Maxime Rivière

Noémie Perret

Damien Delcroix

Amandine Cabiac

Catherine Pinel

Michèle Besson
email: michele.besson@ircelyon.univ-lyon1.fr

Solvent Effect in Hydrogenolysis of Xylitol over Bifunctional Ru/MnO/C Catalysts under Alkaline-Free Conditions

Keywords: hydrogenolysis, glycols, biomass-derived xylitol, bifunctional Ru-MnO catalyst, deactivation

The hydrogenolysis reaction of biomass-derived xylitol to glycols and glycerol has been carried out in different aqueous solvents over a bifunctional Ru/MnO/C catalyst under alkaline-free conditions at 60 bar H 2 and 200°C. In pure water, the retro-aldol reaction takes place. However, decarbonylation and epimerization are the dominant reactions and produce C4 and C5 alditols, which limits the overall selectivity to glycols and glycerol (30%). When 90:10 vol.% water:1,4dioxane and water:2-PrOH solutions are used as solvent, the product distribution is very similar to the one in water. Meanwhile, in 90:10 vol.% water:primary alcohol (ROH with R = Me, Et, nPr, nBu) the overall selectivity to glycols and glycerol is greatly enhanced (up to 70%), whereas the selectivity to C4 and C5 alditols is reduced. In a solution with higher MeOH proportion of 20 vol.%, the glycols are detected with even higher selectivity, however, some deactivation of the catalyst is observed. TGA analysis of the used catalysts shows that during the process some coke is deposited on the catalyst via a dehydrogenation step of ROH. The coke selectively poisons the Ru sites that are active for the undesired reactions of decarbonylation and epimerization.

INTRODUCTION

Inedible biomass is considered to be a promising sustainable alternative resource to supply the increasing demand for carbon-based chemicals, likewise to avert the environmental issues and the dependence on fossil ressources. [START_REF] Besson | Conversion of biomass into chemicals over metal catalysts[END_REF][START_REF] Climent | Converting carbohydrates to bulk chemicals and fine chemicals over heterogeneous catalysts[END_REF] Cellulose and hemicelluloses are the most abundant renewable sources of carbon. Much attention has been given to their conversion to sorbitol and xylitol by hydrolysis and subsequent catalytic hydrogenation in aqueous phase, since alditols are versatile building blocks, precursors of various valuable compounds, including glycols. [START_REF] Petersen | Top Value Added Chemicals from Biomass: Results of Screening for Potential Candidates from Sugars and Synthesis Gas[END_REF] Ethylene glycol (EG) and propylene glycol (PG) are used in a large number of industrial processes: as raw materials in heat-transfer fluids industries, as monomers in the manufacture of polyester resins for clothes, and in cosmetic, pharmaceutical and packaging industries. [START_REF] Yue | Ethylene glycol: properties, synthesis, and applications[END_REF][START_REF] Marinas | Sustainability metrics for a fossil-and renewable-based route for 1,2-propanediol production: A comparison[END_REF] Currently produced in high volume (25 million tons in 2015) from petro-ethylene and propylene via epoxides, [START_REF] Zheng | Selectivity control for cellulose to diols: Dancing on the eggs[END_REF] glycols produced by hydrogenolysis of biomass-derived polyols get a growing interest.

Generally, hydrogenolysis of xylitol and sorbitol requires a supported-noble or transition metal catalyst (i.e., Ru, Ni or Cu) and a homogeneous base (i.e., Ca(OH) 2 , NaOH, Ba(OH) 2 ) to catalyze the C -C and C -O bond cleavages. The reactions are performed in aqueous phase under harsh conditions, namely 160 -230°C and 40 -120 bar H 2 . Through the decades of investigations, the selective catalytic hydrogenolysis of sorbitol and xylitol has improved as concerns the reaction pathways and mechanisms. [START_REF] Jin | Kinetic Modeling of Sorbitol Hydrogenolysis over Bimetallic RuRe/C Catalyst[END_REF][START_REF] Hausoul | Kinetics study of the Ru/Ccatalysed hydrogenolysis of polyols -insight into the interactions with the metal surface[END_REF][START_REF] Jia | Mechanistic insight into the selective hydrogenolysis of sorbitol to propylene glycol and ethylene glycol on supported Ru catalysts[END_REF][START_REF] Montassier | Aqueous polyol conversions on ruthenium and on sulfur-modified ruthenium[END_REF] The hydrogenolysis is a complex system consisting of several reaction steps. In the simplified and most widely accepted scheme for hydrogenolysis of xylitol in the presence of a soluble base, the reaction is initiated by the dehydrogenation of xylitol to the corresponding aldose (i.e., xylose) or ketose (i.e., xylulose or 3-pentulose) over metallic sites and then selective C -C bond cleavage reaction takes place catalyzed by the base via retro-aldol reaction to C2 and C3 intermediates depending on the initial dehydrogenation position (Scheme 1). [START_REF] Sun | Selective hydrogenolysis of biomass-derived xylitol to ethylene glycol and propylene glycol on supported Ru catalysts[END_REF][START_REF] Tajvidi | Hydrogenolysis of cellulose over Cu-based catalysts-analysis of the reaction network[END_REF] These intermediates are then hydrogenated on the metallic sites to the target glycols and glycerol. However, several side-reactions may occur: (i) decarbonylation of the terminal carbonyl bond on the metal which generates CO and C4 compounds (i.e., threitol) [START_REF] Deutsch | Probing the ruthenium-catalyzed higher polyol hydrogenolysis reaction through the use of stereoisomers[END_REF] , (ii) epimerization which forms arabitol and adonitol [START_REF] Hausoul | Kinetics study of the Ru/Ccatalysed hydrogenolysis of polyols -insight into the interactions with the metal surface[END_REF] , and (iii) Cannizzaro-type reaction catalyzed by a base which yields lactate. [START_REF] Deutsch | Probing the ruthenium-catalyzed higher polyol hydrogenolysis reaction through the use of stereoisomers[END_REF] Scheme 1. Simplified reaction scheme for xylitol hydrogenolysis over metal-based catalyst under alkaline conditions and main reactions involved.

The hydrogenolysis of xylitol (10 wt.% aqueous solution, 200°C, 60 bar H 2 ) over 90%Cu-SiO 2 catalyst in the presence of Ca(OH) 2 generated yields of glycols and glycerol of 5 -60%. [START_REF] Liu | Selective Hydrogenolysis of Xylitol to Ethylene Glycol and Propylene Glycol over Silica Dispersed Copper Catalysts Prepared by a Precipitation-Gel Method[END_REF][START_REF] Huang | Selective hydrogenolysis of xylitol to ethylene glycol and propylene glycol over copper catalysts[END_REF] Nibased catalysts, such as 6%Ni-NaY, achieved high conversion of sorbitol in the presence of Ca(OH) 2 (15 wt.% aqueous solution, 220°C, 60 bar H 2 ) and a 70% selectivity to glycols and glycerol. [START_REF] Banu | Hydrogenolysis of sorbitol over Ni and Pt loaded on NaY[END_REF] A high overall selectivity to glycols and glycerol of 85% was recently reported over a [START_REF] Liu | Efficient bimetallic NiCu-SiO2 catalysts for selective hydrogenolysis of xylitol to ethylene glycol and propylene glycol[END_REF] The Ru-based catalysts appear to be the most active catalysts. High conversions of sorbitol were reported over 3%Ru/carbon nanofibers (86%) and 3%Ru/C (71%) in the presence of CaO (20 wt.% solution, 220°C, 80 bar H 2 , 4 h) with a selectivity to glycols and glycerol of 61% and 29%, respectively. [START_REF] Zhao | Hydrogenolysis of sorbitol to glycols over carbon nanofiber supported ruthenium catalyst[END_REF] [START_REF] Rivière | Xylitol Hydrogenolysis over Ruthenium-Based Catalysts: Effect of Alkaline Promoters and Basic Oxide-Modified Catalysts[END_REF] However, the use of large amounts of inorganic homogeneous base limits the economic viability of the process, due to the formation of lactate at the expense of PG and the final neutralization and separation step requirements that generate high amounts of salt. [START_REF] Sun | Selective hydrogenolysis of biomass-derived xylitol to ethylene glycol and propylene glycol on Ni/C and basic oxide-promoted Ni/C catalysts[END_REF] Therefore, recent studies explored the hydrogenolysis reaction of alditol aqueous solutions under alkalinefree conditions over bifunctional catalysts combining both metallic and basic properties. A selectivity to glycols and glycerol of approximately 80% with limited formation of lactic acid was obtained at 68% conversion of sorbitol over Ni-MgO (200°C, 40 bar H 2 ), [START_REF] Wang | Sorbitol hydrogenolysis to glycerol and glycols over M-MgO (M = Ni, Co, Cu) nanocomposite: A comparative study of active metals[END_REF] while a 48% selectivity was achieved over Ni-Ru/Ca(OH) 2 catalysts. [START_REF] Murillo Leo | Selective conversion of sorbitol to glycols and stability of nickel-ruthenium supported on calcium hydroxide catalysts[END_REF] Other formulations such as Ni-Mg-Al, Cu/CaO-Al 2 O 3 and Ni-CaO/C catalysts were reported for sorbitol or xylitol under base-free conditions with a selectivity to glycols and glycerol of 60%, 73% and 69%, respectively. [START_REF] Sun | Selective hydrogenolysis of biomass-derived xylitol to ethylene glycol and propylene glycol on Ni/C and basic oxide-promoted Ni/C catalysts[END_REF][START_REF] Du | Production of C2 and C3 polyols from D-sorbitol over hydrotalcite-like compounds mediated bi-functional Ni-Mg-Al-Ox catalysts[END_REF][START_REF] Jin | Sorbitol Hydrogenolysis over Hybrid Cu/CaO-Al2O3 Catalysts: Tunable Activity and Selectivity with Solid Base Incorporation[END_REF] In our earlier work, we demonstrated that a 3.1%Ru/MnO(4.5%)/C catalyst was very active (220 h - 1 ) and selective to glycols and glycerol (32%) for the hydrogenolysis of xylitol under base-free conditions (10 wt.%, 200°C, 60 bar H 2 ). [START_REF] Rivière | Xylitol Hydrogenolysis over Ruthenium-Based Catalysts: Effect of Alkaline Promoters and Basic Oxide-Modified Catalysts[END_REF] Moreover, the production of glycols was limited by the competition between the desired retro-aldol and the side-reactions in aqueous solution.

Indeed, Ru-based catalysts showed excessive C-C bond cleavage and formed light alkanes under base-free conditions (15% of products in gas phase). As far as we know, only water has been used as solvent for hydrogenolysis of xylitol and little attention has been given to the effect of the solvent on the reaction. It is worth noting that the use of water:ethanol or water:2-propanol (H 2 O:EtOH or H 2 O:2-PrOH) solvents for the hydrogenation of xylose to xylitol resulted in higher rates. [START_REF] Mikkola | Effects of solvent polarity on the hydrogenation of xylose[END_REF] Furthermore, a Cu-ZnO catalyst exhibited a higher activity for hydrogenolysis of glycerol in EtOH and MeOH, which might be due to the in situ generation of hydrogen from the solvent. [START_REF] Wang | Solvent effect on hydrogenolysis of glycerol to 1,2-propanediol over Cu-ZnO catalyst[END_REF] In the present work, the reaction was carried out over a Ru/MnO/C catalyst in water:dioxane and water:alcohol mixtures (MeOH, EtOH, 1-PrOH, 2-PrOH, and 1-BuOH). This catalyst demonstrated a bifunctional character with basic and metallic sites to perform the reaction in base-free aqueous solution, while exhibiting high activity and selectivities. [START_REF] Rivière | Xylitol Hydrogenolysis over Ruthenium-Based Catalysts: Effect of Alkaline Promoters and Basic Oxide-Modified Catalysts[END_REF] We investigated the impact of the reaction media in an attempt to improve its performances. Moreover, water:alcohol mixtures should have low environmental impact, according to the green solvent definition by Fischer et al. [START_REF] Capello | What is a green solvent? A comprehensive framework for the environmental assessment of solvents[END_REF]

EXPERIMENTAL SECTION

Materials. D-Xylitol (99%), 2-propanol (>99.5%), 1-butanol (>99.5%), methyl β-Dxylopyranoside (>99%) were purchased from Sigma Aldrich, methanol (>99.9%) and Catalysts Preparation. The bifunctional 3.1%Ru/MnO(4.5%)/C catalyst (3.1 and 4.5 are the wt.% of Ru and Mn, respectively) was prepared by the successive wet impregnation method. [START_REF] Rivière | Xylitol Hydrogenolysis over Ruthenium-Based Catalysts: Effect of Alkaline Promoters and Basic Oxide-Modified Catalysts[END_REF] Typically, an aqueous solution of the Mn precursor at the desired concentration was added to the suspension of active carbon in water within a round-bottom flask. After stirring for 6 h at RT, water was removed by evaporation and the resulting powder was dried at 110°C overnight and calcined at 200°C (2°C min -1 ) for 4 h under air flow (100 mL min -1 ). Subsequently, the MnO/C powder was wet impregnated with an aqueous solution of the desired amount of Ru precursor.

After evaporation, the solid was reduced at 450°C (2°C min -1 ) for 3 h under H 2 flow (100 mL min -1 ) and, finally, passivated under 1% v/v O 2 /N 2 gas flow for 30 min at RT. The monometallic x%Ru/C catalyst was synthetized using the same impregnation procedure and reduction treatment.

Catalyst Characterization. The metal loadings of the catalysts were determined after dissolution in aqua regia at 150°C for 12 h and analysis by inductively coupled plasma-optical emission spectroscopy (ICP-OES, Activa Jobin-Yvon). The crystallite phases were determined by X-ray diffraction (XRD, Brucker D8A25) with CuK α radiation and multi-channel fast detector LynxEye while the specific surface areas were determined by N 2 physisorption isotherms (ASAP 2020) at 77 K after thermal treatment (350°C for 7 h under vacuum 10 -4 mbar). The amount of basic sites on the catalyst surface were determined by temperature-programmed desorption (TPD, Belcat-M) of CO 2 . Typically, the reduced sample was pre-treated in He gas (50 mL min -1 ) at 500°C for 1 h and then cooled to 100°C, and saturated with 5% v/v CO 2 /He. The TPD profiles were recorded using a thermal conductivity detector (TCD) while heating the sample from 100°C to 500°C under a continuous flow of He (50 mL min -1 ). The used catalysts were analyzed by thermogravimetric analysis coupled with differential thermal analysis (TGA-DTA, Star e System Mettler Toledo). TEM images were taken by using a JEOL 2010 instrument operated at 200 keV.

The average size of Ru particles was determined by measuring approximately 300 particles distributed randomly in the images.

Catalytic testing. Hydrogenolysis of xylitol was carried out in a Hastelloy Parr autoclave (300 mL) at a stirring speed of 1000 rpm. In a typical run, 135 mL of 10 wt.% xylitol solution and 0.5 g catalyst were introduced into the autoclave. After purging with Ar, the reactor was heated to the reaction temperature. It was then pressurized with H 2 , which corresponded to the reaction time t = 0. The same procedure was used for the catalytic tests in solvent mixtures. For instance, for 90:10 vol.% water:MeOH solvent, 15 g of xylitol were introduced in 121 mL water blended with 14 mL MeOH.

During the reaction, liquid samples were periodically collected through a sampling valve and filtered through polyvinylidene fluoride (PVDF) membranes (0.45 µm) before analysis. After reaction, the reactor was cooled to RT and depressurized to atmospheric pressure. The catalyst was separated by vacuum filtration through PVDF membranes (0.45 µm) and dried at 70°C under N 2 atmosphere.

The products in the liquid samples were identified and quantified using two different HPLC instruments (Shimadzu) connected to refractive index diffraction (RID10A) and UV detectors (SPD-M10A). Moreover, the gaseous phase was collected at the end of reaction at RT in a gas collecting bag and analyzed by gas chromatography (µ-GC SRA, Agilent) using a catharometric detector and connected to a mass spectroscopy module (MS 5975, Agilent), as described previously. [START_REF] Rivière | Xylitol Hydrogenolysis over Ruthenium-Based Catalysts: Effect of Alkaline Promoters and Basic Oxide-Modified Catalysts[END_REF] The total organic carbon (TOC) in pure water solvent was measured using a TOC analyzer (Shimadzu TOC-VCHS equipped with ASI-automatic sampler) and compared to initial TOC to estimate the formation of gaseous products. Conversion, selectivities (on a carbon basis %), and initial reaction activity (mol xylitol mol Ru -1 h -1 ) were calculated as previously. [START_REF] Rivière | Xylitol Hydrogenolysis over Ruthenium-Based Catalysts: Effect of Alkaline Promoters and Basic Oxide-Modified Catalysts[END_REF]

RESULTS AND DISCUSSION

Catalyst Characterization. The metal loadings, the specific surface areas, and the total amount of basic sites of the catalysts were determined previously and are summarized in Table S1. [START_REF] Rivière | Xylitol Hydrogenolysis over Ruthenium-Based Catalysts: Effect of Alkaline Promoters and Basic Oxide-Modified Catalysts[END_REF] The XRD pattern (Fig. S1) showed that the MnO phase was obtained with an average crystallite size of 14 nm. No peak attributed to Ru was detected over Ru/C and Ru-MnO/C, suggesting well dispersed Ru particles. 3.1%Ru/MnO(4.5%)/C catalyst exhibits a higher basic sites amount (355 µmol g -1 , determined by CO 2 -TPD) than 2.9%Ru/C (261 µmol g -1 ), according to Ishikawa et al. [START_REF] Ishikawa | Demethoxylation of guaiacol and methoxybenzenes over carbon-supported Ru-Mn catalyst[END_REF] The TEM images of Ru/C and 3.1%Ru/MnO(4.5%)/C catalysts are shown in S2.

The temporal evolutions of the conversion of xylitol, products distribution, and TOC measurements in liquid phase are shown in Figure 2 for the reaction in pure water, taken as a reference experiment. Xylitol was fully converted after 30 h of reaction. The major products in the liquid phase were propylene glycol (PG), ethylene glycol (EG), glycerol (GLY), C4 alditols (mainly threitol and a small amount of erythritol). Butanediols (BDO), notably 1,2-BDO, and C5 alditols (arabitol and traces of adonitol) were also identified. The reaction profile revealed that most products in liquid phase attained a maximum concentration at 80% xylitol conversion after 9 h. Afterwards, the concentrations decreased gradually for EG, GLY, C4 and C5 alditols, whereas the concentrations of PG and BDO were still continuously increasing. The decline of TOC in liquid phase indicates that a large amount of the products was transformed into alkanes and CO 2 in the gas phase. The major products in gas phase were, from the highest to lowest amount (carbon basis): methane, ethane, propane, CO 2 , and traces of butane and pentane. The results confirmed that several reactions were taking place and were competing over the bifunctional catalyst in water under alkaline-free conditions. [START_REF] Rivière | Xylitol Hydrogenolysis over Ruthenium-Based Catalysts: Effect of Alkaline Promoters and Basic Oxide-Modified Catalysts[END_REF][START_REF] Du | Production of C2 and C3 polyols from D-sorbitol over hydrotalcite-like compounds mediated bi-functional Ni-Mg-Al-Ox catalysts[END_REF][START_REF] Jin | Sorbitol Hydrogenolysis over Hybrid Cu/CaO-Al2O3 Catalysts: Tunable Activity and Selectivity with Solid Base Incorporation[END_REF][START_REF] Chen | Hydrogenolysis of biomass-derived sorbitol to glycols and glycerol over Ni-MgO catalysts[END_REF] The glycols and glycerol were mainly produced from the retro-aldol reaction of xylose or xylulose, which are intermediates from xylitol. After the initial step of dehydrogenation of the alditol, the retro-aldol reaction is favored by the basic sites on the catalyst. [START_REF] Sun | Selective hydrogenolysis of biomass-derived xylitol to ethylene glycol and propylene glycol on Ni/C and basic oxide-promoted Ni/C catalysts[END_REF][START_REF] Jin | Sorbitol Hydrogenolysis over Hybrid Cu/CaO-Al2O3 Catalysts: Tunable Activity and Selectivity with Solid Base Incorporation[END_REF] Herein, dehydrogenation of xylitol occurred over metallic Ru, and the MnO species over the surface provide the basic sites.

Moreover, Hausoul et al. [START_REF] Hausoul | Unravelling the Ru Catalyzed Hydrogenolysis of Biomass Based Polyols under Neutral and Acidic Conditions[END_REF] demonstrated that under these reaction conditions of temperature and pressure, Ru particles were also responsible for side-reactions of epimerization, decarbonylation, and dehydroxylation to yield the C5 alditols, C4 alditols, and BDO, respectively. The large extent of decarbonylation and dehydroxylation reactions produced light alkanes that were detected at the end of the reaction. Thus, while Ru is required along with MnO species for the retro-aldol reaction, the Ru particles also inevitably favor the side-reactions. The use of other solvents than pure water was examined in an attempt to modify the catalyst performance.

                             
Hydrogenolysis of xylitol was first run in the presence of 90:10 vol.% H 2 O:1,4-dioxane solvent in order to investigate the influence of an aprotic and non-polar compound, although we were aware of the non-sustainability of dioxane (Table S2). Since 1,4-dioxane is stable under H 2 pressure at high temperature in the presence of Ru-based catalysts, it is often used as solvent for hydrogenation reactions. [START_REF] Al-Shaal | Exploring the ruthenium catalysed synthesis of γ-valerolactone in alcohols and utilisation of mild solvent-free reaction conditions[END_REF][START_REF] Hong | Hydrogenation of succinic acid to tetrahydrofuran (THF) over ruthenium-carbon composite (Ru-C) catalyst[END_REF] As shown in Figure 3.a, xylitol conversion was complete after 30 h and a very similar product distribution as a function of time was observed as the one in pure water (Figure 2). The addition of 1,4-dioxane to water did not affect the activity and the product distribution. Less than 10% of 1,4-dioxane was removed from the liquid phase during the reaction, more particularly in the first few hours because of its partial vaporization under the reaction conditions. Therefore, the use of 1,4-dioxane as aprotic and non-polar solvent did not limit the side-reactions. differed strongly compared to pure water. First, a significant impact was observed on the reaction X and R 1-BuOH (%) rates. The activity decreased from 220 h -1 to 71 h -1 in H 2 O:MeOH, 56 h -1 in H 2 O:EtOH, 57 h -1 in H 2 O:1-PrOH and 39 h -1 in H 2 O:1-BuOH (Figure 4). These results were unlike those reported by Mikkola et al. [START_REF] Mikkola | Effects of solvent polarity on the hydrogenation of xylose[END_REF] S2). Moreover, we assumed that the solubility of H 2 was not the limiting factor in the present system, since the reactor was continuously fed with H 2 to keep the pressure constant and the activity decreased with increasing solubility. Thus, we suggest that the drop of activity might be the consequence of competitive adsorption of the primary alcohol and xylitol on active sites. overall selectivity to C4 and C5 alditols dropped from 38% to 10%. Additionally, the carbon balance in liquid phase was close to 90% in presence of alcohol, which suggests a low production of alkanes.

                                       0
                                                                         0 5 10
                                    0
 C5 alditols  C4 alditols  GLY  EG  PG  BDO R 1,4-dioxane XYL conversion i                                         (a) (c) (d) (b) (e) (f)
                                   0
                                          0
In the other water:primary alcohol (EtOH; 1-PrOH and 1-BuOH) solvents the selectivities to glycols detected were even higher (25% and 34% to EG and PG, respectively) than in water:MeOH, whereas the overall selectivity towards BDO, C4, and C5 alditols decreased to 10%. It is worth to mention that in the case of H 2 O:EtOH quantification of C5 alditols could not be done, since arabitol and EtOH were co-eluted on the HPLC column, however, we presumed that arabitol formation followed the same trend as in the other runs. Therefore, the present results

showed that the use of a 90:10 vol.% water:primary alcohol mixture greatly affected the performance of the Ru/MnO/C catalyst in terms of activity and selectivity. Earlier on we suggested two types of active sites. The number of Ru sites responsible for epimerization and decarbonylation seemed to have decreased, while the ability of MnO species close to Ru associated with the cleavage of the C -C bonds of xylitol to glycols and glycerol was not affected.

Nevertheless, these results (Figure 3.b-e) revealed two important drawbacks associated with the use of primary alcohol: (i) a large fraction of the alcohol was no more analyzed in the liquid phase, and (ii) a deactivation of the catalyst was evidenced. Indeed, during the heating period up to the reaction temperature, the concentration in liquid phase of alcohol decreased: by 10% for MeOH and EtOH, and up to 60% for 1-PrOH. This was due to the vaporization of alcohol to the gaseous phase. Then, the fraction of alcohol in the solvent still decreased during the reaction.

One may note that the loss of primary alcohol in the mixed solvent should increase the concentrations of the analyzed products, and thus enhance the selectivity to the different products. Nevertheless, this overestimation could be neglected owing to the low volume of alcohol (10 vol.%) in the solvent. Furthermore, as mentioned above, a slow-down of xylitol conversion was observed after 9 h, followed by a pseudo-plateau, essentially for EtOH, 1-PrOH and 1-BuOH. Xylitol conversion was not total even upon extension of the reaction time. The products and solvent concentrations remained stable in the liquid phase.

The influence of the fraction of alcohol in the solvent was investigated using MeOH, owing to the lower deactivation when using MeOH. Further insights in catalyst deactivation. The catalytic deactivation can be related with textural modifications or poisoning of the active sites. [START_REF] Hammond | Intensification studies of heterogeneous catalysts: probing and overcoming catalyst deactivation during liquid phase operation[END_REF] The X-Ray diffraction patterns of the used catalysts after reaction revealed a slight increase of Ru crystallite size and disappearance of the diffraction peaks attributed to MnO (Figure S1). These similar results in water or in water:alcohol solvent suggest that the textural modifications were not responsible for the deactivation. Therefore, the deactivation was likely due to the presence of the alcohol. In the same way as xylitol, the alcohol can be dehydrogenated on the Ru active sites and yield the corresponding aldehyde. However, the aldehydes (formaldehyde to butyraldehyde) were not observed, neither in the liquid phase nor in the gas phase. Hence, we concluded that the deactivation was due either to the poisoning of Ru active sites by intermediate species from the alcohol (i.e. hydroxyalkyl or alkoxy species) or to a side-reaction taking place between the alcohol and an intermediate from xylitol. According to computational DFT studies on the dehydrogenation of primary alcohols over Ru or Pt sites, the desorption of alkoxy species is the determining step and the energies are the same whatever the C 2 -C 4 alkoxy species. [START_REF] Garcia-Muelas | Density Functional Theory Comparison of Methanol Decomposition and Reverse Reactions on Metal Surfaces[END_REF][START_REF] Loffreda | Tuning catalytic reactivity on metal surfaces: Insights from DFT[END_REF][START_REF] Sinha | A first principles analysis of the hydrogenation of C1C4 aldehydes and ketones over Ru[END_REF] MeOH behaved differently as compared to the other primary alcohols, since its adsorption energy was ) found higher. According to the present results (Figure 3) the deactivation was less important in the presence of MeOH, suggesting that the methoxy intermediate would be easier to transform to methane. This was confirmed by the high concentration of CH 4 analyzed in the gas phase. The other explanation for deactivation is the occurrence of side-reactions between the alcohol and a derivative issued from xylitol over the Ru active site which will poison the catalyst and limit the access to the surface. Iglesias et al. [START_REF] Iglesias | Dehydration of Xylose to Furfural in Alcohol Media in the Presence of Solid Acid Catalysts[END_REF] noted for instance the formation of an alkyl-xyloside intermediate during dehydration of xylose in MeOH or EtOH solvent at 150°C (Figure S2). This intermediate could be generated under our reaction conditions under mild-acid conditions (pH 4.5-6.5) and deactivate the catalyst, even though it was not detected in the liquid phase.

We further investigated the effect of the alcoholic solvent and of these possible intermediates on the deactivation of Ru/MnO/C catalyst. First, in order to identify the poisoned active sites and the ROH degradation sites, catalytic tests were performed over the monometallic Ru/C catalyst, the MnO/C solid, or the bifunctional Ru/MnO/C catalyst in 90:10 vol.% H 2 O:MeOH solvent in the presence or in the absence of xylitol (Figure 6.a). Regarding MeOH removal from the liquid phase, at t = 0 after the 45 min heating period, 10% of methanol was systematically lost from the liquid phase, which corresponds to the vaporization of a fraction of MeOH. Then, as expected MnO/C was not active for MeOH transformation. Conversely, in the presence of Ru/C or Ru/MnO/C, removal of MeOH was in the range of 50% to 90% after 24 h, and methane was the sole product in the gas phase. Over Ru/C, whether xylitol was present or not, conversion of MeOH was the same (c.a. 50%), which suggests preferential adsorption of MeOH on the surface.

Conversely, over Ru/MnO/C catalyst, degradation was lower in the presence of xylitol (50% vs. 90%), which implies higher adsorption of xylitol. Moreover, as mentioned above, the addition of MeOH influenced the rate of xylitol conversion (Figure 6.b). The initial rate decreased by a factor ca. 17 in the presence of Ru/C vs. a factor ca. 3 in the presence of Ru/MnO/C. This fact confirms distinct active sites on both catalysts with different affinity for MeOH and xylitol. In order to further understand the deactivation phenomenon and to observe the possible deposition of carbonaceous species on the surface of used catalyst, TGA -DTA analysis of used catalysts was performed under air flow. [START_REF] Hammond | Intensification studies of heterogeneous catalysts: probing and overcoming catalyst deactivation during liquid phase operation[END_REF][START_REF] Perret | Catalytic Response and Stability of Nickel/Alumina for the Hydrogenation of 5-Hydroxymethylfurfural in Water[END_REF] After reaction in water, two major weight losses which generated heat were recorded for the used Ru/MnO/C catalyst: one at 80°C (10% weight loss) attributed to the desorption of physisorbed water and a second one at 350 -425°C (90% weight loss) (Figure 7). We assumed that the second one corresponded to the total combustion of active carbon support, principally composed of polyaromatic carbons. [START_REF] Ibáñez | Identification of the coke deposited on an HZSM-5 zeolite catalyst during the sequenced pyrolysis-cracking of HDPE[END_REF] At the end of the analysis, only Ru and Mn oxides remained and represented 7% of the initial mass. After reaction ) in 80:20 vol.% H 2 O:MeOH, the same weight loss was recorded at 350 -425°C and attributed to the combustion of support. Nonetheless, another weight loss of 20% appeared at lower temperature (250°C) along with a heat peak suggesting another combustion reaction. This suggests the presence of a "soft" coke, constituted of oligomers and containing oxygen atoms in their chain. The tendency for TGA-DTA of used catalysts after the catalytic tests in 90:10 vol.% water:alcohol solvents was very similar (Figure S4) to that observed after the experiment in water: MeOH (Figure 7). Since this weight loss was not observed on the used catalyst after reaction in pure water, we may ascertain that the coke originated from species issued from the alcoholic solvent. S4). Only a small amount of 2-PrOH was consumed during the reaction and traces of acetone were temporarily analyzed in the liquid phase, which confirmed that (de)hydrogenation reactions may occur on metallic sites of Ru/MnO/C catalyst. According to the literature, acetone is more rapidly hydrogenated to 2-PrOH than aldehydes to primary alcohols over noble metal catalysts. [START_REF] Van Druten | Hydrogenation of carbonylic compounds[END_REF] Furthermore, we investigated the influence of the addition of 10 vol.% acetone in water for xylitol hydrogenolysis. Acetone was rapidly and totally hydrogenated to 2-PrOH after 5 h, while xylitol was completely converted after 9 h (Figure S5).

Based on these results, acetone was hydrogenated to 2-PrOH and did not form coke on the catalyst surface, contrary to aldehydes. As proposed by Alcalá et al. [START_REF] Alcalá | Density-functional theory studies of acetone and propanal hydrogenation on Pt(111)[END_REF] , dehydrogenation of the α-H next to the carbonyl group of adsorbed aldehydes leads to the formation of an oxametallacycle with metallic sites, which could be the precursor to the oligomer and to the coke formation. The α-H atom is not present for ketones and the cleavage of a C-C bond is much more difficult. A mechanistic pathway was proposed for the degradation of primary and secondary alcohols and aldehydic compounds, which explains that Ru sites are poisoned in primary alcohols (Scheme 2). We assumed that the adsorption mode of the alcohol was independent of the nature (primary or secondary) of the alcohol; the O-H bond cleavage yields an adsorbed alkoxy intermediate (step 1), which is stabilized in the presence of chemisorbed water. [START_REF] Sinha | A first principles analysis of the hydrogenation of C1C4 aldehydes and ketones over Ru[END_REF][START_REF] Michel | Role of water in metal catalyst performance for ketone hydrogenation: a joint experimental and theoretical study on levulinic acid conversion into gamma-valerolactone[END_REF] Then, the alkoxy moves to adopt a parallel mode of adsorption on Ru surface (step 2), to form an oxametallacycle η 1 η 2 (C,O) by releasing α-H atoms (step 3). The energy for adsorption of this compound is the same from primary or secondary alcohols. Then, the reaction pathway is different depending on the nature of the alcohol. Indeed, for secondary alcohols, desorption of the oxametallacycle intermediate (step 4') forms a ketone as observed in our results. Conversely, the intermediate from primary alcohols may release the second α-H to form a less labile complex than ketones (step 4). The latter could then be polymerized to oligomers and lead to the poisoning of active sites. Moreover, we observed a faster deactivation in the presence of acetaldehyde than in the presence of EtOH, suggesting that fewer steps were necessary between the adsorption and the coke formation (step 1'') (Scheme 2). Conversely, in primary alcohol (MeOH, EtOH, 1-PrOH, 1-BuOH) aqueous solutions, a higher overall selectivity to glycols and glycerol (70%) was obtained at the expense of decarbonylation, dehydroxylation and epimerization reactions. Increasing the fraction of MeOH improved the selectivity to glycols, however, it accelerated the catalyst deactivation and the primary alcohol degradation. The deactivation was caused by dehydrogenation of the primary alcohol, which formed light alkanes (i.e. CH 4 from MeOH) and coke on the Ru surface. These results highlight the role of the primary alcohol in selective poisoning of Ru sites that are responsible for the sidereactions. In the meantime, the retro-aldol reaction takes place on the Ru-MnO sites, before being also deactivated.

  2 catalyst for the hydrogenolysis of xylitol in the presence of Ca(OH) 2 (10 wt.% aqueous solution, 200°C, 80 bar H 2 ).

Figure 1 .

 1 Figure 1. Ru nanoparticles on Ru/C are well dispersed over the active carbon surface with a mean particle size of 1.6 nm. On the bifunctional Ru/MnO/C catalyst, large particles (main size around 25 nm) were attributed to MnO by EDX analysis and small particles (2.1 nm) to Ru. The Ru particles are either directly on the support or on large particles of MnO.

Figure 1 .

 1 Figure 1. TEM images of monometallic 2.9%Ru/C [(a) and (b)] and bifunctional 3.1%Ru/MnO(4.5%)/C [(c) and (d)] with the Ru particles size distribution. Hydrogenolysis of xylitol in water:dioxane or alcohol solvents. Hydrogenolysis of xylitol was conducted in different solvents at 200°C and 60 bar H 2 over Ru/MnO/C catalyst under alkaline-free conditions. Some physicochemical properties of the solvents used are given in

Figure 2 .

 2 Figure 2. Temporal evolution of conversion (X), TOC measure, and concentrations during hydrogenolysis of xylitol in pure water over 3.1%Ru/MnO(4.5%)/C. Reaction conditions: xylitol 0.7 mol.L -1 (10 wt.%), 135 mL water, 0.5 g 3.1%Ru/MnO(4.5%)/C, 60 bar H 2 , 200°C.

40 

 40 C5 alditols  C4 alditols  GLY  EG  PG in liquid phase (%)

Figure 3 .

 3 Figure 3. Xylitol hydrogenolysis over 3.1%Ru/MnO(4.5%)/C in different solvents: (a) 90:10 vol.% H 2 O:1,4-dioxane, (b) H 2 O:MeOH, (c) H 2 O:EtOH, (d) H 2 O:1-PrOH, (e) H 2 O:1-BuOH, (f) H 2 O:2-PrOH. X: xylitol conversion, R i : ratio of recovered dioxane or alcohol concentration to the initial concentration. Reaction conditions: xylitol 0.7 mol.L -1 (10 wt.%), 135 mL 90:10 vol.% H 2 O:organic solvent, 0.5 g 3.1%Ru/MnO(4.5%)/C, 60 bar H 2 , 200°C.To survey the impact of primary alcohols (MeOH; EtOH; 1-PrOH; 1-BuOH), they were added as 10 vol.% concentration in water. Primary alcohols are polar with different surface tensions and H 2 solubilities (TableS2). The reaction profiles are shown in Figure 3.b-e, and the data for the activities and selectivities to the main products are summarized in Figure 4. The results

Figure

  Figure 4. Hydrogenolysis of xylitol over 3.1%Ru/MnO(4.5%)/C in water and 90:10 vol.% H 2 O:primary alcohol solvent. Selectivities at 60% xylitol conversion. Reaction conditions: xylitol 0.7 mol.L -1 (10 wt.%), 135 mL 90:10 vol.% H 2 O:ROH solvent, 0.5 g 3.1%Ru/MnO(4.5%)/C, 60 bar H 2 , 200°C.

Figure 6 .

 6 Figure 6. Influence of the catalyst on deactivation. (a) MeOH consumption as a function of time in 90:10 vol.% H 2 O:MeOH in the absence or presence of xylitol. (b) Evolution of xylitol concentration in water or 90:10 vol.% H 2 O:MeOH.

  /C in 100% H 2 O Ru/MnO/C in (90:10) H 2 O:MeOH Ru/C in 100% H 2 O Ru/C in (90:10) H 2 O:MeOH Reaction time (h) Xylitol concentration (mol L -1

Figure 7 .

 7 Figure 7. TGA-DTA under air flow of used 3.1%Ru/MnO(4.5%)/C after xylitol hydrogenolysis in 100%H 2 O and (80:20) H 2 O:MeOH mixture. Reaction conditions: xylitol 0.7 mol.L -1 (10 wt.%), 135 mL solvent, 0.5 g 3.1%Ru/MnO(4.5%)/C, 60 bar H 2 , 200°C.The coke may be initiated by the dehydrogenation of MeOH, even though formaldehyde was not detected in the liquid or the gas phase, or by the degradation of methyl xyloside possibly formed as intermediate. To verify these hypotheses and to identify the species responsible for the catalyst deactivation, xylitol hydrogenolysis was carried out over Ru/MnO/C in the presence of acetaldehyde (CH 3 CHO) or methyl xyloside (Table1). The addition of a small amount (1 g) of

Scheme 2 .coke

 2 Scheme 2. Proposed mechanism pathways for (de)hydrogenation and coke formation on Ru sites of (a) primary alcohols and (b) secondary alcohols leading to selective poisoning of the catalyst. H 2 O and H 2 not shown for clarity. The clear differences in the results obtained in the different solvents confirm the presence of two types of active sites on the catalyst yielding different products. The metallic Ru sites are responsible for (de)hydrogenation and epimerization of xylitol (mostly to arabitol), as well as decarbonylation and dehydroxylation reactions producing C4 products and light alkanes by excessive C-C bond cleavage. The presence of MnO next to Ru particles generates less active bifunctional Ru-MnO active sites, on which successive (de)hydrogenation and retro-aldol reactions take place to produce glycols and glycerol. Therefore, the high selectivity to glycols and glycerol in 90:10 vol.% H 2 O:ROH solvents can be explained by the selective poisoning of Ru sites, while the Ru-MnO sites are not affected. Afterwards, during the reaction the latter sites are also finally poisoned, and total catalyst deactivation is observed.

  Sun and Liu 11 reported a production of 67% glycols and glycerol over 4%Ru/C catalyst in the presence of Ca(OH) 2 (10 wt.% xylitol, 200°C, 40 bar H 2 ) and, in our previous work, we reported an activity of 223 h -1 with a selectivity to glycols and glycerol of 57% over 2.9%Ru/C catalyst in the presence of the same base (10 wt.% xylitol, 200°C, 60 bar H 2 ).

  for hydrogenation of xylose to xylitol: the catalytic activity over Raney Ni in a 75:25 vol.% H 2 O:EtOH solvent (80 -130°C; 40 -80 bar H 2 ) was twice the activity in water, suggesting that the enhancement of H 2 dissociation on Ni active sites was driven by the increasing solubilization of H 2 in the H 2 O:EtOH solvent. However, the H 2 solubilities are similar in H 2 O:ROH (90:10) mixtures or in pure H 2 O solvent (Table

Table 1

 1 Also, it did not affect the product distribution, except a slight increase of the combined selectivity to glycols and glycerol from 34% to 38%, and those to C5 and C4 products from 41% to 44%. Simultaneously to xylitol conversion, methyl xyloside was entirely converted after 9 h. Therefore, we concluded that methyl xyloside, if formed during the reaction in H 2 O:MeOH solvent, could not be responsible for the poisoning. On the contrary, the Ru/MnO/C catalyst was completely inactive in aqueous solution containing 10 vol.% acetaldehyde (Table1). No xylitol conversion was detected after 9 h of reaction. The ratio of recovered acetaldehyde to the initial concentration was 23% after 1 h. Part of the aldehyde was removed by vaporization in the gas phase. Hydrogenation and Cannizzaro-type reactions of acetaldehyde generated small amounts of EtOH and acetic acid (as detected by HPLC) which contributed to this loss, as well as coke formation. After a few hours of reaction, acetaldehyde concentration remained constant and no modification in the product distribution was noted. Taking into account that the poisoning of the catalyst occurred to some extent in H 2 O:EtOH and dramatically in H 2 O:CH 3 CHO solvent, it was suggested that an intermediate formed from the alcohol or the aldehyde was responsible for the coke formation that significantly influenced the reaction.

	). The addition of a small amount (1 g) of

Table 1 .

 1 Hydrogenolysis of xylitol in different solvents and in presence or not of methyl xyloside over 3.1%Ru/MnO(4.5%)/C. a Reaction conditions: xylitol 10 wt.%, 135 mL solvent, 0,5 g 3.1%Ru/MnO(4.5%)/C, 60 bar H 2 , 200°C; b at 60% conversion; c threitol, erythritol, BDO; d R i : ratio of recovered MeOH, acetaldehyde, or methyl xyloside concentration to the initial concentration; e CH 3 -Xylo.: methyl xyloside (1 g) in H 2 O. The catalytic activity increased from 58 h -1 in H 2 O:1-PrOH to 148 h -1 in H 2 O:2-PrOH, which was close to the activity in pure water (220 h -1 ). The distribution of products was different, depending on the alcohol. Best results were achieved in H 2 O:1-PrOH, wherein the overall selectivity to glycols and glycerol was as high as 70%, while the selectivity to C4 and C5 alditols was of 8%. As outlined above, in that case the retro-aldol reaction was favored vs.

	Reaction medium (%vol.%)	Activity (h -1 )	Carbon selectivity b (%) EG PG GLY LA C 4 products c	C 5 alditols	d (%) R i
	100% H 2 O	220	10 12	12	< 1	18	23	-
	CH 3 -Xylo. e	224	12 12	14	< 1	20	24	0
	EtOH : H 2 O (10 : 90)	56	25 34	13	< 1	4	0	40
	CH 3 CHO : H 2 O (10 : 90)	0	-	-	-	-	-	-	23

a decarbonylation and epimerization reactions. Surprisingly, in H 2 O:2-PrOH solvent, the product distribution was the same as in pure water. The selectivity to glycols was of 45% and the one to C4 and C5 alditols of 20%. Thus, competitive decarbonylation, epimerization and retro-aldol reactions took place in H 2 O:2-Pr-OH. Moreover, no deactivation of the Ru/MnO/C catalyst was observed in H 2 O:2-PrOH. Xylitol conversion was complete after 30 h (Figure

3

.f) and no coke was detected on the used catalyst by TGA analysis unlike for the one after hydrogenolysis in H 2 O:1-PrOH (Figure

AUTHOR INFORMATION

Corresponding Author

*to whom correspondence should be addressed. Email: michele.besson@ircelyon.univ-lyon1.fr Tel: +33 472 445 358.

ACKNOWLEDGMENT

The authors gratefully acknowledge financial support from Ecole Doctorale de Chimie Lyon (PhD grant M.R.) and IFP Energies nouvelles. The authors acknowledge the Service Scientifiques of IRCELYON for their help in the characterization of the catalyst.


Mazen Zein
email: mazen.zein@irccyn.ec-nantes.fr

Philippe Wenger

Damien Chablat

SINGULAR CURVES IN THE JOINT SPACE AND CUSP POINTS OF 3-RPR PARALLEL MANIPULATORS

This paper investigates the singular curves in the joint space of a family of planar parallel manipulators. It focuses on special points, referred to as cusp points, which may appear on these curves. Cusp points play an important role in the kinematic behavior of parallel manipulators since they make possible a nonsingular change of assembly mode. The purpose of this study is twofold.

First, it exposes a method to compute joint space singular curves of 3-RPR planar parallel manipulators. Second, it presents an algorithm for detecting and computing all cusp points in the joint space of these same manipulators.

Introduction

Because at a singularity a parallel manipulator loses its stiffness, it is of primary importance to be able to characterize these special configurations. This is, however, a very challenging task for a general parallel manipulator [START_REF] Merlet | Parallel Robots[END_REF] . Planar parallel manipulators have received a lot of attention [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Gosselin | Singularity analysis of closed loop kinematic chains[END_REF][START_REF] Gosselin | Solution polynomiale au problème de la cinématique directe des manipulateurs parallèles plans à 3 degrés de liberté[END_REF][START_REF] Sefrioui | On the quadratic nature of the singularity curves of planar threedegree-of-freedom parallel manipulators[END_REF][START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF][START_REF] Bonev | Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory[END_REF][START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF][START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF] because of their relative simplicity with respect to their spatial counterparts. Moreover, studying the former may help understand the latter. Planar manipulators with three extensible leg rods, referred to as 3-RPR manipulators, have often been studied. Such manipulators may have up to six assembly modes [START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF] . The direct kinematics can be written in a polynomial of degree six [START_REF] Gosselin | Solution polynomiale au problème de la cinématique directe des manipulateurs parallèles plans à 3 degrés de liberté[END_REF] . Moreover, as is the case in most parallel manipulators, the singularities coincide with the set of configurations where two direct kinematic solutions coincide. It was first pointed out that to move from one assembly mode to another, the manipulator should cross a singularity [START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF] . However, Innocenti [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF] showed, using numerical experiments, that this statement is not true in general. More precisely, this statement is only true under some special geometric conditions, such as similar base and mobile platforms [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF][START_REF] Kong | Forward displacement analysis of third-class analytic 3-RPR planar parallel manipulators[END_REF] . In fact, an analogous phenomenon exists in serial manipulators that can change their posture (inverse kinematic solution) without meeting a singularity in general, but not under special geometric simplifications [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Burdick | A Classification of 3R regional manipulator Singularities and Geometries[END_REF][START_REF] Ph | A New General Formalism for the Kinematic Analysis for all Non-Redundant Manipulators[END_REF][START_REF] Omri | How to recognize simply a non-singular posture changing 3-DOF manipulator[END_REF] . The nonsingular change of posture in serial manipulators was shown to be associated with the existence of points in the workspace where three inverse kinematic solutions meet, called cusp points [START_REF] Omri | How to recognize simply a non-singular posture changing 3-DOF manipulator[END_REF] . Cusp points in serial manipulators were determined by looking for the triple roots of the inverse kinematics polynomial [START_REF] Omri | How to recognize simply a non-singular posture changing 3-DOF manipulator[END_REF] or from the equation of the workspace boundary [START_REF] Ottaviano | A cartesian representation for the boundary workspace of 3R manipulators[END_REF] . Likewise, McAree [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] pointed out that for a 3-RPR parallel manipulator (as well as for its spatial counterpart, the octahedral manipulator), if a point with triple direct kinematic solutions exists in the joint space, then the nonsingular change of assembly mode is possible. A condition for three direct kinematic solutions to coincide was established. However, no systematic exploitation of this condition was possible because the algebra involved was too complicated and to the authors' knowledge, the work of McAree [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] has never been pursued yet. Wenger [START_REF] Ph | Workspace and Assembly modes in Fully-Parallel Manipulators : A Descriptive Study[END_REF] showed that to accomplish a non-singular assembly-mode changing motion, a 3-RPR manipulator platform should encircle a cusp point in its joint space. Thus, determination of cusp points is of interest for planning trajectories. A procedure for computing joint space singularities of 3-RPR parallel manipulators was established in a previous paper [START_REF] Zein | Singular curves and cusp points in the joint space of 3-RPR parallel manipulators[END_REF] , where the cusp points were shown on the joint space singularities of these manipulators but no solution was proposed for computing these cusp points.

In this paper, the abovementioned condition for computing cusp points is reviewed and exploited. An algorithm for the systematic detection of cusp points is developed. The method [START_REF] Zein | Singular curves and cusp points in the joint space of 3-RPR parallel manipulators[END_REF] for computing and representing joint space singularities of 3-RPR planar parallel manipulators is first recalled. A descriptive analysis of the singular curves in slices of the joint space of 3-RPR parallel manipulators is conducted. It is shown that the number of cusp points depends on the slice of the joint space in which these cusp points are determined. Moreover, the maximum number of cusp points depends on the geometry of the manipulator. This work helps better understand the topology of the joint space of parallel manipulators and finds applications in both design and trajectory planning.

The following section introduces the 3-RPR manipulator and its constraint equations. Section 3 is devoted to the determination of the singular curves in slices of the joint space. The existence condition of cusp points is derived in section 4 and an algorithm to determine these points automatically is provided. Section 6 is devoted to a descriptive analysis of singular curves in slices of the joint space.

Preliminaries

Manipulators under study

The manipulators under study are 3-DOF planar parallel manipulators with three extensible leg rods (Fig. 1). These manipulators have been frequently studied [START_REF] Sefrioui | On the quadratic nature of the singularity curves of planar threedegree-of-freedom parallel manipulators[END_REF][START_REF] Wenger | Definition sets for the direct kinematics of parallel manipulators[END_REF][START_REF] Bonev | Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory[END_REF][START_REF] Hunt | Assembly configurations of some In-parallel-actuated manipulators[END_REF][START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF] . Each of the three extensible leg rods is actuated with a prismatic joint. The geometric parameters of the manipulators are the three sides of the moving platform d 1 , d 2 , d 3 and the position of the base revolute joint centers defined by A 1 , A 2 and A 3 . The reference frame is centered at A 1 and the x-axis passes through A 2 . Thus, A 1 = (0, 0), A 2 = ( A 2x , 0) and

A 3 = ( A 3x , A 3y ). x A 1 A 2 B 2 B 3 B 1 θ ρ 1 ρ 2 θ 2 d 1 d 3 d 2 θ 1 y A 3 ρ 3 θ 3 α β h Fig 1.
The 3-RPR parallel manipulator under study.

Constraint equations

Let L ≡ (ρ 1 , ρ 2 , ρ 3 ) define the lengths of the three leg rods and let θ ≡ (θ 1 , θ 2 , θ 3 ) define the three angles between the leg rods and the x-axis. The six parameters (L, θ) can be regarded as a configuration of the manipulator but only three of them are independent, so that the configuration space is a 3-dimensional manifold embedded in a 6-dimensional space. The dependency between (L, θ) can be identified by writing the fixed distances between the three vertices of the mobile

platform B 1 , B 2 , B 3 : [ ][ ] [ ][ ] [ ][ ] 2 1 2 1 2 1 1 2 2 3 2 3 2 2 2 3 1 3 1 3 3 ( , ) ( , ) ( , ) ( , ) ( , ) 0 ( , ) ( , ) ( , ) ( , ) ( , ) 0 ( , ) ( , ) ( , ) ( , ) 0 ( , )
T T T d d d ⎧ Γ = - - - = ⎪ ⎪ Γ = - - - = ⎨ ⎪ Γ = - - - = ⎪ ⎩ L θ b L θ b L θ b L θ b L θ L θ b L θ b L θ b L θ b L θ L θ b L θ b L θ b L θ b L θ (1)
where b i is the vector defining the coordinates of B i in the reference frame as function of L and θ. For more simplicity, (L, θ) will be omitted in the following equations.

Expanding each Γ i as a series about the configuration (L, θ) yields

2 3 3 3 3 1 1 1 1 3 3 1 1 2 .. 0 1 ... ! 1 ! i i i i i i i j j j j j j j j n i i i j j j j n θ θ θ ρ θ ρ θ θ ρ ρ ρ ρ = = = = = = ⎛ ⎞ ⎛ ⎞ ∂ ∂ ∂ ∂ ΔΓ = Δ + Δ Γ + Δ + Δ Γ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ∂ ∂ Δ + Δ Γ + = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ + + ∑ ∑ ∑ ∑ ∑ ∑ (2)
If one keeps only the first-order and second-order terms, Eq.( 2) can be written in matrix form as follows:

2 2 2 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 2 2 1 1 0 2 2 T T T T T T T T T ⎡ ⎤ ∂ Γ ⎡ ⎤ ⎡ ⎤ ∂ Γ ∂ Γ Δ Δ Δ Δ Δ Δ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ∂ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ Γ ∂ Γ ∂ Γ ⎢ ⎥ ⎢ ⎥ Δ = Δ + Δ + Δ Δ + Δ Δ + Δ Δ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ∂ ∂ ∂ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∂ Γ ∂ Γ ⎢ ⎥ ⎢ ⎥ ∂ Γ Δ Δ Δ Δ ⎢ ⎥ Δ Δ ⎢ ⎥ ⎢ ⎥ ∂ ∂ ∂ ∂ ⎣ ⎦ ⎣ ⎦ ⎢
is in a regular configuration and the following equation ( 4) can be used instead of Eq.( 3):

( ) ( ) , , 0 ∂ ∂ Δ + Δ = ∂ ∂ Γ L θ Γ L θ θ L θ L (4)
Otherwise the configuration (L,θ) is special and the manipulator meets a singularity. This happens when the constraint Jacobian ∂ ∂ Γ/ θ drops rank so that the second order terms of equation

(3) are required to describe the constraints. The three vertices of the moving platform have the following coordinates in the fixed reference frame,

( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 cos sin ; cos sin ; cos sin . T T x T x y A A A ρ θ ρ θ ρ θ ρ θ ρ θ ρ θ = = + ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ = + + ⎣ ⎦ b b b
Thus, the constraint Jacobian can be put in the following form:

( ) ( ) ( ) ( ) 1 2 1 2 12 2 1 2 1 2 2 2 2 3 2 3 2 3 3 3 23 3 2 2 23 3 3 1 3 1 3 31 3 3 3 1 31 3 1 3 3 0 ( ) ( ) 2 0 ( ( 0 )
)

x x x x x x y y x x y y A s s s A s A A s A A s s A c s A c A s s A s s A c A c ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ⎡ ⎤ ⎢ ⎥ + - ⎢ ⎥ ⎢ ⎥ - - - ⎢ ⎥ ∂ = ⎢ ⎥ ∂ - + - + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ - - - ⎢ ⎥ - - ⎢ ⎥ ⎣ ⎦ Γ θ (5) where (
)

sin i i s θ = ,
( )

cos i i c θ = and
( )

sin ij i j s θ θ = - .

Jointspace Singular Curves

The manipulator is in a singular configuration whenever the axes of its three leg rods are concurrent or parallel [START_REF] Hunt | Geometry of Mechanisms[END_REF] .

x

A 1 A 2 B 1 θ 2 θ 1 y A 3 θ 3 B 2 B 3 d 1 d 3 d 2 α β Axis 1 Axis 3 Axis 2 ρ 2 ρ 1 ρ 3

Fig 2. A 3-RPR parallel manipulator on a singular configuration

We derive the condition for the three leg axes to intersect at a common point (possibly at infinity). We first write the equations of the three leg axes:

1 1 2 2 2 3 3 3 3 3 (Axis 1) : cos( ) sin( ) (Axis 2) : cos( ) ( ) sin( ) (Axis 3) : cos( ) ( ) sin( ) cos( ) x x y y x y x A θ θ θ ⎧ = ⎪ = - ⎨ ⎪ = - + ⎩ (6)
Eliminating x and y in (6) yields the following singularity equation in the task parameters (θ 1 , θ 2 , θ 3 ):

( )

2 2 31 3 3 3 3 12 0 x x y A s s A s A c s + - = (7)
where ( )

sin i i s θ = ,
( )

cos i i c θ = and
( )

sin ij i j s θ θ = - .
It is possible to express Eq (7) as function of the joint space parameters ρ 1 , ρ 2 and ρ 3 by using the constraint equations of the 3-RPR manipulator. However, the resulting equation would be too complicated to yield real insights and difficult to handle. Our approach to compute the singular configurations in the joint space consists in reducing the dimension of the problem. We consider two-dimensional slices of the configuration space by fixing the first leg rod length ρ 1 .

Step 1: We rewrite Eq. ( 7) as a function of ρ 1 , α and θ 1 using the constraint equations of the manipulator:

( ) ( ) ( ) ( ) 2 2 2 1 1 1 2 2 1 1 1 3 3 3 1 1 3 3 3 3 1 1 3 c c cos 0 s s sin 0 c c cos 0 s s sin 0, x x y A d d A d A d ρ ρ α ρ ρ α ρ ρ α β ρ ρ α β + - - = ⎧ ⎪ - - = ⎪ ⎨ + - - + = ⎪ ⎪ + - - + = ⎩ (8)
The first (resp. last) two equations make it possible to express ρ 2 (resp. ρ 3 ) as function of ρ 1 , α and θ 1 . Then, c 2 and s 2 (resp. c 3 and s 3 ) are calculated as function of ρ 1 , α and θ 1 from the first (resp. last) two equations of ( 8) and their expressions are input in Eq. ( 7), which, now, depend only on ρ 1 , α and θ 1 .

Step 2: We fix a value for ρ 1 , so Eq. ( 7) depends now only on α and θ 1 . By varying α or θ 1 , we compute the roots of the equation, to obtain the singular configurations (α s , θ 1s ) for a fixed ρ 1s .

Step 3: For every singular configuration computed in the second step of the approach, we calculate the corresponding values ρ 2s and ρ 3s using equation system (9). We have thus the singular configurations curves in a slice of the joint space (ρ 2 , ρ 3 ) for a fixed value of ρ 1 :

( ) ( ) ( ) ( ) 2 2 1 1 1 2 2 1 1 1 2 3 1 1 3 3 2 3 1 1 3 cos( ) cos( ) sin( ) sin( ) cos( ) cos( ) sin( ) sin( ) x x y A d d A d A d ρ θ α ρ ρ θ α ρ θ α β ρ ρ θ α β ⎧ - + + + ⎪ = ⎪ + ⎪ ⎨ ⎪ - - + + ⎪ = ⎪ - - + ⎩ (9)
Figure 3 shows a slice of the joint space singular configurations for ρ 1 =17 obtained for the same 3-RPR manipulator used by several authors [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] , which has the following geometric parameters: A 1 = (0, 0), A 2 = (15.91,0), A 3 = (0, 10), d 1 = 17.04, d 2 = 16.54 and d 3 = 20.84 in an arbitrary length unit. We refer only to this manipulator in this paper in order to illustrate our work.

Determination of the Cusp Points

Existence condition of cusp points

For serial 3-DOF manipulators, the cusp points can be determined by deriving the condition under which the inverse kinematics polynomial admits three identical roots [START_REF] Omri | How to recognize simply a non-singular posture changing 3-DOF manipulator[END_REF] . However this approach is much more complicated when applied to the direct kinematics polynomial of 3-RPR manipulators because this polynomial is of degree 6.

An interesting alternative approach was proposed in McAree's study 9 by writing the condition under which the manipulator loses first and second order constraints. The resulting condition for triple coalescence of assembly modes was shown to take the following form:

2 2 2 3 1 2 1 2 3 2 2 2 0, T u u u ⎡ ⎤ ∂ Γ ∂ Γ ∂ Γ + + = ⎢ ⎥ ∂ ∂ ∂ ⎣ ⎦ v v θ θ θ ( 10
)
where v is a unit vector in the right kernel of matrix ∂ ∂ Γ/ θ , and u 1 , u 2 , u 3 are the three components of the unit vector u that spans the left kernel. Vectors u and v can be chosen in the set of nonzero rows and columns of the adjoint of matrix ∂ ∂ Γ/ θ (i.e. the matrix of cofactors of the transpose of ∂ ∂ Γ/ θ ), respectively.

Calculating the adjoint of ∂ ∂ Γ/ θ from Eq. ( 5) yields:

k k k k k k k k k k k k k k k k k k - ⎛ ⎞ ∂ ⎛ ⎞ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ∂ ⎝ ⎠ ⎜ ⎟ - ⎝ ⎠ Γ θ (11) where ( ) ( ) ( ) ( )
( )

1 2 3 2 2 3 2 3 3 2 2 3 11 3 3 3 3 3 3 3 3 2 3 22 3 3 3 2 2 2 x x y x y x x y k A A s s A c k s A s A c k A A s s A c ρ ρ ρ ρ ρ ρ = - + - = - + - = - - + - ( ) ( ) (
)

4 1 3 1 3 3 1 3 1 5 2 11 2 2 2 6 1 2 1 2 2 1 2 2 2 x y x x k s A s A c k s A s k s A s ρ ρ ρ ρ ρ ρ = + - = - + = + (12)
Taking u (resp. v) as the first row (resp. column) of ( 11), Eq. ( 10) can be written as ( )

1 2 2 2 2 3 1 2 1 2 3 4 1 4 1 2 2 5 3 5 3 4 2 2 2 1 4 0 k k k k k k k k k k k k k k k k k k ⎛ ⎞ ⎛ ⎞ ∂ Γ ∂ Γ ∂ Γ ⎜ ⎟ - - + = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ⎜ ⎟ - ⎝ ⎠ θ θ θ (13) where ( ) ( ) ( ) ( ) (
)

2 1 2 1 2 21 1 2 2 1 1 1 2 21 2 2 2 1 21 2 2 2 2 3 2 2 2 3 23 2 3 23 3 2 3 2 3 3 2 3 23 2 23 3 3 1 3 1 3 31 3 1 1 3 2 3 2 0 2 0 0 0 0 0 0 0 ( ) 2 0 ( ) 0 0 2 x x x x y x x y x y A c c c c A c c A A c c c A s A A c c c A s A c c A s ρ ρ ρρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρρ + - ⎡ ⎤ ∂ Γ ⎢ ⎥ = - - - ∂ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ - - ⎢ ⎥ ∂ Γ = - ⎢ ⎥ - - ∂ ⎢ ⎥ ⎢ ⎥ - - ⎢ ⎥ + - ⎢ ⎥ ⎣ ⎦ + + - ∂ Γ = ∂ θ θ θ (
)

31 1 3 31 3 1 31 3 3 3 3 0 0 0 0 x y c c c Ac As ρ ρ ρ ρ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ - - - ⎢ ⎥ ⎣ ⎦ (14)
In McAree's study [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] , Eq. ( 13) was not expanded. McAree 9 noted that the expansion of this equation was too complicated to yield any real insight.

We have developed an algorithm to solve this equation for any 3-RPR manipulator and we have implemented it in Maple. This algorithm detects all the cusp points inside the joint space of any 3-RPR manipulators and computes their coordinates. We present it in next section.

Algorithm for calculating cusp points

The presence of cusp points allows the 3-RPR manipulator to undertake non-singular assembly mode changing trajectories; these special trajectories can be executed by encircling a cusp point. In their study [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] , the authors stated that cusp points are pernicious and should be avoided or designed out by judicious dimensioning.

The configuration of the 3-RPR manipulator is given by six parameters: the three rod lengths (ρ 1 , ρ 2 , ρ 3 ), and the platform position variables (θ 1 , θ 2 , θ 3 ). Only three of these parameters are independent. In order to reduce the dimension of the problem, McAree [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] shows that it is possible to consider two-dimensional slices of the configuration space by fixing one of the leg rod lengths.

By doing so, the manipulator configuration can be fully defined by only two parameters. For example, for a fixed value of ρ 1 , a configuration may be fully defined by either (α,θ 1 ) or (ρ 2 , ρ 3 ).

Note that in the first case, the configuration is defined in the output space by the position and the orientation of the moving platform (ρ 1 and θ 1 define the position of B 1 in the plane and α defines the orientation of the moving platform in the plane). In the second case, the configuration is defined in the joint space by the lengths of the three leg rods.

In our work, we have always taken ρ 1 as the fixed parameter. After fixing the value of ρ 1 , we first calculate the singularity curves in (ρ 2 , ρ 3 ), and then we compute all the cusp points of this twodimensional slice.

Algorithm

If we consider equation (7), we notice that it is a function of (θ 1 , θ 2 , θ 3 ). The existence condition of cusp points ( 13) is a function of (ρ 1 , ρ 2 , ρ 3 ) and (θ 1 , θ 2 , θ 3 ). Our first goal is to establish an equation, which is a function of (ρ 1 , α, θ 1 ), and then to solve it to obtain the coordinates of the cusp points. Thus, we first derive a set of equations from the geometry of the manipulator:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 1 2 2 3 1 1 3 3 3 1 1 1 2 2 3 1 1 3 3 3 cos cos cos cos cos s sin sin sin sin sin sin x x y A d A d co d A d ρ θ α θ ρ ρ θ α β θ ρ ρ θ α θ ρ ρ θ α β θ ρ ⎧ - + + = ⎪ ⎪ ⎪ - + + + = ⎪ ⎪ ⎨ + ⎪ = ⎪ ⎪ - + + + ⎪ = ⎪ ⎩ (15)
The algorithm for detecting cusp points is implemented in MAPLE; its steps are presented below:

1. First, the expression of cos(θ 2 ), cos(θ 3 ), sin(θ 2 ) and sin(θ 3 ) in ( 15) are substituted into the singularity equations (7). Then, sin(α) and cos(α) (resp. cos(θ 1 ) and sin(θ 1 )) are written as function of tan(α/2) (resp. of tan(θ 1 /2)). As a consequence we obtain an equation of the form:

( )

1 1 1 , , 0 F t t ρ = (16)
where ( )

tan t α = and
( )

1 1 tan t θ = .
2. Then, the expression of cos(θ 2 ), cos(θ 3 ), sin(θ 2 ) and sin(θ 3 ) in ( 15) are substituted into equations ( 12) and ( 14). Applying then the tan-half substitution as above yields an equation of the form:

( )

1 1 1 , , 0 E t t ρ = (17)
So, we notice that the two equations ( 7) and ( 13) are written now as function of three parameters only.

3. We fix now ρ 1 , and we input the manipulator parameters d 1 , d 2 , d 3 , A 2x , A 3x and A 3y . We have noticed that the direct substitution of the real values of ( )

sin β and
( ) cos β into equations ( 16) and ( 17) make the equations resolution very complicated in the following steps.

we write ( )

sin β and
( ) cos β as a function of an intermediate parameter h, which is the altitude of the moving triangle.

4. The Maple resultant function is used to eliminate ( )

tan t α =
from the two equations ( 16)

and (17). The resulting equation is a polynomial of degree 96 in ( )

1 1 tan t θ =
, which can be factored as follows:

- = ( 18
)
where Q is a 24 th -order univariate polynomial in t 1 and P 1 , P 2 ,…,P n are quadratic and quartic polynomials in t 1 . Note that the factor form could not be obtained without the intermediate parameter h.

5.

We input the parameter h value. We solve equation (18). Each real root t 1i is backsubstituted into (16), which is then solved for t. For every t 1i , we obtain different values for t ij . Finally, we get a number of solution couples (t ij ,t 1i ).

6. We substitute the values of each solution couple (t ij ,t 1i ) into ( 17), and we keep only those solutions that satisfy this equation.

The solutions (t ij ,t 1i ) kept in the last step should give the coordinates of the cusp points. To verify this, we calculate the direct kinematic solutions for each solution (t ij ,t 1i ). In many instances, we have found that some solutions do not yield three coincident solutions. This means that they are not associated with cusp points. So we reject them and we keep only those solutions that give three coincident direct kinematic solutions. These couples are the coordinates of the cusp points, we call them (α ij ,θ 1i ) cusp .

After running our algorithm hundreds of times, we have noticed that in each case all cusp points were determined by the 24 th -order polynomial Q of equation (18), that is, all remaining factors provided spurious solutions. Thus, we may conjecture that the cusp points are determined by Q, although we have no mathematical proof for this fact. All real roots of Q are the cusp points.

With this conjecture, our algorithm simplifies significantly because instead of solving (18) (a 96 th order polynomial), we just have to solve polynomial Q (a 24 th order polynomial).

To implement this result in the algorithm, we must change steps 5 and 6 into the following steps 5' and 6', and eliminate step 7:

5'. We input the real value of parameter h. We solve the polynomial Q. We substitute every real root θ 1i of Q into equation ( 16), and solve it for tan(α/2). For every θ 1i , we obtain different values α ij . Finally, we get a number of couples (α ij ,θ 1i ).

6'. We substitute the values of each couple (α ij ,θ 1i ) into equation (17). The couples that satisfy this equation are the coordinates of the cusp points. We call them (α ij ,θ 1i ) cusp .

Finally, to obtain the coordinates of the cusp points in the joint space (ρ 1 , ρ 2 , ρ 3 ), we use the system of equations ( 9) computed from the geometry of the manipulator. We then obtain the cusp points in a slice of the joint space for a fixed value of ρ 1 .

In step 7 of the algorithm, we have noticed that the existence condition of cusps generates solutions that do not provide triple direct kinematic solutions. This confirms the statement of McAree 9 that the cusp existence condition that he established is a sufficient condition but not a necessary one.

Algorithm Execution

The algorithm execution time slightly depends on the value of ρ 1 and of the 3-RPR manipulator parameters. It highly depends on the number of digits required for the calculation. For 90 digits (which is necessary to guaranty a good accuracy), it is about two minutes on a computer equipped with a 3GHz-Pentium 4 with 512 Mo of Ram.

We present the results of an execution of the algorithm, for the same 3-RPR parallel manipulator introduced in section 3. For the same fixed value ρ 1 =14.98, as the one used in previous papers [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] , the algorithm detects six cusp points instead of five identified in the paper [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] . Figure 4 shows the singular curves of the 3-RPR manipulator for ρ 1 =14.98, and the six cusp points

Descriptive Analysis

In this section, the singular curves are analysed in several slices of the joint space for the manipulator studied in papers [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] and whose geometric parameters were given in section 3. Figure 5 depicts the singular curves for increasing values of ρ 1 and shows that the number of cusp points is not the same for all slices as we may have 0, 2, 4, 6 or 8 cusp points. In Fig. 5, regions with two assembly modes (resp. four, six) are filled in light grey (resp. in dark grey, in black). Zero-cusp slices are obtained for very small values of ρ 1 only (ρ 1 =0.05 in Fig. 5), where the singular curves are made of two separate closed curves that define only one small region with two assembly modes and a large void (note, the two curves are so close that the region cannot be seen in Fig. 5). When ρ 1 is increased, two cusp points appear, the void gets smaller and a four-solution region appears (ρ 1 =2).

Then two more cusp points appear, defining one more four-solution region (ρ 1 =2.8). We have six cusp points and a small void at ρ 1 =6; for ρ 1 =8, 10, 12, 14, 16, 18, 20, 24 and 26, we have always six cusp points but the void is replaced with a six-solution region. Eight cusp points exist in a small vicinity of ρ 1 =27. Then two cusp points and the six-solution region disappear (ρ 1 =29). Finally, the number of cusp points stabilizes to four, defining one central four-solution region surrounded by a two-solution region (ρ 1 =31, …). Interestingly, this last pattern is very similar to the one often observed in a cross-section of the workspace of 3-R serial manipulators [START_REF] Baili | A classification of 3R orthogonal manipulators by the topology of their workspace[END_REF] . However, serial manipulators feature the same pattern in all cross-sections (the sections which passes through the first revolute joint axis), and variation in the number of cusp points arises only from a modification of the manipulator geometry.

The above analysis shows that the joint space topology of 3-RPR manipulators is very complicated. Contrary to serial manipulators, the shape of the singular curves and the number of cusps points depend on which slice is chosen in the joint space. Thus, planning trajectories is not easy. However, we have noticed that the pattern stabilizes for sufficiently large values of ρ 1 . The surface depicted in Figure 6 is of interest:

ρ 2 ρ 2 ρ 2 ρ 2 ρ 3 ρ 3 ρ 3 ρ 3 ρ 1 =27 ρ 1 =29 ρ 1 =31 ρ 1 =26 ρ 2 ρ 2 ρ 2 ρ 1 =16 ρ 1 =18 ρ 1 =20 ρ 1 =24 ρ 2 ρ 3 ρ 3 ρ 3 ρ 3 ρ 2 ρ 1 =8 ρ 1 =10 ρ 2 ρ 2 ρ 1 =12 ρ 2 ρ 3 ρ 3 ρ 3 ρ 3 ρ 1 =14 ρ 2 ρ 1 =0.05 ρ 2 ρ 1 =2 ρ 2 ρ 2 ρ 1 =6 ρ 3 ρ 3 ρ 3 ρ 3 ρ 1 =2.8 ρ 2 ρ 1 =34 ρ 2 ρ 1 =50 ρ 1 =75 ρ 2 ρ 2 ρ 3 ρ 3 ρ 3 ρ 3 ρ 1 =100
i. for planning trajectories in the joint space because it shows clearly the joint space regions that are free of singularities.

ii. for manipulator design, because it offers a tool for defining the values of the joint limits such that the joint space is a singularity-free box.

We can see clearly in figure 6 how the number of cusp points and the topology of the slices stabilize for ρ 1 >31. This feature has been observed in many other manipulator geometries. For example, the 3-RPR manipulator defined by the following geometric parameters: A 1 = (0,0), A 2 = (3,0), A 3 = (1.1,2.7), d 1 =

1.3, d 2 = 0.9 and d 3 = 0.4 in an arbitrary length unit, has a constant pattern as soon as ρ 1 >5 (Fig. 7).

Note that, in contrast with the stabilized pattern obtained for the preceding manipulator, this pattern features a large void.

Most of the research work on parallel manipulators has focused on the analysis and optimization of the workspace. If the workspace is useful for manipulator design, the analysis of singular curve patterns in the joint space can be used as a complementary tool to compare several manipulator geometries.

Conclusions

A procedure for computing joint space singularities of 3-RPR parallel manipulators has been presented in this paper. Then, the existence condition of cusp points defined by McAree 9 has been reviewed and exploited. An algorithm able to detect and to compute cusp points inside any section of the joint space of any 3-RPR parallel manipulator has been established. To the best of the authors' knowledge, such an algorithm had never been proposed before. The algorithm results in a 96 th degree univariate polynomial that can be put in a factored form. We have showed with intensive numerical experiments that the coordinates of the cusp points are the real roots of a 24 th degree univariate polynomial, which is one of the factors of the 96 th polynomial. Finally, a descriptive analysis of the singular curves in slices of the joint space of 3-RPR parallel manipulators has been conducted, with a special focus on the cusp points on these singular curves. It has been shown that, contrary to what arises in serial manipulators where any cross section of the workspace exhibits the same pattern of singular curves and cusp points, this pattern depends on the choice of the slice in the joint space for a given 3-RPR parallel manipulator. On the other hand, we have noticed that it is possible to have a constant pattern by adjusting the joint limits. It has been observed that the

Fig 3 .

 3 Fig 3. Singular curves in (α, θ 1 ) for ρ 1 =17. Number of assembly modes is displayed in each region

Figure 3

 3 Figure3shows the singular curves in the joint space slice (ρ 2 , ρ 3 ) defined by ρ 1 =17. These

2 ρ 3 Fig 4 .

 234 Fig 4. Singular curves and cusp points in a slice of the 3-RPR manipulator joint space (ρ 2 ,,ρ 3 ) for ρ 1 =14.98.

Fig 5 .

 5 Fig 5. Singular curve patterns for increasing values of ρ 1 .

Figure 6

 6 Figure 6 represents the singularities in the joint space of the manipulator studied when ρ 1 varies

Fig 6 .

 6 Fig 6. Joint space singularity surfaces of the 3-RPR manipulator studied when ρ 1 varies from 0 to 50.

Fig 7 .

 7 Fig 7. Stabilized singular curve patterns for another manipulator geometry plotted for ρ 1 =5 (left) and ρ 1 =20 (right).

  The coordinates of the six cusp points are given in the table1 below

		α	θ 1	ρ 2	ρ 3
	Cusp A	50.67 deg	-69.12 deg	0.84	3.77
	Cusp B	-2.59 deg	177.32 deg	13.85	6.26
	Cusp C	-122.89 deg	114.05 deg	31.27	16.17
	Cusp D	57.48 deg	133.77 deg	30.44	26.61
	Cusp E	-0.59 deg	15.46 deg	16.02	29.56
	Cusp F	170.37 deg	-10.65 deg	17.98	26.44
		Tab 1. Coordinates of the six cusp points for ρ 1 =14.98.

L θ L θ

maximum number of cusps depends on the manipulator geometry . No manipulators were found with more than eight cusp points. Most of the research work on parallel manipulators has been focused on the analysis and optimization of the workspace. This paper provides a complementary tool that helps better understand the topology of the joint space of parallel manipulators and finds applications in both design and trajectory planning.


Carl Winsløw

Rolf Biehler

Barbara Jaworski

Frode Rønning

Megan Wawro

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Education and professional development of University Mathematics Teachers

Panellists: Rolf Biehler (Paderborn University, Germany)

Barbara Jaworski (Loughborough University, United Kingdom)

Frode Rønning (Norwegian University of Science and Technology, Norway)

Megan Wawro (Virginia Tech, United States)

Chair:

Carl Winsløw (University of Copenhagen, Denmark)

ABSTRACT

The theme of this panel may surprise some, as university teachers of mathematics typically hold a PhD in mathematics or some adjacent field, and in many places some "pedagogical training" is also foreseen. However, university teaching presents still more challenges (in many places: more inhomogeneous or different student groups to teach), and opportunities (including new technology, andwe hopeuseful resources from research on UME). For all of these reasons, the panel will address the following questions:

1. What is the current, typical preparation of University Mathematics Teachers for their function as teachers? What "in-service" opportunities for teacher development exist? -naturally, answers will depend both on countries and institutions, but sharing experiences could help to provide an updated picture of how the "professional knowledge of UME teachers" is currently built and sustained.

2. Do the current preparation and opportunities for development meet the demands that exist or can be foreseen? Could the preparation and development opportunities be improved, for instance by giving university teachers (more) access to selected parts of current research on UME, and possibly also participate in research and development projects? What initiatives exist, and which could be imagined as beneficialboth to increase the impact and quality of research on UME, and of UME itself?