 h after piranha treatment. PEMs were then deposited on wafers and glass slides [START_REF] Joseph | Up-scaling' potential for polyelectrolyte multilayer membranes[END_REF]45], and were dried with a nitrogen stream and stored at ambient conditions for characterization.

Characterization

Thickness of PEMs by ellipsometry

The dry thickness of PEMs deposited on polished silicon wafers under different pH conditions were measured using an ellipsometer (Woolam EC-400, M-2000V) at ambient conditions. All samples were dried at room temperature with a nitrogen stream prior to measurements.

Methylene blue adsorption on PEMs

Methylene blue (MB) staining technique was used to gain information on the charge of the surface layer (terminal layer) of PEMs. PEMs were assembled on glass slides, dried with a nitrogen stream, and subsequently immersed in aqueous 10 -3 M, pH 7.0 MB aqueous solutions for 10 min. With an immersion time of 15 minutes or less the dye can only diffuse into the outer most layers, making the results independent of the layer thickness [START_REF] Yoo | Controlling Bilayer Composition and Surface Wettability of Sequentially Adsorbed Multilayers of Weak Polyelectrolytes[END_REF][START_REF] Shiratori | pH-Dependent Thickness Behavior of Sequentially Adsorbed Layers of Weak polyelectrolytes[END_REF]. After immersion in the dye solution, the multilayer films were soaked in demi-water for 1 min to remove unbound dye and dried with a mild air flow. UV/Vis spectroscopy was used to determine the amount of MB adsorption on multilayer films. Absorbance was measured at a wavelength of 600 nm.

Fourier transform infrared (FTIR) spectroscopy

FTIR spectra of membranes were acquired using a spectrometer (Varian 670-IR, Varian Inc., USA) in absorbance mode. The samples were dried at room temperature for 24 h prior to FTIR measurement to minimize the influence of water.

Scanning electron microscopy (SEM)

SEM (Philips XL FEG30) was used to study the surface structure and cross section of the membranes.

Before use, the SEM samples were first gold coated.

Atomic force microscopy (AFM)

The surface morphology and roughness of dry membranes were measured by AFM (Agilent 5500) operating in tapping mode at ambient conditions (relative humidity ~30%). Commercial AFM cantilevers (PPP-NCSTR AFM probes from NanoAndMore GmbH) made of Si with a nominal spring constant of 7.4 N . m -1 and with a typical tip radius of less than 7 nm were used. AFM measurements were performed with scan areas of 1×1 μm 2 at 3 different locations for each of the samples. The average roughness (R a ) and the root-mean-square roughness (R rms ), which is given by the standard deviation of the topography height measurement data (Z values), were calculated using Eq. 1:

(1)

Where is the average of the height values within the given area, Z i is the current height, and N is the number of data points taken within the given area.

Contact angle

Optical contact angle measurements of a DI water droplet on the membrane surface were performed using a Krüss goniometer (drop shape analysis system DSA 10 Mk2). For contact angle measurements, the membranes were dried overnight at room temperature. The contact angle of the sessile drop of 3 μL of water on the membrane was measured three times for each membrane at 20°C in static mode, and the average value and standard deviation are reported. The contact angles were measured 5 s after the droplet was placed on the surface.

2.5 Membrane performance

Filtration and rejection

Five dyes (Fig. 2) were selected for the rejection experiments. The dye molecular weight range was chosen to be between 319 to 1100 Dalton. The selection of dyes was made such that it covers neutral, positive, negative, but also small and large molecular sizes. Filtration of membranes with dye solutions was performed using a stainless steel dead-end high-throughput apparatus (HTML, Belgium) [45] with 8 filtration cells, all with a membrane surface area of 1.77 cm 2 . The filtration setup was pressurized with nitrogen to 25 bar. To minimize concentration polarization during filtration, the feed solution was constantly stirred at 500 rpm. Dye rejections were calculated based on the differences between permeate and concentrate concentrations. The retention values were calculated from concentrations of the permeate and of the original feed solutions according to Eq. 2:

% 1 100

p f C R C (2)
Where C p and C f are solute concentration in the permeate and the feed, respectively. The permeation was stopped when the retention reached a constant value. All measurements were based on at least three samples, and the average values were used. The standard deviation on the measurements was about 5%.

All were used in a 35 μM concentration, with most experiments performed in IPA.

Long term filtration studies

To further demonstrate the chemical stability of these weak PEM-membranes for different solvents and for longer time-scales, ACN, THF and DMF were selected for long-term filtrations. Table 2 gives the physico-chemical properties of these solvents. Membranes prepared with 5 bilayers [7.5/3.5] were subjected to filtration with RB in IPA, THF, ACN and DMF separately for more than 50 h.

Results and discussion

The results and discussion section is divided into three major parts. The first part deals with the characterization of PEMs by ellipsometry and MB absorption to study their structure and composition. In the second part, the characterization of the support and the prepared multilayered membranes is given. The third part reports on the SRNF-membrane performance, with membranes prepared under different pH conditions, being tested for their retention of different organic molecules. The results so far show at the chosen pH conditions, three entirely different PEMs are created. For [7.5/7.5], a very thin layer with a high degree of intrinsic charge compensation is found. For [7.5/3.5], the thickest layer is created that is especially highly charged when terminated with PAA (-). Finally, for [3.5/3.5] a thick layer is obtained that always has many free acid groups available, likely an indication of very high extrinsic charge compensation and thus presence of free amine and acid groups. fingerlike pores (Fig. S3), while the thickness of the support layer used is in the range of 150 um.

Surface morphology and surface roughness

AFM images presenting the surface morphology of the membrane support and the multilayered membranes are displayed in Fig. 4. The surface structure seems to become smooth with the deposition of polyelectrolyte multilayers. These images thus suggest that the applied polyelectrolytes form an even layer on top of the support. In images b and c, some small cracks can be observed but as they are much shallower than the thickness of the PEM films these are not believed to be true defects.

Average surface roughness values of the membranes on 1×1 μm 2 scans in root mean square (R rms ) and average roughness (R a ) are presented in Table 4. When the charge density of both PAH and PAA is high (at pH 7.5/7.5), smooth layers are formed due to a flattened conformation of the polyelectrolytes. The layers formed seem to be somewhat more rougher when one of the polymers is only partially charged [START_REF] Shiratori | pH-Dependent Thickness Behavior of Sequentially Adsorbed Layers of Weak polyelectrolytes[END_REF] as at pH 7./5/3.5 and 3.5/3.5, since these thicker layers will have more coiled polymers. The [7.5/7.5] prepared layer is relatively hydrophobic, as could be expected for a PEM layer with a high degree of intrinsic charge compensation (few free -NH 2 and -COOH groups). On the other hand, the contact angle is considerably lower for [3.5/3.5] and a very high degree of extrinsic charge compensation can be expected. It is good to mention that also roughness influences contact angle measurements, however as the determined rough nesses were found to be very low (table 4) this effect could be neglected. 5.0±2.0 l·m -2 ·h -1 ·bar -1 for [3.5/3.5] membranes. The [7.5/7.5] membranes thus have a higher flux.

However, when taking into account the thickness, the [7.5/7.5] membranes have much lower permeability as they are 8 times thinner (Table 3) than [7.5/3.5] and 6 times thinner than [3.5/3.5] membranes. It seems that not only is this layer quite thin, it is also quite dense, a typical behavior expected for PEM membranes with a high degree of intrinsic charge compensation [START_REF] De Grooth | The role of ionic strength and oddeven effects on the properties of polyelectrolyte multilayer nanofiltration membranes[END_REF]. were selected as solutes (Table 1). Filtrations were performed with membranes coated from 50 mM salt solutions and with different pH combinations for PAH/PAA [7.5/7.5, 7.5/3.5 and 3.5/3.5]. PAN-H support used to prepare PEMs based SRNF membranes has RB retention of 46% with a 2.2 l·m -2 ·h -1 ·bar -1 permeance. It is important to mention that the retention of organic dyes in SRNF is a complex mixture of factors including Donnan exclusion, size exclusion and the mutual affinities between the dye, the solvent and the membrane [START_REF] Vandezande | Solvent resistant nanofiltration: Separating on a molecular level[END_REF][START_REF] Marchetti | Molecular Separation with Organic Solvent Nanofiltration: A Critical Review[END_REF][START_REF] Stamatialis | Observations on the permeation performance of solvent resistant nanofiltration membranes[END_REF][START_REF] Gevers | Physico-chemical interpretation of the SRNF transport mechanism for solutes through dense silicone membranes[END_REF]. Furthermore, the dye can potentially foul the SRNF-membrane, either by adsorption on top of the PEM layer, adsorption in the PEM layer or adsorption on the charged membrane support. Such fouling will lead to lower permeance but can substantially increase the initial rejection. This is the first investigation of the SRNF performance of multilayered membranes prepared from weak PEs and because of the mentioned complexity, focus is to show the easiness of the approach in combination with its versatility and on comparing the performance of the differently prepared PEM membranes, rather than determining the exact rejection mechanism for all membrane/dye combinations.

In Fig. 6, the filtration results of multilayered membranes with 5 bilayers of PAH/PAA, prepared from polymer solutions with different pH combinations are given. The retention and permeance results are given for negatively charged solutes (RB and AF) in IPA. For all membranes, more than 90% retention for RB is observed but with a sharp drop in permeability (Fig. 6a). Possibly, a strong solute adsorption in/on the membranes takes place which then densifies (i.e. bivalent dyes can crosslink) the membranes [START_REF] Li | Solventresistant nanofiltration membranes based on multilayered polyelectrolyte complexes[END_REF].

However, the [7.5/7.5] membrane is more permeable than the others with similar rejection. We hypothesize that the lower amount of free -NH 2 and -COOH groups in this layer, also leaves less adsorption sites available for the dyes. Consequently, [3.5/3.5] has the lowest permeability as compared to pure IPA filtration as it has most free -COOH and -NH 2 groups. AF is a smaller molecule than RB and also carries two negative charges. Membranes show slightly less rejection for AF as compared to RB (Fig. 6b), probably because of the smaller size. Especially membrane [7.5/7.5] (Fig. 6b) shows good rejection while retaining a high operational permeability, in agreement with the RB results. It is surprising that the highly negative layer of the [7.5/3.5] membrane adsorbs so many double negatively charged molecules. It is possible that the molecules do not adsorb at the top of the layer, where the excess of charge is localized, but rather deeper inside the layers.

In Table 5, the comparison is given of the SRNF performance of membranes from the present work with previously reported PEM membranes prepared from two strong PEs or one strong and one weak PE [START_REF] Ahmadiannamini | Multilayered polyelectrolyte complex based solvent resistant nanofiltration membranes prepared from weak polyacids[END_REF][START_REF] Li | Morphology and performance of solvent-resistant nanofiltration membranes based on multilayered polyelectrolytes: Study of preparation conditions[END_REF]. The data are given for 5 bilayer thick PEM membranes for RB retention from IPA. It shows that PEM membranes prepared from both weak PEs have a RB retention comparable to that of membranes prepared from both strong PEs or from one of the weak PEs. Present work even shows the highest observed retention for RB within this comparison. To further explore the performance of these membranes, the retention of the relatively large neutral dye molecule BTB (Mw 624 Da) was studied. For BTB, membranes [7.5/7.5] show decent performance both in terms of permeance and rejection (See Fig. 7). Here, surprisingly, the membrane [3.5/3.5] also shows reasonable performance, but the overall retention of BTB by all these three membranes is lower than for the strongly charged AF and RB. Less dye adsorption would lead to less densified films and a lower retention.

Next, MO (with a single negative charge) and MB (with a single positive charge) retentions are given as examples of smaller sized dyes (Fig. 8a andb). For MO (-), the membrane [7.5/3.5] shows good rejection and high permeance. This would be expected for such a highly negatively charged layer, having Donnan exclusion as the main separation mechanism. The other less negatively charged layers show much lower retentions, while [3.5/3.5] again suffering from very low permeabilities. For MB (+), none of the membranes show a good combination of retention and permeance. This is expected as the membranes are all negatively charged. Especially for [7.5/3.5] and [3.5/3.5], this leads to very low permeances, likely because of adsorption and densification of the layers. So by varying the pH of coating PEs, PEM membranes can be prepared with very different performances for different dyes in SRNF.

To see the role of the terminating layer charge on retention performance of membranes in terms of electrostatic repulsion, also filtrations were performed with membranes terminated with positive polyelectrolyte PAH (5.5 bilayers). RB and AF retentions for membranes with 5.5 bilayers of PAH/PAA at different pH combinations with a terminating layer of PAH (+) are given in Table 6. For big dyes, retentions do not significantly change with the type of terminating layer, except for RB and AF for [7.5/3.5], where it is slightly less than compared to PAA (-) terminated layers. Strong interactions of negatively charged solutes with the polycation from the membrane layers might densify the membranes [START_REF] Li | Solventresistant nanofiltration membranes based on multilayered polyelectrolyte complexes[END_REF] and make the membranes less positively charged, which in turn increases the retention and reduces permeance for these layers. The effect of the terminating layer charge is more clear in the retention of small solutes. MO (-) is more rejected by the membranes with PAA (-) terminating layer than by the PAH (+) terminating layers and vice versa for MB (+) retention with membranes prepared at pH 7.5/3.5 (Table 5). For membranes [3.5/3.5], retention of MO (-) is more for a PAH (+) terminating layer than for a PAA (-) terminating layer, but the permeance is also lower, indicating that the higher retention is the result of more adsorption and more densification. For all dyes, [7.5/7.5] membranes show relatively high permeances coupled with relatively good rejections compared to the other membranes (Fig. 6, 7 and 8). The low degree of extrinsic charge compensation is expected to be responsible for this effect. For MO, membrane [7.5/3.5] gives the best performance, likely due to the high negative charge of this layer. Membrane [3.5/3.5] shows only decent performance for the neutral dye BTB. Ionic dyes thus seem to adsorb strongly to this layer, possibly because of a very high degree of extrinsic charge compensation. The main point however, is that by varying the pH, PEM membranes can be prepared with very different performances for different dyes in SRNF. For this system of multilayers, the pH can thus be used as a tuning parameter to optimize the performance for a specific application.

Influence of solvent type and long term stability

To evaluate the membrane performance over long time-scales, filtrations were performed with RB in different solvents such as THF, DMF and ACN. The membranes, coated with 5 bilayers of PAH/PAA at pH [7.5/3.5] were used as they showed the highest RB retention of all tested membranes (Fig. 6a). The membranes exhibited very good performance in terms of RB retention (>90%) and permeance for all of these organic solvents (Fig. 9). The maximum retention of RB is observed in IPA and DMF with a value of 99 and 95% respectively. The permeability was found to be in the following order of ACN>THF>DMF>IPA. This variation in permeation behavior for different solvents may be attributed to solvent-membrane interactions and the physico-chemical properties of the solvents, like molar volume and viscosity. However, it is also important to consider the effect of interactions between membrane-solute, membrane-solvent, and solutesolvent [START_REF] Bhanushali | Advances in Solvent-Resistant Nanofiltration Membranes[END_REF]. A part of the viscosity effect we can see in (Fig. 9) where IPA, with the highest viscosity,

shows the lowest permeance and ACN, with the lowest viscosity, is the highest. The permeability trend observed in the present study is similar to that observed in previously reported SRNF membranes prepared with 20 bilayers of PDDA/PAA [START_REF] Ahmadiannamini | Multilayered polyelectrolyte complex based solvent resistant nanofiltration membranes prepared from weak polyacids[END_REF].

To further evaluate the long term stability of these multilayered membranes, filtrations were performed DMF and the membrane, or RB and the membrane. The membrane seems to take up RB over time, densify and become less permeable, while the rejection shoots up (Fig. 10b).

In our investigation we have focused on pH as a tuning parameter. We stress that the membranes could be further improved by including the ionic strength as a variable. For example, coating at low ionic strength where intrinsic charge compensation between the layers is high and layers are thin [START_REF] De Grooth | The role of ionic strength and oddeven effects on the properties of polyelectrolyte multilayer nanofiltration membranes[END_REF][START_REF] Ilyas | Multifunctional polyelectrolyte multilayers as nanofiltration membranes and as sacrificial layers for easy membrane cleaning[END_REF] would lead to less free charges being available for complexation with the dye molecules, and could further reduce fouling of the PEM layers by the dye molecules.

Conclusions

This manuscript encompasses the first investigation into the use of weak polyelectrolyte multilayers for SRNF applications. PEM-based SRNF-membranes of the weak polyelectrolytes PAH and PAA were successfully prepared by the LbL method with varying the pH of the coating solutions to tune the charge density and structure of PEs, and thus further control the morphology and performance of the resulting membranes for specific dyes from organic media. Membranes [7.5/7.5] have high a permeance coupled with a relatively good retention performance for all the dyes because of a low degree of extrinsic charge compensation combined with a dense structure and a thin layer. Membranes prepared under pH conditions [7.5/3.5] and [3.5/3.5] have more free acid and amine groups and were found to be much more susceptible to dye adsorption, leading to a pronounced film densification, a low permeance, but in some case excellent dye retentions. Specifically for the negative dye methyl orange, a PAA terminated [7.5/3.5] membrane was found to have good performance in terms of retention and permeance, likely due to the high negative charge of that multilayer film. The resulting membranes also showed long term (more than 50 h) stable performance for organic solvents, including troublesome aprotic solvents of industrial importance like THF, DMF and ACN. The results of this study provide a versatile and simple way of using pH to tune performance of weak PEs based multilayered SRNF-membranes for specific applications.

2. 1 .

 1 ChemicalsPolyacrylonitrile (PAN, MW= 150,000 Da) was purchased from Scientific Polymer Products Inc.,

Fig. 1 .

 1 Fig. 1. Overview of the LbL assembly of PEMs of PAH/PAA on a charged membrane support.

MethylFig. 2 .

 2 Fig. 2. Chemical structures of solutes used for rejection experiments.

3. 1

 1 Properties of PEMs 3.1.1 ThicknessThe thickness of multilayer films of PAH/PAA deposited under different pH combinations (7.5/7.5, 7.5/3.5 and 3.5/3.5 for PAH/PAA) at a polymer concentration of 0.1 g•l -1 in a 50 mM NaNO 3 aqueous solution was determined by ellipsometry. The used salt concentration was selected based on a previous study where optimal growth of these PEMs was achieved at 50 mM[START_REF] Ilyas | Multifunctional polyelectrolyte multilayers as nanofiltration membranes and as sacrificial layers for easy membrane cleaning[END_REF]. The measured average thickness for 5 bilayer thick multilayers is reported in Table3. Clearly, the thickness of the layers is influenced by the pH of the coating solution of the polycation and the polyanion. For weak polyelectrolyte pairs such as PAH/PAA, the thickness, composition and organization of the multilayers can be systematically tuned by varying the pH of each PE solution[START_REF] Yoo | Controlling Bilayer Composition and Surface Wettability of Sequentially Adsorbed Multilayers of Weak Polyelectrolytes[END_REF][START_REF] Shiratori | pH-Dependent Thickness Behavior of Sequentially Adsorbed Layers of Weak polyelectrolytes[END_REF][START_REF] Choi | Influence of the Degree of Ionization on Weak Polyelectrolyte Multilayer Assembly[END_REF]. The pH controls the charge density of the weak PEs by controlling the degree of dissociation of the basic and acidic groups on the polymer chains. PAA with pKa of 6.5 and PAH with pKa 8-9[START_REF] Choi | Influence of the Degree of Ionization on Weak Polyelectrolyte Multilayer Assembly[END_REF], are nearly fully charged (80-90% charged groups) at a pH condition of [7.5/7.5] and thus form thin layers upon deposition, requiring just a small amount of material to compensate for all charges from a previously deposited layer, most likely with a high degree of intrinsic charge compensation inside the bulk of the layers. The thickest layers are formed when one of the PEs is only partially charged (PAA at pH 3.5) and other is fully or nearly fully charged (PAH at pH 3.5 and 7.5 respectively). The partially charged polymer segments lead to a more coiled conformation and favor diffusion of counter ions of the employed salt into the multilayer, forming more swollen layers with more mobile chains leading to more extrinsic charge compensation of the polymeric charges.

2

 2 Adsorption of the cationic dye methylene blue (MB) and zeta potential A dye test was performed on PEMs on glass surfaces to determine the excess of free carboxylic acid groups available as binding sites (for positively charged solutes) on the surface of PEMs. Fig.3shows the results of the MB test performed with 5 (ending with PAA) and 5.5 (ending with PAH) bilayers of PEMs deposited on glass slides. A higher absorbance of MB indicates the presence of more free carboxylic acid groups. At pH condition [7.5/7.5] for PAH/PAA, both PEs have a high charge density (fully ionized) producing polycation-polyanion contact pairs and leaving only little free carboxylic acid groups for the adsorption of positive dye. Thus, regardless whether PAH (+) or PAA (-) is the last layer, few free functional sites will be left on the surface with most of the polyions bound with an oppositely charged group. As can be seen from these results, just a small difference in MB absorbance exists for PAH (+) or PAA (-) terminated layers, and these values are always low.However, at the pH combinations [7.5/3.5] and [3.5/3.5], an unbalanced ionization state (PAA only partly charged during deposition at pH 3.5) results in many free acidic groups. For [7.5/3.5], this result in a very high absorbance for the PAA (-) terminated layer, indicating a highly charged layer. Still for the PAH (+) terminated [7.5/3.5] layer, the absorbance drops strongly, which indicates just a small amount of free unbound acid groups. This is an indication of the normal charge inversion leaving just a small amount of free unbound acid groups. For [3.5/3.5], absorbance is always rather high. This is an indication that this layer has many extrinsically compensated ion pairs (free NH 2 and COOH groups).

Fig. 3 .

 3 Fig. 3. Methylene blue absorbance measured at 600 nm of 5 and 5.5 bilayers of PAH/PAA prepared at three different pH combinations with polymer concentration at 0.1 g•l -1 and 50 mM salt.

3. 2

 2 Characterization of support and PEMs based membranes 3.2.1 ATR-FTIR ATR-FTIR was used to confirm the hydrolysis of the PAN support. Fig. S1 presents the ATR-FTIR spectra of a PAN and a hydrolyzed PAN membrane. The peaks at 2243 and 1451 cm -1 are due to stretching vibrations of the CN moiety of the PAN membrane support [48]. For hydrolyzed PAN, most of the CN groups convert to COO -groups, leading to a peak reduction at 2243 cm -1 and the emergence of a new peak at approximately 1674 cm -1 , which corresponds to the C=O bond in the COO -groups. FTIR results thus confirm the hydrolyzation of PAN into a negatively charged membrane support. 3.2.2 SEM Surface images of the PAN-H support and PEMs based SRNF membranes [7.5/7.5], [7.5/3.5] and [3.5/3.5] showed no significant difference in the surface morphology and are given in the supporting information (Fig. S2). A cross section of the PAN-H support shows a very open pore structure with

Fig. 4 .

 4 Fig. 4. AFM images of (a) support; and multilayered SRNF membranes prepared with PAH/PAA pH of (b) [7.5/7.5] (c) [7.5/3.5] (d) [3.5/3.5] with polymer concentration of 0.1 g•l -1 and 50 mM salt.

3. 3

 3 SRNF performance of PEM based membranes 3.3.1 Pure solvent permeability After coating the support with PEMs, the pure solvent permeability of the support and the prepared membranes was measured. The pure IPA permeance values were: 127.0±7.0 l·m -2 ·h -1 ·bar -1 for the PAN-H support, 7.0±0.5 l·m -2 ·h -1 ·bar -1 for [7.5/7.5] membranes, 4.0±1.0 l·m -2 ·h -1 ·bar -1 for [7.5/3.5] membranes, and

Fig. 5 .

 5 Fig. 5. Sessile droplet water contact angle of 5 and 5.5 bilayer membranes [7.5/7.5, 7.5/3.5 and 3.5/3.5] prepared at a polymer concentration 0.1 g•l -1 and 50 mM salt. In the 5 bilayer membranes the terminating layer is of PAA and in 5.5, PAH is the terminating layer.

3. 3 . 2

 32 Influence of solute size and charge Five different dyes with double negative charges (RB, Mw 1017 Da and AF, Mw 558 Da), single negative charge (MO, Mw 327 Da), positive charge (MB, Mw 320 Da) and without charge (BTB, Mw 624 Da)

Fig. 6 .

 6 Fig. 6. SRNF performance of membranes for large sized charged solutes by 5 bilayers of PAH/PAA (prepared at three pH combinations with 0.1 g•l -1 polymer and 50 mM salt) (a) retention and permeance of RB from IPA solution; (b) retention and permeance of AF from IPA solution.

Fig. 7 .

 7 Fig. 7. SRNF performance of membranes for BTB from IPA solution with 5 bilayers of PAH/PAA (prepared at three pH combinations with 0.1 g•l -1 polymer and 50 mM salt).

Fig. 8 .

 8 Fig. 8. SRNF performance of membranes for small sized charged solutes with 5 bilayers of PAH/PAA (prepared at three pH combinations with 0.1 g•l -1 polymer and 50 mM salt) (a) retention and permeance of MO (-) from IPA solution; (b) retention and permeance of MB (+) from IPA solution.

Fig. 9 .

 9 Fig. 9. SRNF properties of weak PEM membranes for RB in different organic solvents. Tested membranes are [7.5/3.5] with 5 bilayers of PAH/PAA prepared with 0.1 g•l -1 polymer and 50 mM salt. (a) Short term (5 h) SRNF performance for polar aprotic solvents.

Fig. 10 .

 10 Fig. 10. Long term SRNF performance of [7.5/3.5] membranes for (a) THF and (b) DMF and (c) ACN (tested membranes are with 5 bilayers of PAH/PAA prepared with 0.1 g•l -1 polymer and 50 mM salt).

Table 1

 1 Properties of solutes used in this study.

Table 2

 2 Physico-chemical properties of solvents used[START_REF] Smallwood | Handbook of Organic Solvent Properties[END_REF].

	Solvent	Molar volume	Viscosity	Molar volume/viscosity	Dielectric
		(cm 3 ·mol -1 )	(cP)	(cm 3 ·mol -1 ·cP -1 )	constant
	IPA	76.92	2.00	38.46	18.30
	DMF	77.43	0.82	94.43	36.70
	ACN	52.86	0.38	139.10	37.50
	THF	81.08	0.55	147.42	7.60

Table 3

 3 Average thickness of PEMs with 5 bilayers of PAH/PAA (prepared at three pH combinations with 0.1 g•l -1 polymer and 50 mM salt).

Table 4

 4 Calculated surface roughness of membranes with 5 bilayers of PAH/PAA (prepared at three pH combinations with 0.1 g•l -1 polymer and 50 mM salt).

			pH[PAH/PAA]
	Roughness	Support	[7.5/7.5]	[7.5/3.5]	[3.5/3.5]
	R rms (nm)	2.6±0.4	0.5±0.1	2.1±0.2	3.5±0.7
	R a (nm)	2.4±0.4	0.4±0.1	1.4±0.1	1.5±0.1

3.2.4 Contact angle with water

Table 5

 5 Comparison of SRNF performance of PEMs based membranes for RB in IPA.

	PEMs membrane	Permeance	Retention	References
		(l·m -2 ·h -1 ·bar -1 )	(%)
	(PAH/PAA, pH 7.5/7.5) 5 50 mM	0.21±0.08	95±4	Present work
	(PAH pH 7.5/PAA pH 3.5) 5 50 mM	0.05±0.09	99±1
	(PDDA/PAA, pH 7/4) 5 0 mM	0.03	97±2	[27]
	(PDDA/PAA, pH 7/4) 5 100 mM	0.06	92±5
	(SPEEK/PDADMAC) 5 0 mM	0.40	94	[31]
	(SPEEK/PDADMAC) 5 100 mM	0.08	98

Table 6

 6 SRNF performance of PAH (+) terminating multilayered membranes with 5.5 bilayers of PAH/PAA (prepared at three pH combinations with 0.1 g•l -1 polymer and 50 mM salt).

			Retention (%)		Permeance l·m -2 ·h -1 ·bar -1
	PEM membranes	[7.5/7.5]	[7.5/3.5]	[3.5/3.5]	[7.5/7.5]	[7.5/3.5]	[3.5/3.5]
	RB	93±2	96±2	91±2	0.22±0.12	0.07±0.01	0.01±0.00
	AF	86±4	86±8	96±2	0.88±0.22	0.45±0.30	0.02±0.00
	MO	23±1	31±4	60±8	1.00±0.30	2.00±0.22	0.10±0.02

Acknowledgments S. Ilyas acknowledges the European Commission -Education, Audiovisual and Culture Executive Agency (EACEA), for her PhD scholarship under the program: Erasmus Mundus Doctorate in Membrane Engineering -EUDIME (FPA N° 2011-0014, Edition III, http://eudime.unical.it/).

Highlights

We present a first example of SRNF membranes based on weak polyelectrolyte multilayers. Within this approach SNRF membrane performance can be tuned by the pH of the multilayer coating solution. Membrane performance is studied in detail with the retention of 5 different organic dyes. Thin and dense membranes with mostly intrinsic charge compensation provide overall better performance. Long term stable membranes for troublesome organic aprotic solvents (THF, DMF and ACN).


Mylène Campana
email: mcampana@laas.fr

Pierre Fernbach
email: pfernbac@laas.fr

Steve Tonneau
email: stonneau@laas.fr

Michel Taïx
email: taix@laas.fr

Jean-Paul Laumond

Ballistic motion planning for jumping superheroes

Keywords: ballistic trajectory, jump, motion planning, animation

We present a framework capable of automatically planning and animating dynamic multi-contact jumping trajectories for arbitrary legged characters and environments. Our contribution is the introduction of an approximate yet efficient multi-contact impulse formulation, used at the motion planning phase. We combine this criterion with a heuristic for estimating the contact locations without explicitly computing them, which breaks the combinatorial aspect of contact generation. This low dimensional formulation is used to efficiently extend an existing ballistic motion planner to legged characters. We then propose a procedural method to animate the planned trajectory. Our approach thus results from a trade-off between accuracy of the law of physics and computational efficiency. We empirically justify this approach by demonstrating a large variety of behaviors for four legged characters in five scenarios.

Introduction

Synthesizing high quality motions for legged characters in arbitrary environments is challenging: the dimension of the problem is high, equal to the number of degrees of freedom, the environment is big and complex, and the motion is additionally constrained by the dynamics of the world. For this reason, motion synthesis is typically addressed with a decoupled approach. First, in the path planning phase a collision-free path is found for the character using a PRM based approach [START_REF] Kavraki | Probabilistic roadmaps for path planning in high-dimensional configuration spaces[END_REF]. Then, in the animation phase a motion is computed and played along the path. Using this decoupled approach, we consider the issue of synthesizing highly dynamic jumping motions for legged characters.

For concision our scope is restricted to the generation of sequences of jumping motions, disregarding alternations with classical walking / running trajectories at the moment. Thus, given start and goal configurations for an arbitrary character in an arbitrary environment, our framework outputs a motion described as a sequence of parabolic jumps [START_REF] Campana | Ballistic motion planning[END_REF], respecting a set of kinematic and dynamic constraints for the character.

Our key idea is the introduction of a simplified multi-contact model within a sampling based planner. We relax the dynamics of the problem with two hypotheses, which assimilate our character with a superhero:

• First, we consider an impulse formulation. Our character is thus assumed to be able to exert a large force instantaneously, as proposed in [START_REF] Campana | Ballistic motion planning[END_REF].

• Then, when verifying the dynamic constraints on the character, we verify the Newton equation of the motion, but assume that the Euler equation is always satisfied. This means that we assume that the character can generate an arbitrary angular momentum.

1

These two hypotheses lead to an efficient, low dimensional formulation of the motion planning problem, solved with a sequence of simple geometric tests, while partially capturing the dynamics model of the character to compute plausible trajectories. To handle the combinatorial aspect of the contact generation problem, we decouple the trajectory planning phase from the contact generation phase, thanks to a heuristic based on the reachable workspace of the character. Finally, to propose a complete framework, we implement a procedural method to automatically animate the computed trajectory. This is achieved by combining a contact generator proposed by [START_REF] Tonneau | Using task efficient contact configurations to animate creatures in arbitrary environments[END_REF] with a dedicated interpolation scheme that we introduce. Our main contribution is thus a computationally efficient ballistic multi-contact motion planner. It extends the PRM planner proposed by the authors of [START_REF] Campana | Ballistic motion planning[END_REF], limited to planning for a single mass point, to arbitrary legged characters generating non-coplanar contact configurations (Sections 3 and 4). Our second contribution is an automatic interpolation method for animating multi-contact jumping trajectories (Section 5).

Related Work

We propose a purely procedural motion synthesis methods, able to generate dynamic motions through multi-contact interactions, with a focus on the motion planning phase. Our work is thus related to physically based motion synthesis methods, as well as motion and contact planning techniques. It is also to be compared with respect to data-driven animation techniques (such as motion capture). This related work provides an overview of these approaches, before focusing to specific contributions to the jumping motion synthesis problem.

A standard motion synthesis technique is to use pre-existing animations, produced either by 3D artists or through motion capture. These approaches are favored because of the high quality of the resulting animations [START_REF] Kovar | Motion graphs[END_REF]. When the environment differs from the original one, local editing methods can be used to adapt the motion [START_REF] Witkin | Motion warping[END_REF][START_REF] Pettré | A 2-stages locomotion planner for digital actors[END_REF][START_REF] Esteves | Animation planning for virtual characters cooperation[END_REF][START_REF] Holden | A deep learning framework for character motion synthesis and editing[END_REF], but they produce unnatural results when the deformation becomes too important.

Rather than using reference animations, physics-based methods synthesize motions with algorithms based on a simplified model of the law of physics [START_REF] Mordatch | Discovery of complex behaviors through contact-invariant optimization[END_REF]. Walking pattern generators combine finite state machines with proportional derivative control to generate periodic locomotion [START_REF] Yin | Simbicon: Simple biped locomotion control[END_REF]. The number of states increases with the complexity of the environment and the desired motions, such that reinforcement based learning techniques are required to overcome the limitations [START_REF] Peng | Dynamic terrain traversal skills using reinforcement learning[END_REF][START_REF] Liu | Guided learning of control graphs for physics-based characters[END_REF]. However, they are based on simplifying assumptions regarding the location and periodicity of the contacts, which do not hold in arbitrary environments.

Sampling based planning algorithms such as PRM [START_REF] Kavraki | Probabilistic roadmaps for path planning in high-dimensional configuration spaces[END_REF] and RRT [START_REF] Lavalle | Algorithmic and Computational Robotics: New Directions, chapter Rapidly-Exploring Random Trees: Progress and Prospects[END_REF] are the standard methods used to find a collision-free path in a complex 3D environment. Our approach follows [START_REF] Tonneau | A reachability-based planner for sequences of acyclic contacts in cluttered environments[END_REF], with an interactive method that breaks the combinatorial by decoupling the planning of a feasible path and the contact generation.

Overall, contributions that have studied jumping motions do not focus explicitly on path planning, rather on the physically accurate adaptation or synthesis of the jump animation, which is complementary with our approach.

As such, most robotics contributions to jumping scenarios assume coplanar contacts on flat surfaces, and do not consider collision avoidance in general [START_REF] Papadopoulos | On robot gymnastics planning with non-zero angular momentum[END_REF][START_REF] Wensing | Development of high-span running long jumps for humanoids[END_REF]. Similarly computer graphics contributions assume the ballistic jumping motion is already precomputed [START_REF] Reitsma | Perceptual metrics for character animation: Sensitivity to errors in ballistic motion[END_REF], and focus on the preparation phase [START_REF] Sulejmanpašić | Adaptation of performed ballistic motion[END_REF][START_REF] Reitsma | Effect of character animacy and preparatory motion on perceptual magnitude of errors in ballistic motion[END_REF], or the reception phase [START_REF] Ha | Falling and Landing Motion Control for Character Animation[END_REF].

To plan highly dynamic motions, recent contributions have proposed hybrid approaches, using both data-driven and physics based methods to generate motions [START_REF] Levine | Physically Plausible Simulation for Character Animation[END_REF][START_REF] Tonneau | Contact re-positioning under large environment deformation[END_REF]. In particular, [START_REF] Yamane | Planning and synthesizing superhero motions[END_REF] propose to deform motion capture trajectories using physics based heuristics. Interestingly, while the authors make similar assumptions to us (they consider characters with surhuman capabilities), their approach is opposite to ours: they constrain a motion adaptation to respect the Euler equation of motion, given customizable bounds on the angular momentum of the character. Because the contact locations are predefined relatively to the center of mass by the reference motion capture animation, the linear part is necessarily validated. However, this limitation once again prevents generalization to arbitrary environments, where the contacts must be changed to obtain a valid motion. We, on the other hand, focus on the Newton equation that regularizes the linear momentum, and do not constrain the angular momentum, making our method able to handle arbitrary contact locations.

Relaxed contact model for impulse based motion planning

Our contact model is based on Coulomb's non-slipping constraint, which we recall briefly in this section. We generalize the constraint to handle an arbitrary number of contacts, by expressing it at the center of mass (COM) of the robot. From this formulation, assuming an impulse formulation of the model, we propose a conservative condition for non-slipping, that is sufficient, but not necessary. This condition, reduced to a simple geometric test, is then used in our motion planner, presented in Section 4.

Definitions. We consider a character of mass m ∈ R, of COM position c ∈ R 3 , assimilated to the geometric root. ċ is the velocity of c.

For the i-th contact point p i , 1 ≤ i ≤ h, K i is the associated convex friction cone, considering a friction coefficient µ i , and a surface normal n i . f i is the contact force applied at p i . Finally, g = [0, 0, -9.81] T is the gravity acceleration.

Non-slipping constraint for an arbitrary number of contacts. The non-slipping condition is given by Coulomb's law: the contact will not slip if the contact reaction force f i lies strictly within the cone K i . Now, if we consider an arbitrary number h of contacts, each reaction force f i must lie in its associated cone K i . The resulting force f c applied at the center of mass c of the character, is defined as the sum of all the forces f i . It follows that the set of admissible resulting forces K such that the non-slipping condition is respected is defined as the Minkowski sum of each individual friction cone:

K = {f 1 + • • • + f h |f i ∈ K i } (1)
As a Minkowski sum of convex cones, K is itself a convex cone [START_REF] Boyd | Convex Optimization[END_REF]. However, the analytical form of K is unknown. Therefore in this work, we use a conservative approximation K c .

We define n

c = h i=1 n i and µ c = min i=1...h µ i .
The cone K c , originating at c, of normal n c , and friction µ c , is included in K by construction, and is thus a conservative approximation of K. Force closure contact configurations, where the resulting normal n c is null, are considered invalid in this formulation. Although this formulation is intuitively really conservative, it has an analytical form that makes it extremely efficient to compute. We justify the interest of this formulation with the variety of the solutions found by our planner.

Impulse model of the non-sliding constraints. We extend the model proposed in [START_REF] Campana | Ballistic motion planning[END_REF] to multicontact motions, by applying it to the centroidal cone K c . Assuming that the motion is determined as a sequence of jumps, we want to verify the non-slipping conditions between the landing and takeoff phases of two consecutive jumps. With an impulse model, we make the hypothesis that the character can only apply large contact forces instantaneously, and assume that the transition between the landing and takeoff phases is instantaneous.

Under the conservative non-slipping condition, Newton's equation is written:

m dċ dt -mg = f c | f c ∈ K c
Under the impulse hypothesis dt is instantaneous, such that the action of gravity is not measurable:

m dċ ≈ f c dt | f c ∈ K c
Thus, dċ and f c are colinear, and dċ ∈ K c . Furthermore, the instantaneous velocity shift dċ is constrained by the relationship

dċ = ċt -ċl (2)
where ċt and ċl are respectively the takeoff and landing velocities.

We now make the observation that, by definition of a convex cone:

-ċl ∈ K c and ċt ∈ K c =⇒ dċ ∈ K c (3)
(3) provides a sufficient condition for satisfying the non-slipping condition. This conservative condition reduces the dimensionality of the motion planning problem, because it removes the relationship between the entering velocity and the exiting velocity of a node, required by classical kinodynamic approaches [START_REF] Kunz | Probabilistically complete kinodynamic planning for robot manipulators with acceleration limits[END_REF]. With a kinodynamic planner, to verify whether a trajectory can be connected with another one, it is required to extend the state space with the velocities. In our case, this is required to verify whether there exists a ċ ∈ K c that satisfies Equation 2 each time we want to connect a new trajectory. Including the velocities doubles the dimensionality of the problem. The conservative condition removes this need: any takeoff velocity ċt that belongs to the cone is automatically achievable from the previous one, independently of ċl .

Path planner

We integrate the multi-contact non-slipping condition within a PRM-based motion planner to consider the dynamics constraints that apply to the character at the planning phase. To reduce the dimensionality of the problem and break the combinatorial, the planner does not consider the complete character model at this phase, nor the explicit computation of contact configurations.

We first present and justify the reduced character model used by the planner. We then introduce a heuristic to estimate the contact location of the configuration without computing them explicitly. Then, we give the details of the planning algorithm.

A reduced character model for contact location estimation. We consider a legged character, described by a kinematic chain R, composed of a root R 0 , and

l limbs R k , 1 ≤ k ≤ l.
The root has a minimum of r ≥ 6 degrees of freedom (dofs), which describe its position and orientation in the world frame. The additional dofs describe the articulations of the back of the character. R is fully described by a configuration q ∈ R r+n . We define q 0 ∈ R r as the configuration of the root of the character.

At the planning phase, to verify the non-slipping condition, we need an estimation of the contact locations. To avoid dealing with the combinatorial and computational complexity of contact generation, we introduce a contact estimation heuristic, based on a dual, low-dimensional representation of the character, introduced by [START_REF] Tonneau | A reachability-based planner for sequences of acyclic contacts in cluttered environments[END_REF]. We recall it here for completeness.

Rather than considering the complete body configuration, the planner only considers the root configuration q 0 . To perform collision detection, the complete body is approximated with a bounding shape W 0 . Additionally, for each limb k, we attach to the root a shape W k , computed as the reachable workspace of the limb. For details on the computation of the W shapes, we refer the reader to [START_REF] Tonneau | A reachability-based planner for sequences of acyclic contacts in cluttered environments[END_REF].

Our contribution is to use the reachable workspace of each limb for contact estimation. Given a configuration q 0 , we assume that if W k is in collision with the environment, it is possible for the character to create a contact (Figure 1) between the limb R k and the environment. The contact location is estimated to be at the center of the intersection between the largest colliding contact surface and W k . This heuristic provides an efficient method to approximate the contact location, and allows the verification of the non-slipping condition without considering the expensive complete model.

Motion planning algorithm for the reduced model. Our algorithm is an extension of the ballistic PRM planner [START_REF] Campana | Ballistic motion planning[END_REF], adapted to generate multi-contact jumping motions. It is based on an oriented version of the PRM planner [START_REF] Kavraki | Probabilistic roadmaps for path planning in high-dimensional configuration spaces[END_REF]. The PRM algorithm processes offline, to generate a graph of connected configurations, which captures the topology of the environment. The resulting graph can be efficiently queried online to compute a path between two specified configurations. The nodes of the graph are configurations. They are connected if there exists a collision-free path between them. The path is computed using a steering method, which takes into account the kinematic (and eventually, the dynamic) constraints of the model to generate a path.

We modify the original algorithm in three ways:

dofs 9 dofs

Figure 1: Illustration of a necessary condition for contact creation. If the trunk bounding box is collisionfree (Left-red), and the reachable workspaces of the limbs are in collision with the environment (Leftgreen), we assume that a contact configuration can be created between the effectors and the environment (Right).

• First, as proposed by [START_REF] Tonneau | A reachability-based planner for sequences of acyclic contacts in cluttered environments[END_REF], the configuration sampling is biased towards configurations that allow to generate contacts. Without loss of completeness, we only consider configurations for which at least one contact creation is possible, that is that if a shape W k is in collision for this configuration (Figure 1);

• Then, to generate a trajectory between two configurations, we use the parabolic steering method introduced by [START_REF] Campana | Ballistic motion planning[END_REF], used to represent jumps;

• Our contribution lies in the validation of the generated trajectory. A trajectory between two configurations is only validated if the multi-contact non-slipping condition is validated.

Algorithm 1 gives the pseudo-code of the motion planning algorithm. It generates a graph, called a roadmap, which can be requested online to compute a sequence of jumps between a start and a goal configuration.

validTrunkRandomSample returns a root configuration q 0 . q 0 is such that W 0 is collision-free, and ∃k, W k is in collision.

findContacts estimates the contact positions on the surfaces resulting of the intersections between the W k shapes and the environment. From the estimated contact locations, the centroidal cone K c is computed with the method computeCone.

validSteer is an extension from the parabola-steering-method of [START_REF] Campana | Ballistic motion planning[END_REF]. To generate a trajectory between two configurations q 0 a and q 0 b , the method determines whether there exists a parabola that verifies the non-sliding condition. This means that the takeoff velocity belongs to the centroidal cone K c (a), while the landing velocity belongs to K c (b). The takeoff and landing velocities are also limited by user-defined bounds ċmax t and ċmax l . These bounds are manually chosen according to the maximal allowed jump range X max of the character (e.g. on a flat ground, X max = ċmax 2 t /9.81). The trajectory is validated if the resulting parabola is collision-free, that is if W 0 is collision-free along the path. If no valid parabola can be found, validSteer returns an empty trajectory T.

Online path request and motion synthesis

Once a roadmap has been generated, requesting a trajectory between two configurations consists in adding them to the roadmap using Algorithm 1. If both configurations have been successfully added, the shortest trajectory connecting them is obtained by computing the shortest traversal of the graph.

A remaining step is to extend the trajectory q 0 (t) computed for the root of the character into a full body animation. In this phase, despite the impulse model formulation, to obtain a plausible animation, we consider that contact duration is not instantaneous. This requires identifying the transition times between contact and flight for each effector.

The whole-body animation technique proceeds as follows for each jump:

Algorithm 1 Probabilistic roadmap planner for ballistic motion planning of a dual-shape system. Input: environment, ċmax l , ċmax t Output: Graph of valid root configurations connected with parabolas f inished ← F alse while not(f inished) do q 0 rand ← validTrunkRandomSample() findContacts(q 0 rand ) K c ← computeCone(q 0 rand ) addToRoadmap(q 0 rand , K c ) for q 0 ∈ Roadmap do T ←validSteer(q 0 , q 0 rand , ċmax l , ċmax t ) addToRoadmap(T) T ←validSteer(q 0 rand , q 0 , ċmax • First, several key configurations are automatically computed at specific times of the trajectory:

two contact configurations at the take-off and landing phases; one configuration at each moment where an effector contact is broken (respectively created); one configuration at the apex of the trajectory (Figure 2).

• Then, a linear interpolation is performed for each dof of the character between each key configurations. It is designed in such a way that collisions are avoided and contact location constraints are maintained.

We first explain how the complete contact configurations are computed, and how the moment where a contact is broken / created is identified. Then, we present the interpolation method to generate the continuous complete animations.

Computation of whole-body contact configurations, and identification of takeoff and landing phases. We first address the identification and animation of the takeoff and landing phases, where contacts occur between the character and the environment. To keep the explanation simple, in this section we consider an input trajectory q 0 (t) which consists in a single jump, and assume that contacts are created and broken simultaneously, although the method handles the general cases.

To identify the duration of the landing phase, we first generate a collision-free, whole-body contact configuration that verifies the non-sliding condition at the impact frame. To generate such a configuration, we use a dedicated contact generator proposed in [START_REF] Tonneau | Using task efficient contact configurations to animate creatures in arbitrary environments[END_REF]. To increase the plausibility of the animation, contact generation is biased towards configurations as close as possible to a manually defined reference configuration (Figure 3). Contact locations may differ from the planned ones as long as the new centroidal cone K c satisfies the criterion (3). We then identify the transition time between the take-off and flight phases using an iterative approach. We go forward in time from t 0 = 0. At each time step t i , we update the root configuration q 0 (t i ), and solve an inverse kinematics problem to maintain the contacts active with the obstacle surface. The last time t transit before the inverse kinematics fails is the transition time with the flight phase. The landing phase is handled similarly, with the exception that we go backwards in time from the impact time.

An additional key configuration is generated at the apex of the parabola. It is sampled to be collision-free, with a bias to lie in the neighborhood of a manually defined reference configuration.

Whole body animation of the jump trajectory. The final animation is obtained by linear interpolation between the computed key configurations (Figure 2). While the root motion is guaranteed to be collision-free, the linear interpolation can result in one limb being in collision with the environment. In this case a local re-sampling phase is performed to find a collision-free trajectory. During this re-sampling only the dofs of the limb in collision are modified. In particular the root motion remains unchanged.

Simulations

Our framework has been implemented using the open source motion planning software Humanoid Path Planner (HPP) [START_REF] Mirabel | HPP: a new software for constrained motion planning[END_REF]. The method source code is available online. Final renderings have been generating with Blender 2.7 using the automatic blender export tool of HPP.

We demonstrate the genericity of our approach with four characters in five environments. The characters have different morphologies with from two to six limbs. They also have different jump abilities, expressed as the friction coefficients of the environment, as well as velocity bounds. The Figure 4 illustrates a few simulation results. All the key qualitative results of the method are detailed in the companion video : they include removals of unnatural jumps by the non-sliding constraints, jump transitions with arbitrary numbers of contacts and collision avoidance. Time performances. Table 1 provides a quantitative analysis of our approach performances in four scenarios. We separate the offline step, including the roadmap construction, from the online step, including the trajectory query and the jump synthesis. Since a full-path motion synthesis is dependent on the number of waypoints found by the planner, we give the average computation times to synthesize one jump and one contact configuration. All benchmarks were run on a PC with 64 GB of main memory and using one core of an Intel Xeon E5-1630 processor running at 3.7 GHz.

From the results we obtain, we observe that the complete computation time for one jump is inferior to the actual animation time. This is also true for our worst case scenario, the houses rooftops with Jumper-man. In this case, the large numbers of triangles that describe both the scene and the character explain the drop in the performances, due to the collision tests. As classically done in video games, using simplified meshes at the planning phase would probably allow us to obtain better performances without impacting the quality of the results. However we chose to preserve the complexity of the scene in this work, to demonstrate that the method scales well with the number of triangles.

More importantly, the critical observation is that both the trajectory query and the contact generation are real time in every scenario. We recall that those are the main contributions of the paper, and the performances obtained allow us to consider interactive applications with more advanced animation methods.

Conclusion

This work introduces an efficient motion planner, able to compute and animate complex trajectories for jumping legged characters, by extending a ballistic motion planner proposed by [START_REF] Campana | Ballistic motion planning[END_REF]. In particular, contrary to previous works based on finite state machines or data-driven animation, the planner is able to automatically compute non-coplanar multi-contact configurations, without making hypotheses about the number of contacts required and their locations. As such, it is able to address arbitrary environments.

Despite the integration of a dynamic impulse contact model, our planner is computationally efficient, thanks to the introduction of a low dimensional conservative criterion for verifying the non-slipping condition without explicitly computing the contact locations. This reduction of the problem's dimension only approximates the kinematic constraints of the character, and in rare cases, the whole body contact generation at the animation phase will fail.

A current limitation of the model is that it does not consider angular momentum in the planning phase. Furthermore, our approximation of the non-slipping condition is highly conservative. While the analytical computation for the Minkowski sum of friction cones is not tractable, we could consider another conservative approximation of the centroidal cone: the Minkowski sum resulting of the linearized shape of the cones [START_REF] Bretl | Testing static equilibrium for legged robots[END_REF][START_REF] Liu | Dextrous manipulation from a grasping pose[END_REF] is algorithmically computable. We wish to compare the quality and computation times of the two approximations for future work.

Regarding the quality of the obtained animations, we believe that the non-slipping criterion increases the plausibility of the obtained motion, despite a current limitation of the model, which does not consider angular momentum in the planning phase. We leave the validation of this hypothesis through a perception user study for future work. To integrate angular momentum constraints within the planning phase, we consider extending the centroidal cone to a 6 dimensional wrench-cone, following the formulation of [START_REF] Caron | Leveraging cone double description for multi-contact stability of humanoids with applications to statics and dynamics[END_REF].

Lastly, we aim at extending our planner to integrate other phases than jumping, to be able to alternate running, rolling or crawling sequences, for a larger spectrum of application.

Figure 2 :

 2 Figure 2: Key configurations computed along a parabola trajectory.

Figure 3 :

 3 Figure 3: Illustration of the generation of a contact configuration. (Left) The root of the reference configuration is placed close to an obstacle so that the obstacle is in the accessible space of the limbs. (Right) Result of the configuration projection on the obstacle to create contacts on hands and feet.

Figure 4 :

 4 Figure 4: Sequences of jumps computed by our planner in three environments.

  Offline columns concern the planning procedure while online columns deal with the path query in the roadmap and the motion synthesis of the whole trajectory. The number of nodes reflects the roadmap memory occupation. Number of triangles: house 449267, desert 47733, pond 6994, Jumper-man 10052. * Skeleton parameters: µ c = 0.6, ċmax t = 18 m/s, ċmax l = 25 m/s. * Jumper-man parameters: µ c = 1.2, ċmax t = ċmax l = 25 m/s.

Table 1 :

 1 Computation time (CT) averages of the method stages for 50 runs of each of the scenarios.

				Offline			Online
	Scenarios	dofs	Roadmap N. of nodes CT (s)	Full trajectory CT (s)	One-jump CT (s) Average Min -Max	N. of jumps	Contact generation (s)
	Skeleton in desert	42	0.51	22.4	2.8718	0.3231	0.2167 -0.9147	9.2	0.0253
	Frog in pond	45	0.76	46.6	0.4742	0.1185	0.1077 -13.25	5.3	0.0326
	Jumper-man on houses *	38	44.93	142.4	7.2497	1.3894	0.3307 -3.9250	6.1	0.0765
	Skeleton on houses *	45	107.13	1256.3	7.2737	0.5057	0.4381 -0.6923	14.4	0.0264

Acknowledgements

This work is supported by the European Research Council through the Actanthrope project (ERC Grant Agreement 340050) and the French National Research Agency Entracte project (ANR Grant Agreement 13-CORD-002-01). We would like to thank the anonymous reviewers for their time and comments.