Pauline Briault

Mathilde Rieu
email: rieu@emse.fr

Richard Laucournet

Bertrand Morel

Jean-Paul Viricelle

Jean- Paul Viricelle

Anode supported single chamber solid oxide fuel cells operating in exhaust gases of thermal engine

Keywords: Single chamber SOFC, energy recovery, unburned hydrocarbons, pollutant emissions reduction

come

Anode supported single chamber solid oxide fuel cells operating in exhaust gases of thermal engine

Introduction

Solid oxide fuel cells are usually described as devices able to convert chemical energy into electrical energy. Conventional solid oxide fuel cells are separated into two compartments containing each electrode split by a gastight electrolyte. Fuel (hydrogen, in most cases) and oxidant (oxygen or air) feed directly each chamber and a special attention is paid to separate carefully the two gases. At the cathode side, oxygen is reduced into oxygen ions (reaction 1) which travel through the electrolyte to the anode side, where they oxidize hydrogen into water (reaction 2). Electrons produced by hydrogen oxidation are then drawn to the cathode through an external circuit. The overall reaction is the chemical oxidation of hydrogen to water (reaction 3). O 2 + 4e-2O 2- (reaction 1)

H 2 + O 2-H 2 O + 2e- ( reaction 2)
H 2 + ½ O 2 H 2 O (reaction 3)

Since the early nineties [1][2], SOFC have been studied in a new configuration where anode and cathode are exposed to the same gas mixture in a single chamber. The Single Chamber SOFCs (SC-SOFCs) are particularly interesting [3][4] due to the absence of sealing between the two usual compartments, removing one of main challenges of conventional SOFCs. However, in SC-SOFC configuration, hydrogen cannot be used as a fuel since hydrogen-air mixtures are highly reactive, thus hydrocarbons, able to form hydrogen through partial oxidation (reaction 4), are most commonly used in the gas mixture.

C n H m + n/2 O 2 n CO + m/2 H 2 (reaction 4)

As there is no difference in the gas composition at each side of the cell, single chamber operation principle is based on the difference of electrodes catalytic activity towards hydrocarbon oxidation. When no current is drawn, the depletion of oxygen at the anode and the maximum concentration of oxygen at the cathode lead to an oxygen partial pressure gradient between both sides of the cell involving an open circuit voltage (OCV) creation.

The single chamber configuration has several advantages in comparison with conventional SOFCs: new cell geometries, stack assembly and cells miniaturization are more easily conceivable. These advantages open the way to new applications such as energy recovery and pollutant emission reduction. In such applications, cells might be embedded at the outlet of engine and could turn unburned hydrocarbons present in the exhaust gas into electricity allowing for instance the energy supplying of electronic devices in vehicles. More largely, such energy conversion device can address all applications where hydrocarbons are burnt such as stationary power plants where environmental and efficiency issues are identified.

To date, only a few authors have investigated such a system. Herman et al. [5] and Uchiyama [6] described equivalent devices in patents. Particularly, N. Uchiyama collaborated on testing such a system in real conditions with T. Hibino's group; two articles dealing with the results were published in 2008 [7][8]. Concerning the gas mixture used for tests, patents and articles expose only evasive or simplified compositions. Herman's patent [5] describes a mixture composed of hydrocarbons (e.g. methane, ethane…), hydrogen and oxygen without any accurate composition.

In Uchiyama's work [6], the studied atmosphere is composed of hydrocarbons, carbon oxides and air without any further details. The only work reporting the concentration of studied stream for single cell test was published by Hibino's group [7][8]. A mixture of four hydrocarbons (methane, ethane, propane and butane) with 1000 ppm of each hydrocarbon specie and with oxygen ranging from 1250 to 15000 ppm is reported. For an electrolyte supported Ni-Ce 0.8 Sm 0.2 O 1.9 (SDC)/ 8%Y 2 O 3 -ZrO 2 (YSZ)/ La 0.8 Sr 0.2 MnO 3 (LSM) single cell, an open circuit voltage and a maximum power density respectively 1100 mV and 20 mW cm -2 were measured at 800°C, a temperature not really considered as realistic for the aimed application. Then, an output power of 1.2 W was delivered by a twelve-cells stack embedded at the exit of a scooter engine, which is still not high enough to provide energy to electronic devices included in a vehicle but proves the feasibility of such energy recovery system. Nevertheless, no exact composition of exhaust gas is given for the stack test at the exit of the scooter.

In the present work, further improvements including gas mixture composition, architecture of the cell and materials are initiated. Indeed, the selected gas mixture is based on a more realistic composition of exhaust gas flowing inside the catalytic converter of a thermal engine. The influence of oxygen content is investigated on the performances of cells. Anode supported architecture is selected instead of the usual electrolyte supported since it has proven an ability to deliver higher power densities by a lower resistance of thinner electrolyte [9]. For the cell components, most performing materials are selected. Ce 0.9 Gd 0.1 O 1.95 (CGO) is used as the electrolyte material due to a high ionic conductivity at intermediate temperature range: 0.025 S cm -1 at 600°C [10]. The anode is a typical cermet composed of nickel and electrolyte material.

Two cathode materials are compared regarding the application, La 0.6 Sr 0.4 Co 0.2 Fe 0.8 O 3-δ (LSCF), a well-known cathode material [11][12] and Pr 2 NiO 4+δ (PNO) only used up to now for conventional SOFCs [13]. Finally, electrolyte microstructure is improved in order to enhance performances.

Experimental

Exhaust stream composition

The gas mixture used in this project is based on data provided by an engine manufacturer corresponding to the gas composition after the catalytic converter. However, it has been slightly modified to fit to laboratory conditions. The gas stream is hence composed of hydrocarbons that there is more than 90 % of nitrogen in the composition meaning that the mixture contains very low levels of reactive species.

Materials and cells preparation

All raw materials used for electrochemical tests and cells manufacturing are commercial powders. Suppliers and powder specifications are given in table 2.

Anode supported cells are prepared by screen printing the electrolyte on the green anode made by tape casting. The slurry for tape casting is prepared according to the formulation previously described in the literature [14]. The powders mixture used in this paper is composed of 60 wt% of NiO from Novamet and 40 wt% of CGO from Rhodia. Electrolyte ink is composed of 70 wt% of CGO from Fuel Cell Materials and 30 wt% of a solvent-binder mixture (terpineol mixed with 5 wt% of ethyl cellulose). Various thicknesses of electrolyte are prepared by screen printing with respectively 3, 5 or 7 layers. The half cells are then co-sintered in two main steps. At first, a slow heating rate (60°C h -1 ) is performed up to 600°C with a 1h-dwell time to remove the binders contained in the anode tape. Then, a faster heating rate towards 1400°C and a 6h-dwell time are respected to sinter the half cells. The reduction of the anode is carried out at 650°C for 4 hours in a 6% H 2 atmosphere. Two cathode layers are then screen-printed on the reduced half-cell using an ink composed of 70 wt% cathode material and 30 wt% CGO (Fuel Cell Materials) as solid content. A first heating at 300°C for 30 min in air is done to remove cathode ink binder and then cathode are annealed at 1100°C in argon to avoid any anode reoxidation. A gold mesh is screen printed on top of the cathode to collect correctly the current during the experiment. Same heating treatment in air (300°C, 30 min) as for the cathode is carried out and the gold mesh is finally annealed at 900°C for 2h under argon.

Fuel cells characterization

In SC-SOFC system, the geometry of the cell but also of the whole test chamber have a great influence on performances. That's why a schema of the experimental test bench and photographs of the sample holder are reported in Figure 1. Experiments are performed in a horizontal tubular furnace. The investigated cell is placed in an alumina sample holder between two gold meshes.

The alumina sample holder is composed of two symmetrical parts to maintain the cell parallel to the flow of gas inlet. The cell is just put between the gold meshes and there is no load applied to press the contact except the weight of the alumina top (i.e. 40 grams). Furthermore, the position of the cell into the sample holder also allows to divide the flow by half between anode and cathode sides.

The incoming gas mixture feeding the reactor is monitored by mass flow meters. Water is generated by nitrogen bubbling at controlled temperature. At the beginning of each test, temperature is increased to 600°C under nitrogen at 10 L h -1 . Then, gas mixture is supplied to the cell during the isothermal stage at 60 L h -1 . This flow rate was fixed to avoid homogeneous catalysis in the gas phase itself. Indeed, at high temperature and at lower flow rate, the gas compositions studied were not stable and gas decomposition occurred without any cell.

Various temperature dwells are done every 50°C between 600°C and 450°C to perform characterizations. Then the cooling is done under nitrogen to avoid other reaction.

Four wires electrical measurements are performed at each temperature dwell. Gold meshes on each side of the cell are used to collect current and two gold wires are stuck on each gold mesh.

Electrical and electrochemical measurements are performed by a VersaSTAT3 from Princeton Applied Research. Impedance spectroscopy measurements were done at open circuit voltage with frequency ranging from 100 kHz to 0.1 Hz using an electrical perturbation of 20 mV.

In parallel, the gas composition is monitored at the outlet of the set-up by an in line micro gas chromatograph (SRA Instruments microGC3000). The water content is measured thanks to a humidity sensor and condensed before the gas flows to the micro-chromatograph.

After tests, cells microstructures are characterized by a scanning electron microscope (SEM) Jeol JSM 6500. ImageJ software is used to evaluate the porosity of electrolyte. Micrographs obtained by SEM (about 10 micrographs) are analyzed with this image processing program to get a porosity average.

X-ray diffraction (XRD) analyses are carried out on a Siemens D5000 using Cu Kα radiation at 40 kV and 40 mA. After all performed experiments, no nickel oxidation is evidenced in the studied conditions.

Specific data calculation

Fuel utilization is calculated from electrical measurements, by assuming it is defined as the faradaic efficiency: ε f (Equation 1).

Equation 1:

= with I: current delivered by the cell and I th : theoretical current obtained if the fuels are completely consumed.

In order to calculate this efficiency, the four fuels present in the gas mixture have to be taken into account: H with HC in and HC out : hydrocarbons volume content at the inlet and the outlet, respectively.

Results and discussion

Influence of ratio R=HC/O 2 on Ni-CGO/CGO/LSCF-CGO cell performances

LSCF is employed for these tests due to a more suitable electrochemical behavior as cathode material [11][12] and a better chemical stability under the gas mixture [15].

As illustrated in Figure 2, a typical cell used for experiment is composed of a 15µm cathode (corresponding to 2 screen printed layers) and a 20 µm (3 screen printed layers) electrolyte on top of a 400 µm thick anode.

Polarization curves are measured at different temperatures ranging from 600°C to 450°C, each 50°C, for each ratio R=HC/O 2 defined in table 1.

R=0.67

For the gas mixture with the lower oxygen content, i.e. R=0.67, electrical measurements show low performances reaching only 2.5 mW cm -2 at 450°C as a maximum power density. SEM analysis after test, reported in Figure 3, revealed carbon whiskers and nodules in the anode layer.

Furthermore, carbon deposit is also visible after the cell test, with black particles detached from the anode. These carbon particles may be a product due to carbon cracking (reaction 8) and Boudouard equilibrium (reaction 9).

C n H m x C + C n-x H m-y + y/2 H 2 (reaction 8) 2 CO C + CO 2 (reaction 9)
The carbon particles formation explain the poor performances obtained during cell operation for the R=0.67 ratio. Indeed, carbon particles may reduce the efficiency of nickel catalytic sites [16].

Comparison of R=0.21 and R=0.44

Electrical performances of Ni-CGO/CGO/LSCF-CGO cells operated under R=0.21 and R=0.44 are reported in Figure 4. For easier comparison, OCV and maximum power densities are determined from the polarization curves and their evolutions are reported in Figure 5 as a function of the temperature. OCV values are relatively low, reaching 700 mV at 450°C. This is attributed to the porous electrolyte letting hydrogen free to go through the open porosity from anode to cathode where it consumes oxygen. Therefore, the gradient of oxygen partial pressure between electrodes decreases and the OCV as well [17][18]. Moreover, for both ratios, an evolution of OCV values is visible with the temperature. Indeed, OCV increases with the decrease of the temperature. This is in agreement with the catalytic activity of cathode towards hydrocarbons oxidation that decreases with the temperature decrease [15,17] and so leads to a higher gradient of oxygen partial pressure between the two electrodes.

Concerning the power density, it increases with a temperature decrease. The maximum values measured are 15.2 mW cm -2 and 8.2 mW cm -2 at 500°C for R=0.21 and at 450°C for R=0.44, respectively. A low decrease of performances is observed for the cell tested under R=0.21 at 450°C.

Finally, OCV and maximum power densities are higher with the cell tested under R=0.21.

Therefore, only this ratio is kept for the following experiments.

Overall, performances obtained here are relatively low compared to the literature for single chamber SOFC operating with various hydrocarbon/air mixtures. Some hundreds of mW.cm -2 are generally reached [9,19]. In the present study, as for Hibino's study for exhaust application (20 mW cm -2 at 800°C, [8]), the main difference is the high dilution of reactive species. Indeed, there is more than 90% of nitrogen and only about 1% of fuel.

Influence of cathode materials under R=HC/O 2 =0.21

Only tests with R=0.21 ratio are performed with PNO as cathode and are exposed here in comparison with LSCF.

Polarization curves of Ni-CGO/CGO/PNO-CGO cells operated under R=0.21 at different temperatures are shown in Figure 6. For easier comparison, evolutions of maximum power density and OCV are reported in Figure 7 for both cathodes as a function of the temperature.

Cell with PNO cathode has shown a better OCV than the one with LSCF material for both ratios.

However, no greater power density is delivered. At 600°C, maximum power densities of both materials are equivalent. But, power densities of cells with PNO cathode decrease with temperatures decrease whereas they increase with LSCF cathode.

In literature, it is known that PNO is a non-stable structure, able to decompose in oxidizing atmospheres, into the Ruddlesden-Popper-type Pr 4 Ni 3 O 10 and praseodymium oxide [20]. XRD analyses are carried out before and after cell test on this cathode and the diffractograms are reported in Figure 8. Other phases than the initial PNO are observed, mainly praseodymium oxide Pr 6 O 11 .

The formed phases could explain the low performances, firstly by limiting the grip of the current collector on the cathode. Indeed, it is observed that, at the end of experiments, the screen-printed gold mesh is completely delaminated from the PNO cathode. Secondly, the new phase may also reduce PNO electrochemical active phase content, thus lowering power density. Moreover, the detected parasite phases might justify the higher OCV of the Ni-CGO/CGO/PNO-CGO cell.

Indeed, the catalytic activity towards HC oxidation of these oxides is uncertain and may be lower than the one of pure PNO, which could lead to a higher oxygen partial pressure gradient between the two electrodes and thus to higher OCV.

To conclude, PNO cathode material is considered as non-suitable for long term cell operation in the conditions investigated in the present study, and following experiments are carried out by using only LSCF as cathode.

Modification of the electrolyte microstructure of a Ni-CGO/CGO/LSCF-CGO cell and characterization under R=HC/O 2 = 0.21

Even if porous electrolyte is not an issue in a single chamber SOFC configuration, Suzuki et al. [18] showed the influence of porosity on the OCV. Indeed, open porosity lets hydrogen and carbon monoxide diffuse freely from anode side to cathode side and consume oxygen by oxidation. Thus, oxygen partial pressure at the cathode side would be decreased and thereby OCV as well. Rembelski et al. [17] showed that a reduction of open porosity by increasing the electrolyte thickness led to higher OCV values and better electrochemical cell performances.

Nevertheless a compromise has to be found because the ohmic loss is directly related to the electrolyte thickness [21].

As the electrolyte used in this work is not fully dense, cells with 3 different electrolyte thicknesses were prepared and characterized.

Microstructure of the cells

Electrolyte characteristics (thickness and porosity) of the 3 different cells, reported in table 3, are observed and measured by SEM micrographs analyses and ImageJ image processing program.

Cells are named in reference with electrolyte thickness for the following results. The CGO-20 cell is the one used during the previous experiments (comparison of R ratios and cathode materials). Microstructures of electrolytes are reported in Figures 2 and9.

The increase of thickness leads to an increase of the electrolyte density. Indeed, porosity decreases from 12% for the cell with a 22 µm thick electrolyte to 8% with a 43 µm thick electrolyte. The performances of these different cells are evaluated under the R=0.21 atmosphere.

Comparison and performances

Experiments are thus performed for cell with thicker electrolytes (CGO-30 and CGO-40) and polarization curves are reported in Figure 10. In order to compared these cells to the previous ones made with a CGO-20 electrolyte, OCV and maximum power density obtained under R=0.21 are reported in Figure 11 for the three cells.

As expected, a significant improvement of OCV values is noticed with the increase of electrolyte thickness by decreasing the open porosity from one electrode to the other and limiting gas diffusion through pores [17][18]. Open circuit voltage is thus enhanced thanks to an increased oxygen partial pressure gradient between electrodes. A higher maximum power density is thus induced especially at the highest temperatures. The best power density obtained is 25.1 mW cm -2 at 550°C for the CGO-40 cell. It shows that an electrolyte microstructure optimization carried out here, a pertinent selection of materials and architecture of the cell lead to better performances at lower temperature than work reported by Yano et al. and Nagao et al. [7][8]. However, further improvements could still be made by densification of the electrolyte. Indeed, the use of a thinner dense electrolyte would lower ohmic losses and also avoid gas cross diffusion problem.

Electrochemical characterizations

In order to complete these results and to evaluate the impact of the resistance of the cell on its performances, electrochemical impedance spectroscopy measurements are performed.

A Nyquist diagram corresponding to a typical Ni-CGO/CGO-20/LSCF-CGO cell tested under R=0.21 at 600°C is reported in Figure 12.

Firstly, even if measurements are performed down to low frequencies (25 mHz), the semicircle corresponding to electrode resistance contribution is not completed. That is probably due to gas diffusion limitations; as active species are largely diluted in nitrogen.

Secondly, considering the intercept of the diagram with real axis, which is usually considered as the ohmic resistance, it is important to mention that the measured value 0.3 Ω cm 2 is much larger than expected from the cell. Indeed, with a 20 µm thick CGO electrolyte, the ohmic resistance should be around 0.1 Ω cm 2 if we consider a conductivity of 2.10 -2 S cm -1 for dense and even porous CGO electrolyte at 600°C [22]. This phenomenon has already been reported in the literature [23] where the authors mentioned that this response in a relatively high frequency range (above 10 kHz) cannot be attributed only to electrolyte resistance but also to contact resistance.

Indeed, in the present study, the cell is only placed between the gold meshes and there is no load applied to press the contact.

For all the studied cells, the shape of Nyquist diagrams is the same. The main conclusion is that the major resistance contribution comes from electrodes and not from electrolyte or contact resistance. Mass transfer and/or kinetics of electrode processes dominate the total resistance of the cell. That can be due to the low content of reactive species. Indeed, the atmosphere is highly diluted in N 2 to fit exhaust gas stream composition.

In single chamber configuration, it is important to remind that the resistance of the cell is not only responsible of the overall performances. Indeed, the open circuit voltage, created thanks to an oxygen partial pressure gradient between the two electrodes, has a strong influence. Thus, even if the cells got nearly the same internal resistance, the higher the OCV is, the higher the performances will be, if no other phenomenon, such as electrode degradation, occurs.

Focus on the energy recovery and emissions reduction results

As the aimed application is to convert exhaust gases into electricity and reduce hydrocarbons emission, fuel utilization and HC conversion rate are calculated from results obtained from cell CGO-40 which has delivered the best performances at 550°C under R=0.21.

Fuel utilization

In order to evaluate the ability of the cells to convert gas into electricity, fuel utilization is calculated according to equation 1 (with I: the delivered current at maximum power density) and is reported in Figure 13 for the CGO-40 cell versus temperature.

From 450°C to 600°C, fuel utilization increases from about 2 to 4.5%. Even if these values seem low, they are in accordance with literature for single chamber SOFC [24]. In the present work, it is noticeable that flow rate is quite high (60 L h -1 ) with a large N 2 dilution. Therefore, the flow velocity around the cell is high (~ 0.6 m s -1 ). Furthermore, gas distribution is not optimized and a large amount of reactive gas species flowed around the cell without being used for electrochemical reactions.

Conversion rate of unburned hydrocarbons

The conversion rate of hydrocarbons is another important factor for the application. Indeed, it allows to show the hydrocarbons emissions reduction released to the atmosphere. To show the capability of cells to decrease pollutant emission, HC conversion rate is calculated according to equation 3. The results obtained at the OCV with the Ni-CGO/CGO-40/LSCF-CGO cell are reported in Figure 14.

HC conversion rate increases from 9 to 17% as temperature increased from 450°C to 600°C. This is in accordance with catalytic activity measurements of nickel. Indeed, the higher the temperature is, the more active towards HC conversion the nickel becomes [15]. The studied cell, that delivered the best performances at 550°C, allowed to convert 15 ± 3 % of unburned hydrocarbons at this temperature.

Conclusion

This paper deals with an innovative concept of energy recovery from exhaust gas produced by a thermal engine. For this project, most performing cell architecture and cell materials were used in exhaust gas conditions at a realistic temperature range. Three different gas mixtures were investigated for an anode supported Ni-CGO/CGO/LSCF-CGO cell. The lowest oxygen containing ratio (R=0.67) was ruled out because of a huge carbon formation on the anode side.

However, the other ratios (R=0.44 and R=0.21) have delivered high power densities, reaching 15.2 mW.cm -2 at 500°C for the oxygen richer ratio (R=0.21). PNO was also investigated as a new cathode material for single chamber SOFC. Although a greater open circuit voltage was created, the maximum power densities were lower than with LSCF material especially when the temperature was decreased from 600 to 450°C. The formation of insulating phases on the PNO cathode during the cell operation has been enlightened and might be the cause of lower performances. Highest performances were obtained with a Ni-CGO/CGO-40/LSCF-CGO cell by increasing the electrolyte thickness up to 40 µm in order to reduce the open porosity; a reduction of 15% of hydrocarbons emissions was possible and 25.1 mW cm -2 were delivered with a fuel utilization of 4.3% at 550°C. Therefore, improved results compared to literature were reached at lower temperatures (500-550°C), and with a gas mixture closer to real conditions.

Figure 1 :Figure 2 .Figure 3 .Figure 4 .Figure 6 .Figure 7 . 21 Figure 8 . 21 Figure 9 .Figure 10 .Figure 11 .Figure 12 .Figure 13 . 21 Figure 14 .

 123467218219101112132114 Figure 1: Schema of experimental test bench (up) and schema and photographs of sample holder

FiguresFigure 1 :Figure 2 .Figure 3 .Figure 4 .

 1234 Figures

Figure 5 .Figure 6 .Figure 7 . 21 Figure 8 . 21 Figure 9 .Figure 10 .Figure 12 .Figure 13 .

 567218219101213 Figure 5. OCV (on left) and maximum power density (on right) versus temperature for Ni-CGO/CGO/LSCF-CGO cell under the ratios 0.21 and 0.44

Figure 14 .

 14 Figure 14. Hydrocarbons emissions reduction (HC conversion rate) of the Ni-CGO/CGO-

Table captions Table 1 .

 captions1 Experimental composition of gas species mixtures simulating an exhaust gas (R=

	HC/O 2 )

Table 2 .

 2 Powders specifications (use for (A): Anode, (C): Cathode, (E): Electrolyte)

Table 3 .

 3 Thickness and porosity of electrolytes of Ni-CGO/CGO-XX/LSCF-CGO cells (-XX refers to electrolyte thickness)

Table 1 .

 1 Experimental composition of gas species mixtures simulating an exhaust gas (R= HC/O 2 )

	40/LSCF-CGO cell versus temperature for ratio R=0.21

Table 2 .

 2 Powders specifications (use for (A): Anode, (C): Cathode, (E): Electrolyte)

Number of screen-printed layers Thickness (µm) Porosity (%) Cell reference

	3	22 ± 1	12 ± 1	CGO-20
	5	31 ± 1	10 ± 1	CGO-30
	7	43 ± 1	8 ± 1	CGO-40

Table 3 .

 3 Thickness and porosity of electrolytes of Ni-CGO/CGO-XX/LSCF-CGO cells (-XX refers to

	electrolyte thickness)

50 / µm Specific surface area, S BET / m² g -1 LSCF (C)

  La 0.6 Sr 0.4 Co 0.2 Fe 0.8 O 3-δ

			Fuel Cell Materials	2	3.9
	PNO (C)	Pr 2 NiO 4+δ	Marion Technologies	0.5	4.2
	NiO (A)	NiO	Novamet	1.5	3.5
	CGO (A)	Ce 0.9 Gd 0.1 O 1.95	Rhodia	0.2	12
	CGO (E+C)	Ce 0.9 Gd 0.1 O 1.95	Fuel Cell Materials	3.2	8.6

Acknowledgments

The project was supported by ARC Energie, la Région Rhône-Alpes, France and by the CNRS via the Cellule Energie -INSIS.


Olivier Chesneau

Florentin Millour

Orsola De Marco

S N Bright

Alain Spang

Eric Lagadec

Djamel Mékarnia

W J De Wit

The RCB star V854 Cen is surrounded by a hot dusty shell

Keywords: high angular resolution, individual: V854 Cen, Stars: circumstellar matter, Stars: mass-loss

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

Introduction

The R Coronae Borealis (RCB) stars are rare hydrogendeficient carbon-rich supergiants, best known for their spectacular declines in brightness at irregular intervals [START_REF] Clayton | [END_REF]De Marco et al. 2002). Two evolutionary scenarios have been suggested for producing an RCB star, a double-degenerate merger of two white dwarfs (WD), or a final helium-shell flash in a planetary nebula central star. However, the discovery that RCB stars have large amounts of 18 O is interpreted as a serious argument in favor of the merger scenario (Clayton et al. 2011(Clayton et al. , 2007)). One may speculate that the merger scenario leads to some observational consequences, such as a fast rotation for the remnant star that leads to a circumstellar environment with an axis of symmetry. Polarimetry has been the main technique to reveal these symmetries, yet the results have so far never been fully conclusive (Yudin et al. 2003;Clayton et al. 1997;Stanford et al. 1988) Send offprint requests to: Florentin.Millour@oca.eu ⋆ Based on observations made with the VLTI at Paranal Observatory under program 091.D-0030 and 093.D-0056.

O. Chesneau passed away shortly after submitting this letter. We express our profound sadness on this premature demise and convey our deepest condolences to his family. V854 Cen is an unusual member of the RCB class owing to its relatively large hydrogen content and also to the PAHs detected in the mid-IR (among other oddities). Extensive polarimetric observations have been reported (Kameswara Rao & Raveendran 1993;Whitney et al. 1992). Another striking aspect of this star is the fast wind that was spectroscopically detected, which reaches several hundreds of km s -1 (Clayton et al. 2013(Clayton et al. , 2003;;Lawson et al. 1999;Clayton et al. 1993;Kameswara Rao & Lambert 1993;Lawson 1992). The key aspect of the polarimetric observations reported in Whitney et al. (1992) is that the emission lines were unpolarized, which implies that they were formed outside regions of high dust concentration. Kameswara Rao & Lambert (1993) also provided strong arguments that V854 Cen may be at the origin of the bipolar nebula based on high-resolution spectroscopy. HST long-slit, far-ultraviolet spectra of V854 Cen showed that the C II emission region around V854 Cen is significantly extended by about 2.5" (Clayton & Ayres 2001).

With the advent of optical interferometry, many dusty environments around evolved stars were resolved (Chesneau 2011;[START_REF] Chesneau | Asymmetric Planetary Nebulae 5 Conference[END_REF]. Some first attempts to monitor the dust production of the RCB RY Sgr at high spatial resolution were performed with the MIDI/VLTI instrument (Leão et al. 2007). A few observations of V854 Cen are reported in Bright et al. (2011). But the limited (u, v) coverage still hampered detecting clear departure from spherical symmetry.

This letter presents optical interferometry measurements obtained with the Very Large Telescope Interferometer (VLTI) and reports the discovery of a moderately elongated structure around V854 Cen. The observations are presented in Sect. 2. In Sect.

3 we analyze the AMBER H, K continuum measurements by means of simple geometrical spectrally dependent models together with VISIR mid-IR images. The results are then discussed in Sect. 4. The interferometric observations were obtained with the low spectral resolution mode of AMBER (R=35), a three-telescope combiner located at the VLTI (Petrov et al. 2007). The observations were performed with the 1.8m auxiliary telescopes (ATs) under photometric conditions, using the compact configuration (B ≤ 35m) in 2013 and the long configuration (B ≤ 140m) in 2014. The observations log is presented in Table 1 and the (u, v) plan coverage of the compact configuration is plotted in Fig. 1. We reduced the data using the standard Amber data reduction software amdlib v3.0.3b1 (Chelli et al. 2009;Tatulli et al. 2007). The visibilities secured in 2013 and 2014 are represented in Fig. 2. Noteworthy, the American Association of Variable Star Observers (AAVSO) database shows that V854 Cen was at maximum light, V∼7.5-7.2 when the 2013 and 2014 AMBER observations were obtained.

Observations

The AMBER visibilities, a proxy of size and shape of the system, show strong variations as a function of spatial frequencies and as a function of wavelengths. This means that the object is resolved by AMBER and that its shape spectrally varies between the H and K bands. The closure phases, a proxy of asymmetries in the image of the system, are equal to zero within the uncertainties. This means that this probably is a centrosymmetric object.

V854 Cen was observed on 30 of June 2008 with the mid-infrared imager VISIR/VLT (Lagage et al. 2004) as part of an imaging survey of post-AGB stars and related objects (Lagadec et al. 2011). The images were obtained with a pixel scale of 0.075"and a field of view of 19.2 × 19.2 arcsec through the SiC filter (λ = 11.65µm, ∆λ 2.34µm). Narrow-band filter images were also obtained, but had a too low S/N for scientific use. The data were processed as described by Lagadec et al. (2011), with an additional step to remove horizontal stripes produced by the mid-infrared detector, wor which we used filtering in the Fourier space. Fig. 3 displays an image of V854 Cen obtained through the SiC filter at 11.65µm (∆λ 2.34µm) for which the contrast of the faintest regions was enhanced. The bright pointsource is clearly surrounded by a fainter structure with a largest extent of ∼3".

Analysis

We analyzed the AMBER data with the fitOmatic routine (Millour et al. 2009a), which enabled us to introduce different spectra for different simple geometric components. Various combinations of geometrical models were tested, including uniform disks, Gaussian disks, rings, power-law and exponential-lawprofile disks. Our best-match model for the compact array is a two-component model, consisting of an unresolved uniform disk (Θ ≤ 2.5 mas, star component), and a flattened Gaussian (shell component) with a FWHM of the minor axis of 8 ± 1 mas, and a major axis of 11 ± 3 mas. The orientation of the major axis is 126±29 • . But it must be kept in mind that the (u, v) coverage is not ideal for an accurate determination of the P.A. angle and flattening of the structure. The quality of the fit is relatively good with a reduced χ 2 of 1.5. Combining the 2013 compact-array data with the 2014 long-array data brings consistent new information. The star remains unresolved (Θ ≤ 1.0 mas), and a slight increase of the apparent diameter with a minor axis of 12±2 mas, and a major axis of 15 ± 5 mas is measured (reduced χ 2 of 4.9). The orientation of the major axis is 133±49 • . The aspect ratio of the shell is moderate, and we note that a round structure of Θ = 12.3 ± 2 could also fit the data, albeit with a larger reduced χ 2 of 6.0. Assuming a disk-like geometry, the observed flattening would correspond to an inclination of i ≤ 60 • . The spectra of both components are spatially and spectrally isolated. The star flux relative to the total flux steadily decreases from 85% at 1.53µm to 42% at 2.49µm, while the shell relative flux increases from 15% to 58%, respectively, with photometric errors at the level of 25%. Assuming a temperature of 6750 K for the central star (Bright et al. 2011), a hot temperature T ≤ 1400K is inferred for the shell, which explains its direct detection in the near-IR.

In Fig. 2, we show an illustration of the data fit with our best geometrical model. We tested for a companion star or a single clump of dust in the shell given the closure phase signal we observed on V854 Cen (null within the error bars). The test showed point-like sources whose flux were always lower than 3% of the total flux, which means that this is well below the detection limit for such asymmetries in the system with the current data down to a typical spatial resolution of 10 mas. We also tested a fully clumped uniform shell that contained up to 30 clumps, to see whether the high H-band visibilities could be explained this way, and this may indeed be an explanation of these high visibilities, although we cannot clearly infer the clump structure (because there are too many parameters to fit). The clumpy model is shown in Fig. 4. Importantly, the K-band visibilities (Fig. 2) at the highest spatial frequencies show a striking increase, after a deep decrease of visibilities near the spatial frequency of 100 arcsec -1 . We interpret this feature as a first visibility lobe followed by a second visibility lobe that is clearly seen in the data. This is a typical feature of objects whose brightness distribution contains a sharp edge (see, for instance, an example in Millour et al. 2009b). Indeed, such a sharp edge would produce wiggles in the Fourier transformation of the image of the object (and hence in the visibilities). This supports the idea that the shell around V854 Cen is slightly better described by a (truncated) uniform disk (reduced χ 2 of 4.9) than by a (smooth) Gaussian disk (reduced χ 2 of 7.0), hence the sharp edges in the geometrical models shown in Fig. 2). The lower contours of the VISIR image were analyzed to study their geometry by fitting 2D Gaussian at different flux levels. This yielded a good estimate of the nebula's ellipticity and orientation as a function of the distance to its center. An elon-gated structure is discovered with a P.A. of 92±19 • and an ellipticity of ∼1.2. The resolved parts of the image represent 5.4% of the total flux in the SiC filter. This P.A. is only marginally constistent with the P.A. derived with AMBER. It is possible that the spatial structure of the mass loss of V853 Cen is inhomogenous and randomly variable, which would explain this difference. We attempted to check the consistency of the P.A. determined by AMBER and VISIR and those detected by polarimetry. This information is not systematically published, therefore we refer to a few other publications. According to Kameswara Rao & Raveendran (1993), a position angle of ∼65 • is observed in the V band for maxima episodes during which the direct starlight is less affected by dust clumps and the polarimetric signal is lowest (such as at JD=2448299.0 with V=7.2). The U band is also highly polarized (2.82 ± 0.1%) with a measured P.A. angle of 44 ± 1 • . Kameswara Rao & Lambert (1993) also proposed a polarized P.A. angle of 65±15 • that corresponds to the scattering from a disk. Unfortunately, V854 Cen is very active and produces dust clumps at a high rate, which affects the photometric bands differently and hampers detecting a polarimetric signal from a stable circumstellar environment.

Discussion

We used a compact configuration (baselines limited to ∼50m) optimized for studying the circumstellar environment of V854 Cen. A moderately flattened dusty environment was discovered around V854 Cen. This shed new light on the evolutionary scenarios that have been suggested for the production of RCB stars: the double-degenerate merger of two white dwarfs, which may lead to an axis of symmetry that promotes equatorially enhanced mass-loss and disks, or the final helium-shell flash in a planetary nebula central star, which may retain a central symmetry and promote no disk (Clayton et al. 2011(Clayton et al. , 2007;;Webbink 1984).

Below we elaborate further on the importance of this discovery within the context of RCB star formation scenarios. What is the origin of this slightly elongated environment? The closure phases are close to zero, which excludes a close companion or single dust clump in the vicinity of the star during this period of maximum light that was brighter than 3% of the total flux, that is, a flux difference of at least 4 magnitudes. This may imply that the dust is distributed in a disk that possibly is the relic of a past event, or that the observed spatial distribution reflects a field of randomly launched clumps in free-fall around the central star.

The disk, or circumstellar structure, which would be constantly replenished by a discrete or continuous process, would explain the very hot temperature of the dust. An inspiring comparison may be made with the so-called Be stars (Rivinius et al. 2013), for which the fast rotation of the central star together with its pulsational properties triggers the formation of a dense circumstellar disk. But, this does not imply that the dust clumps are exclusively formed in the equatorial plane, because if they were, the light declines would be much less frequent this very active star. Furthermore, even though the original merged star might have been rotating quickly, when it became a giant and became larger by a factor of about 10 4 , the surface speed would have dropped by that factor. The RCB stars are known to be slow rotators, and the narrow emission lines of V854 Cen are unresolved (≤ 20km s -1 , Kameswara Rao & Lambert 1993). V854 Cen has a well-known single-pulsation period (43.2 days) whose phase is related to the formation of the dust (Crause et al. 2007). Convection may also contribute to the launching process of the dust clumps, like in cool supergiant stars.

The intermediate inclination of the system (we recall that i ≤ 60 • ) implies that the dust clumps that are regularly launched must originate from the high latitudes of the star to intercept the line of sight and dim its visual flux this deeply! This indicates a non radial formation and launching process of the carbonaceous dust clumps.

The polarization properties of V854 Cen are similar (albeit weaker) to those of the intermediate-luminosity red transient (ILOT) V4332 Sgr (Kamiński & Tylenda 2013;Berger et al. 2009) that suffered an outburst in 1994 and later developed the characteristics of a large giant. ILOTs are outbursts with energies intermediate between those of novae and supernovae (Kasliwal 2012) examples of which are V838 Mon (Bond et al. 2003) or V1309 Sco (Tylenda et al. 2011). Kamiński & Tylenda (2013) show that the polarized continuum of V4332 Sgr in 2011 disagrees with the unpolarised emission lines. This is interpreted as evidence for a disk seen at high inclination that completely obscures the central source. The star light would then reach us after scattering on dust located above and below the disk. The emission lines, on the other hand, would derive from photoexcitation by stellar radiation of the molecules located in the material above and below the plane. This configuration is also observed for the Sakurai object (Hinkle & Joyce 2014;[START_REF] Chesneau | Asymmetric Planetary Nebulae 5 Conference[END_REF], another object thought to have suffered a very late final helium flash. Noteworthy, strong abundance similarities between Sakurai object and V854 Cen sources were detected (Asplund et al. 1998).

If the merger scenario for the RCBs applies, then they too may be in the ILOT range if observed at the time of the merger. Two 0.5M ⊙ WDs will deliver quite a substantial amount of gravitational energy ( ∼ < 5 49 erg), more than would be the case for a main-sequence star merger, therefore they may cluster in the upper region of the ILOT locus on the energy-time diagram. The stellar expansion that ensues would make the object a giant, as is the case for ILOTs. WD-WD merger simulations of RCB stars are still relatively crude (Staff et al. 2012) and do not simulate the possible formation of a disk. However, the post-merger is a fast rotator before the radial expansion. Magnetic fields, which were not included in the simulations probably play an important role, because they potentially affect many aspects of the star activity, such as its pulsations and whether an axial-symmetry is established. An interesting possibility is that the dust clumps launched by the star from its high latitudes are not able to ballistically leave the system and accumulate in the equatorial plane in a process reminiscent of the so-called wind-compressed disk proposed by Bjorkman & Cassinelli (1993). The launching conditions of dust around cool stars have recently been investigated in depth by Höfner (2009[START_REF] Höfner | Why Galaxies Care About AGB Stars: Their Importance as Actors and Probes[END_REF]. More observations are needed to link the extended nebula detected by VISIR and the compact structure resolved with the VLTI. High angular resolution (∼20-50mas) polarimetric and coranagraphic imaging in the optical such as provided by the SPHERE/VLT instrument will be a key to achieve this goal and better understand the dust ejection mechanism in V854 Cen.

Fig. 1 .

 1 Fig. 1. Spectrally dispersed (u, v) coverage of the compact 2013 and extended 2014 configuration of the AMBER observations.

  Fig. 2. Top: Best geometrical models of the environment of V854 Cen at two selected wavelengths. Middle: 2013/2014 combined H (blue) and K (red) bands interferometric visibilities obtained with AMBER/VLTI compared with the signal from the polychromatic geometrical modeling. The wavelength label is in µm. Bottom: Closure phases. Colors are the same.

Fig. 3 .

 3 Fig. 3. 11.65µm VISIR image with enhanced contours of the external regions.

Fig. 4 .

 4 Fig. 4. Top: One of the best models for a clumpy environment of V854 Cen at two selected wavelengths with 30 clumps. Middle: 2013/2014 combined H (blue) and K (red) band interferometric visibilities obtained with AMBER/VLTI compared with the signal from the polychromatic geometrical modeling. The wavelength label is in µm. Bottom: Closure phases. Colors are the same.

Table 1 .

 1 Log of V854 Cen AMBER observations.

	Date	Stations	Wavelength	Nb. Obs.
	14/04/2013 A1-B2-D0 1.54 -2.40 µm	5
	15/04/2013 A1-C1-D0 1.54 -2.40 µm	3
	06/05/2014 A1-G1-K0 1.54 -2.40 µm	2
	06/05/2014	A1-G1-J3	1.54 -2.40 µm	1
	Notes. Calibrator angular diameters from SearchCal@JMMC (Bonneau et al.
	2006): HD 124433 0.916 ± 0.065 mas, HD 127214 0.57 ± 0.04 mas.

Acknowledgements. A great thanks for the ESO staff for the help with these observations, and in particular Thomas Rivinius. We also thank the referee, Pierre Kervella, for his help in improving this paper. This work has made use of JMMC, CDS and AAVSO resources.


Michel Coste
email: michel.coste@univ-rennes1.fr

Mady Kartoue
email: kartoue@hotmail.com

Demdah

Extra modes of operation and self motions in manipulators designed for Schoenflies motion

Keywords: Parallel manipulators, Schoenflies motion, mode of operation, self-motion

We study 4-UPU parallel manipulators performing Schoenflies motion and show that they can have extra modes of operation with 3 degrees of freedom, depending on the geometric parameters of the manipulators. We show that the transition between the different modes occurs along self-motion of the manipulator in the Schoenflies mode.

Introduction

Several architectures have been proposed and studied for parallel manipulators with 4 d.o.f. (degrees of freedom) in order that they perform Schoenflies motion: 3 d.o.f. in translation and 1 d.o.f. in rotation with fixed direction of axis (in general taken to be vertical); see for instance [START_REF] Rolland | The Manta and the Kanuk: Noved 4 dof parallel mechanism for industrial handling[END_REF][START_REF] Pierrot | H4: a new family of 4-dof parallel robots[END_REF][START_REF] Zhiao | Geometric Analysis of Overconstrained Parallel Manipulators with Three and Four Degrees of Freedom[END_REF][START_REF] Company | A new high speed 4-dof parallel robot. Synthesis and modeling issues[END_REF][START_REF] Kong | Type synthesis of 3T1R 4-dof parallel manipulators based on screw theory[END_REF][START_REF] Li | Mobility analysis of a novel 3-5R parallel mechanism family[END_REF][START_REF] Zhao | Novel Four-DOF Parallel Manipulator Mechanism and Its Kinematics[END_REF][START_REF] Gogu | Structural synthesis of fully isotropic parallel robots with Schoenflies motions via theory of linear transformations and evolutionary morphology[END_REF][START_REF] Amine | Singularity Analysis of the H4 Robot using Grassmann-Cayley Algebra[END_REF][START_REF] Amine | Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures[END_REF][START_REF] Solazzi | Kinematics analysis and singularity loci of a 4-UPU parallel manipulator[END_REF]. Among these manipulators are 4-UPU which have been proposed in [START_REF] Zhiao | Geometric Analysis of Overconstrained Parallel Manipulators with Three and Four Degrees of Freedom[END_REF][START_REF] Zhao | Novel Four-DOF Parallel Manipulator Mechanism and Its Kinematics[END_REF][START_REF] Solazzi | Kinematics analysis and singularity loci of a 4-UPU parallel manipulator[END_REF]. We show in this paper that these 4-UPU may also have other modes of operation, according to the geometric parameters of the architecture. The fact that parallel manipulators can have several modes of operation is well known and has been studied in the case of 3-RPS in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF]. In the case of the 4-UPU, the extra modes of operations, when they exist, have 3 d.o.f., that is one less than the Schoenflies mode. We also show that the transition poses between the Schoenflies mode and the extra mode are poses from which the manipulator has a circular self-motion in the Schoenflies mode.

The paper is organized as follows. We describe the manipulator under study in Section 2. We study its different modes of operation in Section 3. Then we establish in Section 4 the relationship between self-motion in the Schoenflies mode and transition poses to the extra modes of operation, when they exist. Finally we study briefly in Section 5 the influence of modification of the architecture on the existence of the extra modes of operation.

The 4-UPU manipulator

The manipulator has four identical legs. The basis is a horizontal rectangle with vertices B 1 = (a, b, 0) T , B 2 = (-a, b, 0) T , B 3 = (-a, -b, 0) T , B 4 = (a, -b, 0) T . The platform is also a rectangle with vertices of coordinates P 1 = (c, d, 0) T , P 2 = (-c, d, 0) T , P 3 = (-c, -d, 0) T , P 4 = (c, -d, 0) T in an orthonormal frame attached to the platform. We assume a, b, c, d > 0. The coordinates of the vertices of the platform in the fixed orthonormal frame are Q i = R P i + T for i = 1, . . . , 4, where R is the rotation matrix giving the orientation of the platform and T the translation vector giving the position of the centre of the platform. The vertical unit vector V = (0, 0, 1) T is orthogonal to the base and N = R V is a unit vector orthogonal to the platform (see Fig. 1, left). Each leg joining B i to Q i has a universal joint in B i (decomposed in two rotoidal joints rj i,1 and rj i,2 ), a prismatic joint on the segment [B i Q i ] and another universal joint in Q i (decomposed in two rotoidal joints rj i,3 and rj i,4 ). The axis of rj i,1 is vertical (parallel to V ), the axis of rj i,4 is orthogonal to the platform (parallel to N ) and the axes of rj i,2 and rj i,3 are parallel, both orthogonal to V , to N and to the vector Q i -B i (see Fig. 1, right).

The disposition of the axes yields the constraints that V, N, Q i -B i have to be coplanar for i = 1, . . . , 4. These constraints are satisfied if N = V , i.e. if the manipulator stays in the Schoenflies mode of operation with 3 d.o.f. in translation and 1 d.o.f. in rotation, the rotations having vertical axis. The constraints are also satisfied if N = -V , i.e. if the platform is horizontal and upside down. This mode of operation has also 4 d.o.f. (3 in translation and 1 in rotation) and one can move from one pose in this mode to any other by a Schoenflies motion with vertical axis of rotation. So we call this mode the reverse-Schoenflies mode of operation.

We shall see in the next section that the constraints may also be satisfied in other modes of operation of the manipulator, depending on the architectural parameters a, b, c, d.

3 Different modes of operation

The constraint equations

In order to write down the constraint equations, we choose a rational parametrization of the group of spatial rotations, namely the Cayley parametrization:

R = 1 ∆   1 + p 2 -q 2 -r 2 2(pq -r) 2(rp + q) 2(pq + r) 1 -p 2 + q 2 -r 2 2(qr -p) 2(rp -q) 2(qr + p) 1 -p 2 -q 2 + r 2   where ∆ = 1 + p 2 + q 2 + r 2 .
(

) 1
The vector (p, q, r) T gives the oriented axis of the rotation and p 2 + q 2 + r 2 is the tangent of the half-angle of rotation; the matrix R is the rotation matrix associated to the quaternion 1 + ip + jq + kr. So this parametrization misses the half-turns. In particular, we shall not obtain the poses in the reverse-Schoenflies mode of operation. This drawback is the price we pay for simple computation and output. We shall see in subsection 3.4 how to circumvent this drawback.

We set T = (x, y, z) T for the translation vector; so we have six pose variables x, y, z, p, q, r. The constraint equations are det(V, N, Q i -B i ) = 0 for i = 1, . . . , 4. Chasing the denominator 1 + p 2 + q 2 + r 2 , we obtain

(qr -p)x -(rp + q)y + (-(b + d)r -a + c)p + ((a + c)r -b + d)q = 0 , (qr -p)x -(rp + q)y + ((b + d)r -a + c)p + ((a + c)r + b -d)q = 0 , (qr -p)x -(rp + q)y + (-(b + d)r + a -c)p + (-(a + c)r -b + d)q = 0 , (qr -p)x -(rp + q)y + ((b + d)r + a -c)p + (-(a + c)r + b -d)q = 0 . (2)

The components of the set of admissible poses

The four polynomials listed in the system (2) generate an ideal Const in the ring of polynomials in x, y, z, p, q, r with coefficients depending on the parameters a, b, c, d. Actually z does not appear in the constraint equations. The different modes of operation are studied by decomposing the ideal Const into primary components, as done for instance in [START_REF] Schadlbauer | The 3-RPS parallel manipulator from an algebraic viewpoint[END_REF]. This decomposition over the field R(a, b, c, d) (performed with Maple, or with Singular) gives two components Const = Schoenflies ∩ Extra, where

Schoenflies = p, q , Extra = (a + c)(b + d)r 2 + (a -c)(b -d), (b + d)rp + (b -d)q, c(b + d)rx + d(a -c)y . (3)
This decomposition means, from the geometric viewpoint, that the set of poses satisfying the constraint equations (and whose rotation part is not a half-turn) splits in two parts:

• The poses for which p = q = 0, i.e. the poses for which the rotation axis is vertical. These are the poses corresponding to the Schoenflies mode of operation.

• The poses satisfying the three equations generating the ideal Extra.

We now examine more closely the Extra component. The first equation in r has no real solution if (c -a)(b -d) < 0; in this case, there is no pose satisfying this equation and the only mode of operation is the Schoenflies mode (besides the reverse-Schoenflies mode, see below in subsection 3.4). If (c -a)(b -d) > 0, there are two real solutions

r sol± = ± (c -a)(b -d) (a + c)(b + d) . (4)
For each of these solutions r sol+ and r sol-, we have a mode of operation (indexed by + or -) consisting of the poses such that

r = r sol± , (d -b) q = (b + d)r sol± p , d(c -a) y = c(b + d)r sol± x . (5)
In each of these modes of operation the translation vector (x, y, z) T ranges in a vertical plane and the rotation vector (p, q, r) T ranges in a line. There are two d.o.f. in translation and one in rotation. Note that the symmetry w.r.t. the plane x = 0, or the plane y = 0, carries the mode of operation + to -and vice-versa.

It remains the case (c -a)(b -d) = 0. In this case there is only one extra mode of operation. In order to determine this extra mode, one has to recompute the primary decomposition in the special case since the computation giving ( 3) is made for generic parameters a, b, c, d.

• If a = c and b = d, this mode of operation consists of the poses such that q = r = x = 0. There are three d.o.f.: the centre of the platform can move in the (y, z) plane, while the platform can rotate around an axis parallel to the x-axis.

• If a = c and b = d, this mode of operation consists of the poses such that p = r = y = 0. There are three d.o.f.: the centre of the platform can move in the (x, z) plane while the platform can rotate around an axis parallel to the y-axis.

• Finally when a = c and b = d (congruent base and platform), the extra mode of operation consists of the poses such that r = 0 and px + qy = 0; there are four d.o.f. for this mode.

Except for the case of congruent base and platform, the extra modes of operation can be described in a unified way. Assume for instance a ≤ c and b ≥ d and set

δ 1 = c 2 -a 2 and δ 2 = b 2 -d 2 . ( 6
)
Then the extra modes of operation are described by:

r = ± δ 1 δ 2 (a + c)(b + d) , δ 2 (a + c) q = ∓δ 1 (b + d) p , δ 1 d y = ±δ 2 c x . (7)
In each of these extra modes, the axes of the rotoidal joints rj i,2 and rj i,3 remain parallel to the fixed vector (δ 2 c, ∓δ 1 d, 0) T .

An example

We give an example of the extra mode of operation for a manipulator with parameters a = 1, b = 3, c = d = 2. Then δ 1 = √ 3 and δ 2 = √ 5. We take r = 1/ √ 15. We fix x = y = 0, z = 5. We let p vary from 0 to +∞, keeping q = -5/3 p. The motion is represented in Fig. 2. It starts with a pose in the Schoenflies mode and finishes with a pose in the reverse-Schoenflies mode. The second row of Fig. 2 shows the motion from a viewpoint orthogonal to the vector ( √ 5, -√ 3, 0) T . It shows that the legs remain in vertical planes orthogonal to this vector, because the axes of rj i,2 and rj i,3 remain parallel to this vector during the motion.

Recovering the half-turns

The parametrization we have chosen for the rotation group misses the halfturns. Hence, the analysis of the ideal generated by the constraint equations misses the reverse-Schoenflies mode of operation. It could be the case that it also misses other modes of operation. We check now that this is not the case.

We can change the initial pose for a pose where the platform is already upside down, for instance we can perform an initial half-turn of the platform around the x-axis. This has the effect of exchanging d and -d. Once this change performed, we can proceed with exactly the same analysis as before, denoting by p ′ , q ′ , r ′ the new rotation variables. The relations p ′ = -1/p, q ′ = -r/p and r ′ = q/p between the two sets of rotation variables are obtained by writing that the quaternion (1 + ip ′ + jq ′ + kr ′ ) i is a real multiple of 1 + ip + jq + kr.

The primary decomposition of the ideal generated by the constraint equations in the variables x, y, z, p ′ , q ′ , r ′ is obtained from (3) by exchanging d with -d. It gives the two components

RevSchoenflies = p ′ , q ′ , Extra ′ = (a + c)(b -d)r ′2 + (a -c)(b + d), (b -d)r ′ p ′ + (b + d)q ′ , c(b -d)r ′ x -d(a -c)y . (8)
The first equation in r ′ of the Extra ′ component has real solutions if and only if (c -a)(b -d) ≥ 0, and the solutions in r ′ then give the extra modes of operation already identified.

We can also repeat the same analysis starting now with an initial half-turn around the y-axis, which amounts to exchanging c and -c. We have then recovered all half-turns, except the one around the z-axis which belongs to the Schoenflies mode. The analysis of the modes of operation is indeed complete.

We could also have used the parametrization of the rotation group using unit quaternions. The computation of the primary decomposition of the ideal generated by the constraint equations is then more difficult, but can still be performed with Singular; it gives three components corresponding to the Schoenflies mode, the reverse-Schoenflies mode, and the possible extra modes. Also, the output of this computation is less easily readable for the extra component. However, it confirms the results explained in this section.

Self-motions and transition between the modes 4.1 Poses of transition from the Schoenflies mode to an extra mode

In this section we shall always consider the manipulator in Schoenflies mode of operation. The rotation matrix is here restricted to have a vertical axis, so we can write it as

R =   cos(θ) -sin(θ) 0 sin(θ) cos(θ) 0 0 0 1   (9)
The pose variables for the Schoenflies mode are θ, x, y, z. Comparing with the (p, q, r) orientation variables of the preceding section, we have here p = q = 0 and r = tan(θ/2).

We also restrict our attention to the case (c -a)(b -d) > 0 where there are two extra modes of operation with 3 d.o.f. besides the Schoenflies and reverse-Schoenflies modes. We assume a < c and b > d, so that we can use the quantities

δ 1 = √ c 2 -a 2 and δ 2 = √ b 2 -c 2 .
According to the analysis of the preceding section, each extra mode (denoted with + or -) intersects the Schoenflies mode in the set of poses which satisfy

θ = ± arccos ab + cd ad + bc , δ 1 d y = ±δ 2 c x . ( 10
)
These transition poses form two vertical planes TR ± (of dimension 2). We shall now compare the transition poses with actuation singularities in the Schoenflies mode.

Actuation singularities in the Schoenflies mode

The actuated joints are the four prismatic joints on the legs. We use as coordinates in the joint space

ρ i = Q i -B i 2
, the squares of the lengths of the legs, for i = 1, . . . , 4. The inverse kinematic mapping for the Schoenflies mode is (θ, x, y, z) → (ρ 1 , ρ 2 , ρ 3 , ρ 4 ). The Jacobian determinant of the inverse kinematic mapping is, up to a constant factor (cf [13, (4.97)] or [START_REF] Solazzi | Kinematics analysis and singularity loci of a 4-UPU parallel manipulator[END_REF]):

J = (ad -bc) cos(θ) ((ad + bc) cos(θ) -(ab + cd)) z . ( 11
)
The structure of the inverse kinematic mapping and its singularities are more easily understood after a linear change of coordinates in the joint space. Set

σ 1 = (ρ 1 -ρ 2 + ρ 3 -ρ 4 )/8, σ 2 = (ρ 1 -ρ 3 )/4, σ 3 = (ρ 2 -ρ 4 )/4, σ 4 = ρ 4 . ( 12
)
Then the inverse kinematic mapping is rewritten as

σ 1 = sin(θ) (ad -bc) , σ 2 = (cos(θ) c -sin(θ) d -a) x + (sin(θ) c + cos(θ) d -b) y , σ 3 = (-cos(θ) c -sin(θ) d + a) x + (-sin(θ) c + cos(θ) d -b) y , σ 4 = (x + cos(θ) c + sin(θ) d -a) 2 + (y + sin(θ) c -cos(θ) d + b) 2 + z 2 , (13)
which shows that σ 1 depends only on θ and σ 2 and σ 3 depend linearly on x, y with coefficients depending on θ. The factor (ad-bc) cos(θ) of J is the derivative of σ 1 w.r.t. θ, the factor (ad + bc) cos(θ) -(ab + cd) is the determinant of the linear mapping (x, y) → (σ 2 , σ 3 ) and the factor z is half of the derivative of σ 4 w.r.t. z. We shall assume in what follows that ad -bc = 0, i.e. that the base and the platform are not similar (in this case the manipulator is completely singular and there is self-motion from every pose).

We concentrate our attention on the actuation singularities satisfying the equation (ad + bc) cos(θ) = ab + cd. This equation has two solutions precisely when (c -a)(b -d) > 0, and the solutions in θ are

θ sol± = ± arccos ab + cd ad + bc . (14)
The poses satisfying θ = θ sol± form two components AS ± (which are spaces of dimension 3) of the actuation singularities of the Schoenflies mode. We are interested in these components of the actuation singularities because, according to [START_REF] Amine | Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures[END_REF], each component AS ± contain a plane TR ± of intersection with an extra mode.

Self-motion in the Schoenflies mode

The image of a component AS ± of the actuation singularities in the joint space is contained in a plane P ± . Indeed we have

sin(θ sol± ) = ± δ 1 δ 2 ad + bc (15)
so that setting θ = θ sol± in (13) yields

σ 1 = ±(ad -bc) δ 1 δ 2 ad + bc , σ 2 = δ 1 ∓ δ 2 ad + bc (d δ 1 x ± c δ 2 y) , σ 3 = δ 1 ± δ 2 ad + bc (-d δ 1 x ∓ c δ 2 y) , σ 4 = x - d δ 1 (-δ 1 ∓ δ 2 ) ad + bc 2 + y - c δ 2 (∓δ 1 -δ 2 ) ad + bc 2 + z 2 . ( 16
)
This shows that the image of AS ± is contained in the plane P ± defined by the equations

σ 1 = ±(ad -bc) δ 1 δ 2 ad + bc and (δ 1 ± δ 2 ) σ 2 + (δ 1 ∓ δ 2 ) σ 3 = 0 (17)
Since AS ± is of dimension 3 and its image in the joint space is of dimension 2, there is self-motion from almost every pose in AS ± . The equations ( 16) of the inverse kinematic mapping restricted to AS ± give the information about this self-motion: it occurs along the vertical circle Γ ± which is the intersection of the vertical plane Π ± given by the second or third equation of ( 16) and the sphere Σ ± given by the fourth equation. This intersection is reduced to a point for (σ 1 , σ 2 , σ 3 , σ 4 ) belonging to a parabola in the plane P ± , and it is empty outside of the parabola; the equation of the parabola is obtained by writing that the square of the distance of the center of Σ ± to Π ± (a quadratic expression in σ 2 or σ 3 ) is equal to the square of the radius of Σ ± , which is σ 4 .

Observe that the vertical plane TR ± (equation ±δ 2 c x -δ 1 d y = 0) is perpendicular to the vertical plane Π ± (second or third equation of ( 16)). Observe also that the center of the sphere Σ ± of coordinates

d δ 1 (-δ 1 ∓ δ 2 ) ad + bc , c δ 2 (∓δ 1 -δ 2 ) ad + bc , 0
belongs to TR ± . It follows that each circle Γ ± of self-motion (intersection of Σ ± with Π ± ) contains two poses which allow transition to the extra ± mode; these poses are those with maximal and minimal values of z on the circle. The transition between modes of operation in self-motions was already observed for 3-RPS in [START_REF] Husty | Self-motions of 3-RPS manipulators[END_REF]. See Fig. 3 which shows the geometry of the objects mentioned above in an example.

Example continued

We give an example of self-motion in the Schoenflies mode for the same manipulator as the one in Fig. 2. The self-motion from the pose x = y = 0, z = 5, θ = arccos(7/8), which is a transition pose to an extra mode, is shown in Fig. 4. The picture on the right shows the self-motion from a viewpoint in the direction of the vector ( √ 3, √ 5, 0) T (as in the second row of Fig. 2); the legs are always in parallel planes and a transition pose is attained when these planes are vertical. Note that the direction of the vertical planes of self-motion is perpendicular to the direction of the vertical planes of motion in the extra mode.

Rhombus base and platform

The case when the base and the platform are rhombi instead of rectangles (as in [START_REF] Solazzi | Kinematics analysis and singularity loci of a 4-UPU parallel manipulator[END_REF]) can be dealt with in exactly the same way. Taking B 1 = (a, 0, 0) T , B 2 = (0, b, 0) T , B 3 = (-a, 0, 0) T , B 4 = (0, -b, 0) T , P 1 = (c, 0, 0) T , P 2 = (0, d, 0) T , P 3 = (-c, 0, 0) T , P 4 = (0, -d, 0) T leads to the same condition (c -a)(b -d) ≥ 0 for the existence of extra operation modes. We shall not comment further this case

Moving one anchor point

The reason for the existence of extra modes of operations lies in the fact that one can rotate the platform rectangle in such a way that the corresponding vertices of the base and the platform lie on parallel lines (see Fig. 5). If the condition (c -a)(b -d) > 0, there are two opposite angles of rotation θ given by formula [START_REF] Amine | Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures[END_REF] leading to this situation. Figure 5 5), then the extra mode of operation labelled + persists, but the mode labelled -disappears; the same happens for D -, exchanging + and -. If B 1 is moved out of the union of the two lines D + and D -, both extra modes of operation disappear. This shows that the existence of these extra modes of operation is a non generic situation which does not occur if the base and platform are general quadrilaterals.

Conclusion

We have studied a 4-UPU manipulator which is designed to perform Schoenflies motion. We have shown that, besides the Schoenflies and reverse Schoenflies (upside down) modes of operation with 4 d.o.f., this manipulator can offer two extra modes of operation with 3 d.o.f.. We have characterized the manipulators which have this property in terms of the dimensions a, b of the base rectangle and c, d of the platform rectangle: a necessary and sufficient condition is ((ca)(b -d) > 0.

We have studied the poses in the Schoenflies mode for which there is a transition to an extra mode, when this condition is satisfied. We have shown that these transition poses belong to components of the actuation singularities. The manipulator has self-motion along vertical circles on these components, and on each circle of self-motion there are two transition poses. We have also shown that the existence of these extra modes of operation is a non generic situation, which disappears for small general modifications of the architecture.

Figure 1 :

 1 Figure 1: Left: the 4-UPU. Right: geometry of a leg; dashed lines are the axes of rotoidal joints

Figure 2 :

 2 Figure 2: Motion from the Schoenflies mode to the reverse-Schoenflies mode from two different viewpoints

Figure 3 :Figure 4 :

 34 Figure 3: Left: picture of AS + , 3-d component of actuation singularities in the workspace. Right: Picture of P + , plane in in the joint space containing the image of AS + under IKM

  represents the two rotations in the case of the example a = 1, b = 3, c = d = 2; The rotation leading to the operation mode labelled + (resp. -) is represented in blue (resp. brown).

Figure 5 :

 5 Figure 5: Rotations of the platform explaining the extra modes of operation