Miao Zhang

Frédéric Averseng

Francia Haque

Patrizia Borghetti

Jean-Marc Krafft

Benoît Baptiste

Guylène Costentin
email: guylene.costentin@upmc.fr

Slavica Stankic
email: slavica.stankic@insp.jussieu.fr

Defect-related multicolour emissions in ZnO tetrapods: from violet, over green to yellow

Keywords: ZnO, nanoparticles, defects, PL emission, EPR

INTRODUCTION

The properties and, thus, technological potential of any semiconductor (chemical, optical, magnetic, etc.) are highly governed by the atomic-scale defects it contains. This applies to both the nature and quantity of defects that are present. A semiconducting metal oxide, ZnO, finds applications as a transparent conducting oxide for solar cells, 1,2 piezoelectric for various electronic devices, 3 transparent thin film for gas sensors, 4,5 nanowire-based gas sensor for NOx or H2, 6 UV optical emitter in LEDs, [7][8][9] and dilute magnetic semiconductor for spintronics. 10,11 The presence of lattice defects is known to impact the performance of ZnO in most of these application fields. In particular, defects are highly undesiderable in ZnO used as UV optical emmiter in LEDs since their presence lead to lower luminescent efficiency when compared to a defect-free material. Hence, a fundamental knowledge on ZnO defects is required in order to tune the ZnO performances for a variety of novel high-tech applications.

Point defects are present in all crystalline materials in forms of missing (vacancies), interstitial (interstitials) or substituted (anti-sites) ions and are, generally, electronically charged. They induce local changes and lattice relaxations and new electronic states are generated within the band gap. The identification of crystal defects is, however, not a simple task. This can be illustrated by highly controversial assignments of photoluminescence (PL) and electron paramagnetic resonance (EPR) data reported for ZnO, with several defects proposed for a given signal or several signals ascribed to a given defect. This concerns in particular, the highly discussed green luminescence (centered 2.17-2.5 eV) that was controversially assigned to a various ZnO defects (Vo, OZn, Oi, VZn). [12][13][14][15][16][17][18][19] The absence of consensus results from both, an inadequate comparison of samples of different origins (for instance single crystals vs.

nanostructured ZnO) and/or the variability of experimental conditions applied during the synthesis, processing and measurements. The nature and concentration of a given defect depend strongly on all stages of the sample's history (from synthesis, over processing to measurements) and the corresponding spectroscopic properties are thus expected to vary. This particularly concerns nanostructured ZnO (nanoparticles, thin films) where, for instance, the measurements in air can strongly differ from those in vacuum. 20,21 Another illustration is the transformation of an existing defect into a new one through the change of its oxidation state or the interaction with surrounding molecules, leading to the disappearance of its specific spectroscopic signal.

For instance, oxygen vacancies can be efficiently passivated by wateras clearly demonstrated in FTIR studies on ZnO nanoparticles. [22][23][24] Therefore, exposing ZnO nanoparticles to a watercontaining environment is likely to severely hinder the identification of native oxygen vacancies initially present in the sample. Also, while theoretical calculations may provide useful information for the identification of lattice defects in ZnO, one should keep in mind that they are commonly performed at T = 0 K, i.e. far from the conditions usually achieved in experiments. Additionally, nanostructured systems are usually obtained and studied far from thermodynamic equilibrium. Thus, the straightforward application of such theoretical results for discussing the existence of given defects is at least questionable. For example, because the formation energy of singly charged oxygen vacancies (Vo + ) was calculated to be rather high in ZnO, 25 the Vo + is therefore considered to be a defect rather difficult to create. Yet the present study will unambiguously prove otherwise.

In this contribution, by using highly controlled experimental conditions, we will demonstrate a close correlation of PL/EPR signatures to corresponding point defects observed on ZnO smoke nanopowder at different stages (as-synthesized, in the presence of O2 and H2O molecules, after thermal post-growth treatments). In that context, we applied the following strategy: i)

Use of clean synthesis route to exclude extrinsic defects thus limiting the investigation to the sole intrinsic ones.

ii) Application of high-vacuum conditions (P < 10 -5 mbar) during the storage, handling, and spectroscopic characterizations.

Such strategy allowed us to identify the nature of defects that are natively present in ZnO smoke and to follow their evolution as well as the appearance of new ones at different processing stages.

EXPERIMENTAL DETAILS

Synthesis and storage. ZnO smoke nanopowder was produced by metal combustion method in a glove box where the ignition of metallic Zn-foil of high purity (99.99%, 0.125 mm thickness, Advent Research Materials Ltd) was resistively induced. 26 As confirmed by XPS analysis, these synthesis conditions ensure that the formation of extrinsic defects was avoided. The white powder was collected on a glass plate and, then, transferred into a quartz-glass cell dedicated to PL or EPR measurements. The powder was constantly maintained under dynamic vacuum (P < 10 -5 mbar) during the storage and in situ spectroscopic measurements.

Electron Microscopy. For the purpose of microscopic studies, as-synthesized ZnO powders were directly deposited on TEM grids, preventing the use of any solvent. Transmission electron microscopy (TEM) measurements were performed on a JEOL 2100 field emission transmission electron microscope operating at 200 kV with a 0.18 nm resolution. Scanning electron microscopy (SEM) images were recorded on a field emission Zeiss Supra 40 scanning electron microscope.

X-Ray Diffraction. X-Ray powder diffraction measurements were carried out at the X-ray diffraction platform of IMPMC on a Panalytical XpertPro MPD diffractometer using Co Kα radiation source (λKα1 = 1.78897 Å, λKα2 = 1.79285 Å) and an X'Celerator detector. The samples were prepared within a glove box and mounted in a homemade chamber allowing XRD measurements in controlled, anoxic conditions.

Cell parameters, mean coherent domain sizes and microstrain were refined using the Rietveld method as implemented in the Fullprof software. Since the peak width was larger than the instrumental resolution, lorentzian isotropic size (Y) and lorentzian isotropic strain (X) parameters were refined. The two effects can be decoupled on the diagram measured over a wide angular range because the first induces a peak broadening as Y/cos(theta) whereas the second as X/tan(theta).

Electron Paramagnetic Resonance (EPR).

In order to strictly control the different thermal treatments and subsequent EPR measurements, the following procedure has been followed: (i) a homemade setup equipped with a tubular oven has been used for all thermal treatments. High-vacuum was reached (P < 10 -5 mbar) using a turbo-molecular pump. Connections also permitted the controlled addition and removal of gases, such as O2 and H2O vapor, during or after the thermal treatments. (ii) The ZnO sample was treated and measured in a suprasil quartz tube (5mm external diameter), combined with a highvacuum bellow-type metal-glass stopper and ball joint, which allowed for a continuous connection with the homemade setup during all treatments, measurements, avoiding any unwanted external contamination.

The EPR measurements were performed on a JEOL FA300 computerized spectrometer working at ~9.3 GHz (X-band). All spectra were recorded placing the powders under dynamic vacuum (P < 10 -5 mbar) at 77 K, using an insertion dewar containing liquid nitrogen, applying a microwave power of 2 mW and 100 kHz field modulation and presented, as is usual, as the first derivative of the absorbance. In order to avoid the overmodulation and subsequent alteration of narrow EPR signals (resulting in signal broadening, loss of close peaks or multiplets, etc.), a modulation width/amplitude of 0.1mT was systematically used. The g-factors were calibrated by using solid 2,2-diphenyl-1-picrylhydrazyl (DPPH, g = 2.0037) ) and care was taken to always keep the same height of powder in the EPR tube (2 cm).

Additionally, g-factor of solids can be anisotropic (value differs among the considered direction), and is thus described as a tensor (3x3 matrix with zero non-diagonal terms, and gx, gy and gz diagonal terms). Three cases may then occur: rhombic (gx ≠ gy ≠ gz), axial (gz ≠ gx = gy), or isotropic (gz = gx = gy = g) symmetry. [START_REF] Weil | Electron Paramagnetic Resonance-Elemental Theory and Practical Applications[END_REF] Computer simulations of the spectra were performed using the EPRsim32 program. [START_REF] Spalek | [END_REF] Diffuse reflectance UV-vis (DR UV-Vis). The spectra were acquired at room temperature using a Varian Cary 5000 spectrometer equipped with an integrating sphere, while a sample of barium sulfate (BaSO4) with ~100 % reflectance was used as a reference.

Photoluminescence (PL). Room-temperature experiments were performed on a Flurolog II fluorometer (Jobin-Yvon) equipped with a 450 W Xe lamp as an excitation source.

Photoluminescence emission spectra were obtained fixing the excitation energy at 4.13 eV (300 nm) and using a WG-320 cut-off filter on the emission side to prevent the contributions of the first-and second order coming from excitation light. The PL band fitting analyses were done by using IGOR.

Photoluminescence and UV-vis diffuse reflectance measurements were carried out in a quartz glass tube that was connected to the same homemade high vacuum line (P < 10 -5 mbar) as that used for EPR measurements.

To investigate the impact of annealing steps in oxygen rich atmosphere (Section 3.2.2), the sample was alternatively annealed in PO2 = 100 mbar, then outgassed at room temperature to P < 10 -5 mbar for 1 h before the PL / EPR spectra were recorded under dynamic vacuum as usually.

The same procedure was repeated in the second and third cycle.

Raman. The same quartz glass cell as described for PL and UV experiments was used for Raman investigations. Raman spectra were recorded at room temperature on a Jobin-Yvon triple-stage T64000 Raman spectrometer that is equipped with a CCD camera. An Argon laser monochromatic source (λ= 514.5 nm) was used.

X-ray photoelectron spectroscopy (XPS). XPS spectra were obtained with a nonmonochromatic Al K source (photon energy, hν = 1486.7 eV) and a hemispherical analyzer Phoibos 100 with a pass energy of 20 eV at normal emission. The stoichiometric ratio n1/n2 between two species distributed homogeneously in the sample is obtained through the following equation: n1 / n2 = I1(22T2) / I2(11T1), where I is the integration areas of the peak Zn 3s and that of the component of O 1s corresponding to lattice oxygen in ZnO, T is the transmission function of the analyzer tabulated for the analyzer Phoibos 100, Zn and O are provided by the NIST standard reference database for the ZnO material, [START_REF] Powell | NIST Electron Effective-Attenuation-Length Database -Version 1.3[END_REF][START_REF] Llovet | NIST Database of Cross Sections for Inner-Shell Ionization by Electron or Positron Impact[END_REF] while Zn and O are taken from references. [START_REF] Yeh | Atomic calculation of photoionization cross-sections and asymmetry parameters[END_REF] 3. RESULTS Hall method 33 to be 3 %, indicates that the distances between the relevant crystal planes are not identical, pointing out some stress probably induced by the presence of defects. Raman spectrum (Figure 2b) reveals two intense peaks centered at 100 cm -1 (E2 low ) and 439 cm -1 (E2 high ).

E2 low and E2 high Raman modes are particularly sensitive to the motion of zinc and oxygen in the lattice, respectively, 34,35 and represent a direct measure of ZnO crystal quality. Additionally, two less intense peaks are observed at  490 and at  600 cm -1 . The former peak is usually detected on nanostructured ZnO and attributed to the surface optical phonon (SOP). 36 The broad and weak signal at 600 cm -1 can be correlated with the E1(LO) peak previously detected on ZnO single crystals at  590 cm -1 and assigned to intrinsic lattice defects. 35 In ZnO thin films grown in Zn-rich conditions, such a Raman mode at  580-590 cm -1 was systematically attributed to an excess of zinc in the ZnO lattice, in line with the growth conditions. 37, 38 . The width of the excitonic band and, so the necessity of two Gaussian curves for its adequate reproduction come from the morphological heterogeneity of the sample (Figure 1a). The visible emission can also be successfully fitted by a pair of Gaussians. The first one is centered at 2.50 eV (495 nm), which corresponds to green luminescence (GL) commonly reported in ZnO and, so far, attributed to all defects likely to be present in ZnO. [41][42][43][44] We will not discuss this PL band in the present paper since it is a subject of another ongoing study. The second Gaussian band reveals a slightly higher contribution and peaks at 2.88 eV (430 nm), an energy that matches with violet luminescence (VL). VL is less frequently measured on ZnO and observed on samples produced in non-equilibrium conditions followed by an ultra-rapid reactive quenching. 45 Unlike GL, for which no uniform recombination model was calculated, transitions between 2.75 and 2.90 eV have been quite consistently calculated to arise from energy states related to interstitial zinc (Zni). [46][47][48] The same assignment can also be found in several experimental studies. 38,49,50 VL observed in PL spectrum of as-synthesized ZnO smoke fits well to this interval suggesting Zni to be the defect specific of the applied synthesis method. This is not surprising since metal combustion synthesis occurs far from the equilibrium and includes rapid quenching conditions that, as mentioned above, favor Zn-rich ZnO. Such an excess of Zn, in at least the first few nanometers ( 5 nm), of as-synthesized ZnO smoke surface was confirmed by XPS analysis (Zn/O = 1.26 ± 0.15).

EPR spectrum (Figure 3b) of the as-prepared sample consists of two signals: an intense and narrow signal at g = 1.956 (species A) and a weaker one at g = 1.960 (species B). EPR signals with g-value close to 1.960 have been often observed on ZnO yet without a consensus on their origin, so far. [51][52][53][54][55][56][57] Only a few studies report an EPR signal at g = 1.956 on ZnO, 58-60 yet

Morazzoni and coworkers assigned it to Zni + . 60 One should notice that this EPR signal prevails in the EPR spectrum of as-prepared ZnO smoke in the same way that VL was shown to dominate the visible domain of PL spectrum.

Considering the Zn-excess determined by XPS, the maximum of the visible PL and the synthesis conditions, we assign the two main PL/EPR features of as-grown ZnO smoke (Couple I: 2.88 eV / g = 1.956) to Zni + . Additionally, the proximity in g-values of the two EPR signals (1.960 vs. 1.956) may point to the same type of species (i.e. Zni + ) yet experiencing a somewhat different crystallographic environment. In order to get more insights on that point, the impact of molecular oxygen on these two signals was further investigated since only defects located in the surface can be available for such interaction that would modify their corresponding signals.

Adsorption of O2 at RT

Figure 4a compares the PL spectra of the as-synthesized ZnO (violet, solid line), in the presence of O2 (orange, PO2 = 10 mbar) and after pumping out oxygen (violet, dotted line). Overall, the PL intensity decreases in the presence of O2. It is well known that the adsorption of an e - acceptor (such as O2) on semiconductors surface leads to an upward band bending that, consequently, results in the generation of a depletion layer near the surface. 61,62 Such a depletion layer determines the depth of the active PL and, so, the photoluminescence yield. This explains why the otherwise dominant light emission mechanism measured for the as-prepared

ZnO smoke (UV-emission) is almost one order of magnitude weaker in the presence of O2. In the case of nanoparticles such effect is even more pronounced, since much of the material is close to the surface and, thus, more affected by band bending. Similar effects occurring upon O2 admission were reported for ZnO nanowires by Reshchikov and coworkers 63 who measured a much higher intensity of excitonic emission in vacuum than in ambient air pressure. Such O2 effect known to affect the intensity and shape of the PL spectra illustrates how important it is to record PL spectra under dynamic vacuum to prevent misleading interpretation due to O2 interfering effects. On the other hand, a strong decrease of VL upon interaction with O2 implies that the underlying defect (i.e. Zni + ) transforms into an optically inactive center. To be able to interact with gaseous O2, Zni + defects must be located at the surface. Additionally, a PL band emerges at 2.80 eV (VLB) upon O2 adsorption. This band belongs also to the violet emission domain and was most probably already present in the spectrum of the as-synthesized sample yet, its contribution was hardly distinguishable from the close and much more intense (four times) band at 2.88 eV.

Hence, in a similar way to the coexisting EPR signals at rather close g-factors (g = 1.956 and 1.960), the two close violet bands (2.80 eV vs. 2.88 eV) are associated with defects of the same nature, yet localized in different environmentswhich is confirmed by their different reactivity towards oxygen.

In the EPR spectrum recorded under 10 mbar of O2 (Figure 4b), the two close EPR signals (g = 1.956 and 1.960) also show opposite behaviors: while the one at g = 1.956 (species A)

disappears almost completely (similarly to VL contribution in PL spectrum) the signal at g = 1.960 (species B) remains mainly unaffected (similarly to VLB contribution in PL spectrum).

After pumping O2 out, the signal at g = 1.956 recovers back to its former state (dotted violet curve in Figure 4b), which points out a reversible process. Such a reversible broadening or, in our case, a complete disappearance (and recovery) of an EPR signal in the presence (and removal) of physically adsorbed O2i.e. close enough to interact with the related paramagnetic speciesis well reported in literature. 64,65 Hence, the intensity decrease of the signal at g = 1.956 can be explained by strong dipolar interactions between adsorbed O2 and paramagnetic Zni + from the surface.

Apart of being the dominant contributions in the initial spectra, VL at 2.88 eV and EPR signal at g = 1.956 show also a similar sensitivity towards O2. These two contributions (Couple I) are therefore both assigned to surface Zni + and identified as the main native defects in ZnO smoke.

We also note the recovery of Couple I signals to their initial state after pumping the O2 out (dotted violet curves in Figure 4). This points towards rather weak O2 adsorption and reversible electron transfer.

The pairing of the second couple (2.80 eV / g = 1.960; Couple II) is based on a similar reasoning: relatively lower contributions in initial PL/EPR spectra compared to those associated with couple I, and similar tendencies upon O2 addition. The couple II exhibits PL maxima and gvalues only slightly different from those of couple I while not being affected by added O2, and, therefore, can be attributed to the same type of defect, i.e. Zni + , yet present deeper in the bulk. This is in agreement with a recent work by Erdem et al., 66 in which an EPR signal at g = 1.96 was suggested to originate from core defects of ZnO. 67

Effects of annealing treatments

In order to follow the outcome of the Zni + native defects and the eventual creation of new defects, the as-synthesized sample was annealed in either O2-poor or O2-rich conditions. The newly observed spectroscopic features and the associated species or defects assignments are gathered in Table 1. The related processing protocols were adopted in line with formation energies calculated for various point defects in ZnO over a wide range of O2 partial pressures. 25, 68

Annealing in low PO2

Annealing in high vacuum. Compared to the as-synthesized ZnO smoke, no substantial morphological or structural changes could be observed on the sample annealed at T = 773 K in high vacuum (P < 10 -5 mbar) (Figure SI-2). Though the crystalline mean size domain noticeably decreases, the lattice parameters remain unchanged and the microstrain moderately increased from 3 to 6 % (Table 2), which may be due to the formation of new defects in ZnO crystal. In the corresponding Raman spectrum (Figure 5, green curve), we observe that E2 high peak (439 cm -1 ), specific of lattice-oxygen motions, is strongly affected by such thermal treatment whereas a new E1(LO) peak, specific of intrinsic defects, emerges at  560 cm -1 .

The initially intense UV-emission (see Figure 3a) is strongly reduced upon annealing ZnO smoke to 773 K under high vacuum (P < 10 -5 mbar) (Figure 6a) implying that a significant amount of defects is formed in the crystal where they provide new channels for electron-hole recombination. More interestingly, such annealing treatment results in a new PL band (Figure 6a) that can be reproduced by a single Gaussian curve centered at 2.41 eV (515 nm). Although mathematically not necessary for the resulting fit, one can notice that the VL ~2.80 eV may nevertheless be present in a small contribution within the new PL band. On the contrary, the VL ~2.88 eV remains rather undetectable. The corresponding EPR spectrum is shown in Figure 6b. When compared to that obtained on as-synthesized sample (zoom in Figure 6b vs Figure 3b), the signal at g = 1.956 is not detected anymore, whereas that at signal at g = 1.96 is comparatively not modified. The behavior of the former signal upon high-vacuum annealing is comparable to that of VL (2.88 eV), as it is also the case upon interaction with O2. This strongly confirms the attribution of both spectroscopic features (couple I) to the same defect.

Simultaneously with the disappearance of the EPR signal at g = 1.956, a new signal is observed at g > 2.00. The EPR spectrum is rather complex in this range and the simulation (Figure 5b)

shows contribution of three overlapping signals: The origin of an isotropic EPR signal (species C) detected on ZnO with the g-value close to 2.00 remained a matter of great debate until present. Besides resonating close to the value of free electrons (g = 2.002)attribution adopted by some authors 69,70 defects such as VZn:Zni complexes, 71 Zni + , 72 Vo +69 and Os -73 were also suggested. The other two signals detected in EPR spectrum of vacuum-annealed sample (species D and E) present a pseudo-axial rhombic symmetry consistent with superoxide species (O2 -) adsorbed on ZnO. 60,[74][75][76] The gz component is known to be the most sensitive to the environment of the paramagnetic species, so that its variation in species D and E may point out different localizations of O2 -yet both on the surface.

I. giso =
The presence of superoxide anions on sample only annealed in high vacuum and measured upon continuous pumping (P < 10 -5 mbar) is, at first, surprising since the generation of such radicals generally occurs when a previously reduced surface is contacted with gaseous O2. [75][76][77] It is therefore evident that, here, for the creation of notable amount of O2 -upon vacuum annealing, oxygen must have an intrinsic origin and, therefore, comes from the lattice of ZnO smoke. The evident loss of lattice oxygen can proceed via one of the following processes, each of them involving the concomitant formation of an oxygen vacancy (VO 0 , VO + or VO ++ ):

(1) ZnO

∆,𝑣𝑎𝑐 → ZnO1-n + nVo 0 + n/2 O2 (2) ZnO ∆,𝑣𝑎𝑐 → ZnO1-n + nVo + + n/2 O2 -+ n/2 e  ∆,𝑣𝑎𝑐 20ℎ → ZnO1-n + nVo + + n/2 O2 + ne  (3) ZnO ∆,𝑣𝑎𝑐 → ZnO1-n + nVo ++ + n/2 O2 -+ 3n/2 e  ∆,𝑣𝑎𝑐 20ℎ → ZnO1-n + nVo ++ + n/2 O2 + 2ne 
Pathway (1) considers the formation of a neutral oxygen vacancy. There is no associated electron release required for the formation of O2 -, although observed via corresponding EPR signals, nor any paramagnetic species that might stand for the EPR signal at g = 2.002 observed along with those of O2 -. Pathway (3) implies the formation of a doubly charged oxygen vacancy VO ++ , fulfilling the conditions of e -release and potentially explaining the formation of O2 -. Yet, it still does not explain the signal at g = 2.002. Thus, only pathway (2) can completely explain the changes observed in EPR spectra upon high-vacuum annealing: departure of oxygen from the lattice and release of electrons associated with the formation of O2 -and of paramagnetic Vo + species.

Considering the calculated formation energies of oxygen vacancies in ZnO 47,78,79 the absolute values greatly differ as a function of the calculation method. However, irrespectively of the calculation method used, the trend in Ef is rather constant in these studies: while the neutral oxygen vacancy was systematically reported to exhibit the highest Ef value, the formation of Vo ++ was shown to be favorable in O-poor conditions. 25,68,78 The formation energy of a neutral oxygen vacancy VO 0 was calculated in references 25 and 80 to be  3.5 eV that should prohibit its presence even in an extremely O-poor condition. Nevertheless, when passing to nanostructured systems the formation energy of the defect can be considerably loweredas recently shown for Vo in ZrO2 81 and on ZnO. 82 Still, one has to be careful not to rely too strictly on these results since calculations are performed at T = 0 K, generally far from most experimental conditions, and especially ours. For instance, although the Vo + is consistently reported as instable and observable only under irradiation 80 , thus seemingly contradicting at first our experimental evidence, the O2 -formation shown in our EPR spectrum strongly supports its presence in annealed ZnO smoke.

In conclusion, we cannot exclude any of the reaction pathways represented by (1-3), but only pathway (2) can explain the changes observed in EPR spectra upon high-vacuum annealing: (i) release of electrons necessary for the formation of O2 -(specific EPR signals observed) and (ii)

formation of an additional paramagnetic species which resonates at g = 2.002. Thus the EPR signal at g = 2.002 as well as its PL counterpart, measured on the same sample with maximum at 2.41 eV, can be quite assuredly assigned to Vo + (Couple III). We also note that the PL peak is rather broad and, although it can be represented by a lone Gaussian curve, it cannot be completely excluded that other contributions take also partfor instance, those that can be correlated to Vo and Vo ++ , and which can be simultaneously formed following 1-3 pathways.

This will be discussed in relation with the 2.5-2.2 eV PL emissions in a forthcoming paper.

Similar assignments for peaks of Couple III can be found in several studies. 83,84 For instance, an emission band peaking at 2.43 eV was proposed as a result of the recombination between an e -from an isolated Vo + vacancy and a photoexcited hole. 85 More recently, PL band at 2.45 eV was suggested to result from the Vo +  valence band transition. 86 However, since it was detected in extremely different and mostly uncontrolled experimental conditions, no consensus is yet reached for its assignment. In addition, different transitions matching this energy were calculated by several groups. [87][88][89] Finally, this led to the fact that several defects likely to exist in ZnO were suggested as candidates for this PL band. The attribution of the EPR signal at g=2.002 is, on the other hand, hindered by the fact that it lies very close to the value usually reported for free electrons (g  2.00). However, the in situ (dynamic vacuum, P < 10 -5 mbar) detection of O2 -specific EPR signals in our experiment confirms that oxygen is released from the Zn-O network and O-vacancies must have subsequently been formedand related to the signals of Couple III.

We have furthermore noticed that the O2 -specific EPR signals, detected in the course of highvacuum annealing, disappear when the sample is subjected to a longer pumping time at room temperature (t > 20 h). Evidently, O2 -species can be fully removed in the form of O2 through a fully reversible electron transfer at room temperaturea typical behavior of superoxide species.

In contrast, the 2.41 eV and g = 2.002 couple (couple III) were still detected after 20h pumping at RT, indicating that Vo + vacancy is stable upon time.

The loss of lattice oxygen occuring upon annealing under vacuum explains also the intensity decrease of the signals related to surface Zni + (couple I). Indeed, apart from the fraction of released electron necessary for O2 -formation, pathway (2) also provides electrons that may interact with surface Zni + and transform them into Zn 0 . Part of this Zn 0 is most probably sublimated at these temperatures and, thus, removed from the system. Indeed, once the annealing treatment was completed a greyish deposit was observed on the cold part of either the PL or EPR cells. This is supported by XPS indicating a decrease of the Zn/O ratio of the annealed sample (1.10) compared to that obtained on as-synthesized ZnO smoke (1.26). Figure 7 reports a synoptic proposal for the interconnected redox processes occurring upon vacuum annealing treatment and that are responsible for the disappearance of some native defects and generation of new ones. The assignement of couple III to Vo + is also supported by investigating the expected reactivity of water on such defect 23,88 , as shown in figure 8. Water splitting fills up the Vo + vacancy, leading to the intensity decrease of couple III. Moreover, the associated released electron interacts with surface Zn 2+ and forms Zn + , explaining the concomitant intensity increase of the couple I (VL at 2.88 eV and g= 1.956), as described in equation ( 4): Annealing in Zn-vapor. PL and EPR spectra of ZnO smoke annealed in Zn-rich atmosphere are shown in Figure 9a and b, respectively. Both PL and EPR spectra show signals (at 2.41 eV (515 nm) and g = 2.002 (species C)) identical to those of the couple III observed on vacuum annealed ZnO smoke. This is not surprising since the formation of oxygen vacancies is expected under such experimental conditions appliedi.e. Zn-rich atmosphere. As in the case of the vacuum annealed sample, the electron release upon formation of Vo + may once more explain the intensity decrease of spectroscopic fingerprints related to surface Zni + (couple I). The main difference compared to annealing in vacuum is that upon Zn-vapor annealing, D and E EPR species attributed to O2 -species are not detected (Figure 6b, Table I). Such absence of the O2 - specific signals is yet not surprising since Zn being present in excess in the gas phase so that the newly formed O2 may rapidly react with gaseous Zn to form new ZnO entities (figure 7).

Annealing in high P O2

In another set of experiments, the sample was repeatedly annealed in oxygen and then outgassed.

The PL spectra obtained after each of the 3 cycles are presented in Figure 10a. After the first cycle (brown curve), the intensity of VL (2.88 eV, surface Zni + ) decreased along with an intensity increase in the lower energy part (< 2.5 eV) of the spectrum. Both evolutions are further pronounced with every additional O2 annealing/outgassing cycle. After the third cycle, VL is completely absent, which can be explained by the oxidation of Zni + that is thus transformed into optically inactive Zn 2+ centers. The intense PL band that develops at 2.0 eV i.e. in the yellow part of the optical spectrum (600 nm, yellow luminescence (YL)) is rather broad and can be decomposed into more than one PL emission (Figure SI-3). Besides YL contribution at 2.07 eV, the one at 2.41 eV is also present, which is also confirmed by EPR signal at g = 2.002 (not shown) (Table 1). Interestingly, YL disappears after longer pumping time (24 h instead of usual 1h) (Figure 10b), suggesting that the underlying defect is weakly bound to ZnO. Whereas the evolution of yellow luminescence (YL) is not surprisingsince it was systematically measured in oxygen rich conditions and on all forms of ZnO (single crystals, 90 thin films 91 and nanoparticles 20,92,93 ) the coexistence of the PL / EPR fingerprints attributed to Vo + remains unusual at first glance (Table 1). However, considering the applied experimental conditions the competition between three processes should be assumed: i) formation of Vo (due to high annealing T), ii) passivation of Vo (due to the presence of O2) and iii) formation of defects related to YL emission (due to high PO2) (Figure 7). Indeed, though O2 is present during annealing, a competition between the generation and passivation of oxygen vacancies occurs, as demonstrated by Drouilly et al. 94 Despite the rather high formation energies calculated for Oi-type defects, DFT stability diagrams show 25,78 that these defects can be significantly favored in O2 rich atmosphere.

Therefore, the yellow luminescence was principally assigned to oxygen interstitialsmore precisely to singly charged ones (Oi -). [95][96][97] Based on the absence of a new EPR signal in the sample exhibiting YL, we have to exclude a paramagnetic nature of this defect. According to theory, 25,98 Oi may not only be isolated, yet located close to lattice oxygen ions, giving rise to a molecular-like diatomic entity. Such entity would exhibit fully occupied ppπ* orbitals 25,98 , with an electronic structure similar to that of O2 2-peroxide species. Therefore, in contrast to the generally accepted picture of singly charged oxygen interstitials (Oi -) being assigned to yellow emission, 95,97 we instead propose a peroxide-like entity stabilized at the surface of ZnO nanoparticles where it acts as the recombination center for YL. A similar scenario is demonstrated also in the case of NiO surface in Ref. 99 . Once at the surface, the oxygen molecule can be reduced to surface bidentate O2 2-by electrons provided either via oxidation of Zni + or from oxygen vacanciesthe formation of which was shown to be one of the competing processes in this experiment. Not dissociated, such peroxide-species can easily leave the surface as was confirmed by the disappearance of YL upon more efficient pumping (light violet curve in Figure 10b). In the same spectrum, the reversibility of such a process can be observed through the recovery of the Zni + characteristic signal. The attribution of YL to such O2 2-bidentate, localized at the surface instead of being embedded in ZnO crystalas suggested by Janotti et al. 25 may be questioned except if one considers the nano-nature of ZnO (Table 1, figure 7).

At the same time, it opens up basis for new computational strategies.

CONCLUSIONS

By combining electron paramagnetic resonance (EPR) and photoluminescence (PL) spectroscopies, we have studied the nature of native defects in ZnO nanopowders prepared via metal combustion technique. Several processing protocols have been applied to follow the outcome of the existing defects and the eventual creation of new ones. Thanks to the strict control of the experimental conditions during all stages of sample's storage, processing and measurements, crystal defects are identified in correlation with the conditions-specific PL/EPR fingerprints.

We demonstrate here how important it is to precisely detail the experimental conditions in which spectra were recorded. Combining EPR and PL in controlled conditions allows the spectroscopic fingerprints to be unambiguously attributed to the corresponding defects. An example is the couple I (2.88 eV / g = 1.956) that otherwise escapes detection on nanostructured ZnO as due to O2 or H2O containing atmosphere applied during the measurements. The same is true for the identification of Vo + that could be missed if the corresponding measurements were not performed in conditions that ensure the detection of O2 -specific signals i.e. in dynamic high vacuum (P< 10 -5 mbar). Moreover, thanks to the detection of O2 -, the doubtless assignment of EPR at g = 2.002 to Vo + species provided a basis to exclude these defects to be present in assynthesized sample and to attribute the observed couple I exclusively to Zni +as additionally supported also by XPS results. We also showed that the use of probe molecules, such as O2, may provide efficient discrimination between same defects yet localized in different crystallographic environment, namely surface or bulk.

3. 1 .

 1 Figure 1a, b shows representative TEM images of as-prepared ZnO smoke nanoparticles. Rods

Figure 1 .

 1 Figure 1. Representative TEM (a, b) and HRTEM (c) images of as-synthesized ZnO smoke sample. HRTEM image (c) corresponds to ZnO tetrapod arm viewed along a [021] zone axis of the wurtzite structure. The inset in (c) shows SAED of the selected HRTEM zone in violet.

Figure 2 . 40 3. 1 . 2 .

 24012 Figure 2. XRD pattern (a), Raman (b) and DR UV-Vis spectrum (c) of as-synthesized ZnO smoke.

Figure 3 .

 3 Figure 3. As-synthesized ZnO smoke: (a) PL spectra (black curve) and corresponding bandfitting analysis (red). The contribution of each emission is represented by dotted curves in black, magenta, violet and green. (b) Experimental (black curve) and simulated (red curve) EPR spectra. Violet and magenta dotted curves show the contribution of the two paramagnetic species (A and B).

Figure 4 .

 4 Figure 4. PL (a) and EPR (b) spectra of as-synthesized ZnO smoke in vacuum (solid violet curve), in presence of O2 (solid orange curve, PO2 = 10 mbar) and after pumping out O2 (dotted violet curve, P < 10 -5 mbar).

Figure 5 .

 5 Figure 5. Raman spectra of as-synthesized and annealed (773 K, P < 10 -5 mbar) ZnO smoke.

Figure 6 .

 6 Figure 6. PL spectrum with corresponding band fitting (a) and experimental and simulated EPR spectra (b) of ZnO smoke annealed at 773 K in high vacuum (P < 10 -5 mbar). Despite being shown in arbitrary units, the y-scales in Figure 6 are almost one order of magnitude lower than in Figure 3.

Figure 7 :

 7 Figure 7: Diagrammatical representation of the defects detected along the history of the smoke ZnO sample, starting from as-synthesized to post annealing treatments either in low PO2 (vacuum and Zn-vapor atmospheres) or in high PO2 atmospheres, in relation with the qualitatively interconnected redox processes occurring, depending on the operating conditions. Native defects in ZnO smoke, i.e. bulk and surface Zni + are reduced upon annealing in low PO2 by electrons released by the formation of oxygen vacancies. During the annealing in high PO2, antagonistic processes occur leading, on the one hand, to the oxidation of Zni + defects and formation of peroxide-like species, and, on the other hand, to oxygen vacancy formation. Annealing at 773 K in low PO2 Vacuum (P < 10 -5 mbar) Zn-vapor O 2-latt → VO + + ½ e -+ ½ O2 -ads Zni + + e -→ Zn 0 After long time pumping: O2 -ads → O2g + e -

( 4 )Figure 8 :

 48 Figure 8 :(a) PL spectra and (b) EPR spectra of high vacuum annealed ZnO smoke (773 K and P < 10 -5 mbar) and after exposing this powder to water (PH2O = 10 mbar).

Figure 9 .

 9 Figure 9. PL (a) and EPR (b) spectra obtained on ZnO smoke annealed at 773 K in the presence of Zn-vapor (PZn  100 mbar).

Figure 10 .

 10 Figure 10. PL spectra (a) of ZnO smoke repeatedly annealed in O2 (Tann = 773 K, PO2 = 100 mbar) then outgassed at RT. (b) Comparison of PL spectra of the ZnO after three cycles recorded after 1h (yellow) and 24h pumping (light violet).

Figure SI- 2 .

 2 Figure SI-2. XRD and Rietveld refinements of the (a) as-synthesized and (b) annealed (773 K and P < 10 -5 mbar) ZnO smoke. For both, the red crosses are the experimental points, the black line is the calculated pattern, blue vertical tick marks refer to Bragg reflections, and the blue line is the calculated difference pattern. The insets represent TEM and SEM images for assynthesized and annealed sample, respectively.

Table 1 :

 1 List of the correlated PL and EPR signals (Couple) and of the EPR or PL signals (not associated with any PL or EPR feature) that have been detected depending on the history of the smoke sample (as-synthesized and / or upon subsequent post treatments) and their proposed assignment.

		Eemiss (Ev)/ g value	Conditions for detection	assignment
			as-synthesized smoke and Zn annealed
	Couple I	2.88 eV / g = 1.956	sample (in lower relative amount)	Surface Zni +
			(sensitive to O2 adsorption)
			as-synthesized smoke and Vacuum
	Couple II	2.80 eV / g = 1.960	annealed and Zn vapor annealed samples	Bulk Zni +
			(insensitive to O2 adsorption)
			Vacuum and Zn-vapor annealed samples
	Couple III	 2.41 eV / g  2.002	(high intensity) and oxygen annealed sample (weak intensity)	Oxygen vacancy Vo +
			(sensitive to water adsorption)
	EPR	gz = 2.047-2.043; gy = 2.009; gx = 2.003	Vacuum annealed sample (disappearance upon long time pumping)	Superoxide species (O2 -) adsorbed on the ZnO surface
	PL	2.0 eV	Oxygen annealed sample (disappearance after long time pumping associated with simultaneous recovery of surface Zni + features)	Peroxide-like species O2 2-in interaction with the ZnO surface
	PL	2.5 eV	as-synthesized smoke	to be discussed in a forthcoming paper

Table 2 .

 2 Comparison of lattice parameters, microstrain and mean coherent domain size of assynthesized and vacuum-annealed (773 K, P < 10 -5 mbar) samples, obtained via Rietveld analysis.

ACKNWOWLEDGMENT Dr. Miao Zhang wants to thank the China Scholarship Council (CSC) for the three years Ph.D. founding. The authors are also grateful to: Vincent Losinho from Laboratoire de Réactivité de Surface, Sorbonne Université, (LRS) for his help in designing and building the set up used for EPR and PL in situ studies and Stephane Chenot from Institut des NanoSciences de Paris, Sorbonne Université, (INSP) for the maintenance of synthesis chamber and high-vacuum storage lines.

Bulk and surface Zni + were identified as native defects specific of the applied synthesis method.

Upon annealing in O2-poor conditions, we undoubtedly demonstrate the formation of singly charged oxygen vacancies (Vo + ) on ZnO surface. Annealing in high PO2 was shown to lead to antagonism effects relative to high temperature and atmosphere, respectively. Apart from relatively small contribution of Vo + as due to high-annealing temperature applied, defects specific of O2-rich conditions are produced in the form of peroxide-like diatomic oxygen. The various defects identified in this study were shown to produce color specific PL emissions that range from violet (Zni + ), over green (Vo + ) to yellow (O2 2-) visible domain. Hence, under strictly controlled conditions, ZnO smoke can represent an appropriate model system for studying crystal defects. A great deal of cross-agreements between strictly controlled experimental conditions and PL/EPR fingerprints provides a simple approach that can be used for studying and/ or identifying defects in other ZnO types.

Electronic Supplementary Information for


Ricardo Soto
email: ricardo.soto@ucv.cl

Stéphane Caro
email: stephane.caro@irccyn.ec-nantes.fr

Broderick Crawford
email: broderick.crawford@ucv.cl

Eric Monfroy
email: eric.monfroy@univ-nantes.fr

Robust Solutions for a Robotic Manipulator Optimization Problem

In robotics, pose errors are known as positional and rotational errors of a given mechanical system. Those errors are commonly produced by the so-called joint clearances, which are the play between pairing elements. Predicting pose errors can be done via the formulation of two optimization models holding continuous domains, which belong to the NP-Hard class of problems. In this paper, we focus on the use of constraint programming in order to provide rigorous and reliable solution to this problem.

Introduction

Accuracy is one of the key features that favor robotic manipulators for many industrial applications. Superior levels of accuracy are achieved by controlling or measuring all possible sources of errors on the pose of the moving platform of a robotic manipulator. Joint clearance is one of most important sources of errors. It introduces extraneous degrees of freedom between two connected links. When present, they generally contribute importantly to the degradation of the performance of a mechanism. Various approaches have been proposed to compute and quantify the errors due to joint clearances [START_REF] Wang | Position errors due to clearances in journal bearings[END_REF][START_REF] Lin | Accuracy analysis of planar linkages by the matrix method[END_REF][START_REF] Zhu | Uncertainty analysis of planar and spatial robots with joint clearances[END_REF][START_REF] Innocenti | Kinematic clearance sensitivity analysis of spatial structures with revolute joints[END_REF][START_REF] Voglewede | Application of workspace generation techniques to determine the unconstrained motion of parallel manipulators[END_REF][START_REF] Venanzi | A new technique for clearance influence analysis in spatial mechanisms[END_REF][START_REF] Meng | Accuracy analysis of parallel manipulators with joint clearance[END_REF], however none of them focuses on the reliability of solutions, which is mandatory to provide an accurate prediction of the pose error.

In this paper, we investigate the use of constraint programming in conjunction with interval analysis with the aim of guaranteeing the reliability of solutions. To this end, we combine a branch and bound algorithm with interval analysis, which allow drawing firm bounds on the pose errors given possible ranges for the clearances. We illustrate experimental results where the proposed approach generally outperforms the well-known solvers GAMS/BARON [START_REF]GAMS[END_REF] and Ecl i ps e [START_REF] Wallace | ECLiPSe: A Platform for Constraint Logic Programming[END_REF], while providing reliable solutions.

The remaining of this paper is organized as follows. Section 2 presents an error-prediction model used to characterize the influence of joint clearances on the end-effector pose of any robotic mechanical system. An overview of constraint programming -including the implemented approach-is presented in Sect. 3.

The experiments are presented in Sect. 4, followed by the conclusions and future works.

The Problem Formulation

In order to generalize the application of our approach, in this paper we consider robotic mechanical systems with the following configuration: n revolute joints, n links and an end-effector. Hence, the manipulator is assumed to be composed of n + 1 links and n joints; where 0 is the fixed base, while link n is the end-effector. Next, a coordinate frame F j is defined with origin O j and axes X j , Y j , Z j . This frame is attached to the (j -1)st link for j = 1, . . . , n + 1. Figure 1 depicts an instance of such system considering n = 2 involving joints with join clearance as illustrated on the right side of Figure 1. Our goal is to find the maximal positional (translational) and rotational error for a given robotic configuration. Let δp r and δp t representing the rotational and translational errors of the manipulator end-effector, respectively. Then, the maximal pose error of the end-effector for a given manipulator configuration can be obtained by solving the following two optimization problems:

maximize δp r , (1)
subject to δr 2 j,X + δr 2 j,Y -∆β 2 j,XY ≤ 0, δr 2 j,Z -∆β 2 j,Z ≤ 0, j = 1, . . . , n

maximize δp t , (2)
subject to δr 2 j,X + δr 2 j,Y -∆β 2 j,XY ≤ 0,

δr 2 j,Z -∆β 2 j,Z ≤ 0, δt 2 j,X + δt 2 j,Y -∆γ 2 j,XY ≤ 0, δt 2 j,Z -∆γ 2 j,Z ≤ 0, j = 1, . . . , n
where δr a,b corresponds to the small rotation in joint a with respect to axis b, and δt a,b corresponds to the translation in joint a with respect to axis b. β and γ are simply constants used to limit the pose errors depending on the given configuration (an extended explanation of this model can be found in [START_REF] Binaud | The kinematic sensitivity of robotic manipulators to joint clearances[END_REF]). Let us notice that the two problems are nonconvex quadratically constrained quadratic (QCQPs). Indeed, all constraints of both problems are convex, their feasible sets are convex, and their objectives functions are non-convex. Then, both problems belong to the category of NP-Hard problems [START_REF] Lin | Accuracy analysis of planar linkages by the matrix method[END_REF].

3 Constraint Programming for Global Optimization

Definitions

A Constraint Satisfaction Problem (CSP) is a formal problem representation, which mainly consists in a sequence of variables holding a domain and a set of relations over those variables called constraints. The idea is to find values for those variables so as to satisfy the constraints. The software technology devoted to tackle this problem is named Constraint Programming (CP). In this work, due to the presence of continuous decision variables, we focus on Numerical CSP (NCSP), which is an extension of a CSP devoted to continuous domains. Formally, a NCSP P is defined by a triple P = x, [x], C where:

-x is a finite sequence of variables (x 1 , x 2 , . . . , x n ). -[x] is a finite set of real intervals ([x 1 ], [x 2 ], . . . , [x n ]) such that [x i ] is the domain of x i . -C is a set of constraints {c 1 , c 2 , . . . , c m }.
A solution to a NCSP is a set of real intervals that satisfy all the constraints. Optimization problems are handled in the same way. Hence, the 4-tuple P = x, [x], C, f (x) is employed in this case, where f (x) is the cost function to be maximized or minimized.

NCSP Solving

In order to guarantee accurate solutions, NCSPs cannot be handled in the same way that CSPs mainly due to the presence of constraints over real numbers. Indeed, the representation of reals in numerical computations is not exact since it is commonly done by means of floating-point numbers, which are a finite set of rational numbers. This inaccuracy may lead to rounding errors and as a consequence to reach wrong solutions. One solution for rigorously dealing with real numbers relies on the integration of interval analysis on the solving process. The idea is to compute approximations over domains represented by intervals bounded by floating-point numbers [START_REF] Benhamou | Applying Interval Arithmetic to Real, Integer and Boolean Constraints[END_REF]. A detailed presentation of interval analysis can be seen in [START_REF] Moore | Interval Analysis[END_REF], and some examples devoted to robotics in [START_REF] Jaulin | Applied Interval Analysis with Examples in Parameter and State Estimation[END_REF].

Then, the core idea for solving NCSPs relies in combining a branch and prune algorithm with interval analysis for handling continuous domains. A tree-data structure that holds intervals as the potential solutions is built on the fly by interleaving branching and pruning phases. The branching phase is responsible for creating the branches of the tree by splitting real intervals, while the pruning tries to filter from domain intervals that do not conduce to any feasible solution. The idea is to speed-up the solving process. This is possible by applying consistency techniques for continuous domains such as the hull and the box consistency [START_REF] Lhomme | Consistency Techniques for Numeric CSPs[END_REF][START_REF] Benhamou | CLP(Intervals) Revisited[END_REF], which are similar to the arc-consistency [START_REF] Mackworth | Consistency in Networks of Relations[END_REF] for finite domain CSPs.

Figure 2 depicts an algorithm for rigorously handling NCSPs. The procedure begins by receiving as input the set of constraint and domains of the problem. Then, four actions are embedded in a while loop. The Contract operator is responsible for pruning the tree, and Split applies a dichotomic division of intervals in order to carry out the branching process. Every computation of elementary operations {+, -, ×, /} is done by using interval arithmetic. The process stops when the real values of the solution have reached the precision required of the problem. A slight modification to the previous algorithm is required to handle optimization problems. Indeed, here a CP-based branch and bound algorithm is combined with interval analysis (see Figure 2). The corresponding cost function f has been added to the input set. A variable m is initialized to +∞ in order to maintain an upper bound on the global minimum. In this way, potential solutions exceeding this bound are discarded by adding to the set of constraints. Five instructions are embedded in the same while loop. Now, Contract takes into account the cost function in order to prune the tree. The Update function has been added for updating the upper bound once better solutions are found. The branch and bound algorithm described above has been implemented on top of the RealPaver solver [START_REF] Granvilliers | Algorithm 852: RealPaver: an Interval Solver Using Constraint Satisfaction Techniques[END_REF]. This implementation is used for the experiments presented in the next section.

Algorithm 1 Input: C = {c1, . . . , cm}, [x] 1 L ← {[x]} 2 While L = ∅ and ¬stop criteria do 3 ([x], L) ←Extract(L) 4 [x] ←ContractC([x]) 5 {[x ], [x ]} ←Split([x]) 6 L = L ∪ {[x ], [x ]} 7 End While 8 Return(L) Algorithm 2 Input: f , C = {c1, . . . , cm}, [x] 1 L ← {[x]} 2 m ← +∞ 3 While L = ∅ and ¬stop criteria do 4 ([x], L) ←Extract(L) 5 [x] ←Contract C∪{f (x)≤m} ([x]) 6 m ←Update([x], f ) 7 {[x ], [x ]} ←Split([x]) 8 L = L ∪ {[x ], [x ]} 9 End While 10 Return(L, m)

Experiments

As previously mentioned the RealPaver solver has been used as base for our branch and bound algorithm. In this section, we verify the reliability of solutions and we compare the performance of such implementation with the state-of-theart solvers GAMS/BARON [START_REF]GAMS[END_REF] and Ecl i ps e [START_REF] Wallace | ECLiPSe: A Platform for Constraint Logic Programming[END_REF]. Ecl i ps e is a widely used solver in CP and one of the few having support for continuous domains. GAMS/BARON is a popular solver from the mathematical programming field devoted to particularly hard optimization problems.

The following experiments take into account 4 rotation and 4 translation models (see Table 1). We consider from two to five joints for each model. The information related to the problem size (number of variables, number of constraints, and number of operators) is also given. Then, we provide the solving times in milliseconds for BARON; Ecl i ps e , and for the proposed algorithm. The experiments were run on a 3 Ghz Intel Pentium D Processor 925 with 2 GB of RAM running Ubuntu 9.04. All solving times are the best of ten runs.

The results show that our implementation is in general faster than its competitors. For smaller problems involving two and three joints, our algorithm exhibits excellent performance, being 100 times faster that BARON. This performance is clearly reached by the efficient filtering work done by the HC4. This is also influenced by the fact that there is no need for highly accurate computations for this particular problem: we actually do not have to reach the usual precision, which makes the search process less costly than they usually are.

In the presence of rotational problems considering four and five joints, our approach remains faster. However, once the number of joints increases, in particular when the number of variables is greater than 20, the searching begins to be slow. This is obviously explained by the exponential complexity in the problem size that leads to complete search methods like our branch and bound to a slower convergence. Despite of this common phenomenon, we estimate that the presented solving times are reasonable regarding the problem complexity as well as the solving techniques involved. Let us remark that although RealPaver as well as Ecl i ps e use CP-based solving techniques. The performance of our approach is significantly better, this is not surprising due to RealPaver was originally designed for tackling problems with continuous domains.

Besides, solving times may actually not be the only point to emphasize here: It may also be important to discuss the reliability of the solutions given by BARON. We could indeed verify using RealPaver that the optimum enclosure computed by BARON is unfeasible on some of the above tested problem instances. It was already noted in [START_REF] Lebbah | An Efficient and Safe Framework for Solving Optimization Problems[END_REF] that BARON may not always be reliable, where the following example illustrates how BARON can fail to give the correct value and return a an inconsistent point instead. Let x, y be two real variables, x, y ∈ [-10, 10], and two constraints y -x 2 ≥ 0 and y -x 2 (x -2) + 10 -5 ≤ 0. When BARON tries to find the lowest x for which both constraints are satisfied, it returns the point (0, 0) although it is inconsistent w.r.t. the constraints of the problem (cf Figure 3, reproduced from [START_REF] Lebbah | An Efficient and Safe Framework for Solving Optimization Problems[END_REF]).

Finally, let us focus on the results obtained for the manipulator composed of two revolute joints illustrated in Fig. 1. Let us assume its geometric parameters are defined as follows: RealPaver and BARON were used to solve the optimization problem for all manipulator poses. Figures 5 and6 show the value of the maximum positioning error of the end-effector obtained with RealPaver and BARON, respectively. This value is expressed in millimeters and plotted throughout the manipulator workspace. We can notice that the maximum obtained with BARON is always higher than the one obtained with RealPaver. The difference in any point between the results given by the two solvers is around one millimeter and decreases when positioning error rises: As we already mentioned it, we could notice that the solutions obtained with BARON may not be feasible.

a 1 = 1 m b 1 = 0 m α 1 = 0 rad a 2 = 0.

Conclusions and Future Works

In this paper we have modeled and solved the complex problem of predicting the maximum pose error in robotic mechanical systems. To this end we have employed constraint programming techniques and interval analysis. Experimental results demonstrate the effectiveness of our approach where it is able to in general outperform well-known solvers such as BARON and Ecl i ps e , providing reliable solutions. Let us also mention that the presented approach is not only devoted to robotics, it in fact applies to whatever problem requiring rigorous computation and reliability in the solutions.

In this context, there are several directions for future work. In the future we plan to design new pruning criteria for accelerating the convergence of solutions. In this way, it would be easier to handle the exponential growth of the space of solutions. Finally, it would be interesting to pay an extended attention to the unreliability of solutions obtained by BARON. Such a problem could be also present in additional solvers.

Fig. 1 .

 1 Fig. 1. Left: A serial manipulator composed of two revolute joints. Right: Clearanceaffected revolute joint.

Fig. 2 .

 2 Fig. 2. Left: The branch and prune algorithm. Right: The branch and bound algorithm

7 m b 2

 2 = 0 m α 2 = 0 rad and the joint clearances are equal to: ∆β j,XY = 0.01 rad ∆β j,Z = 0.01 rad ∆b j,XY = 2 mm ∆b j,Z = 2 mm

Fig. 3 .

 3 Fig. 3. BARON fails to find the global minimum and returns an inconsistent point instead.

Fig. 4 .

 4 Fig. 4. 3D plot of the maximum positioning error of the manipulator with two joints throughout its Cartesian workspace.

Fig. 5 .

 5 Fig. 5. Isocontours of the maximum positioning error in millimeters of the manipulator with two joints throughout its Cartesian workspace obtained with RealPaver

Fig. 6 .

 6 Fig. 6. Isocontours of the maximum positioning error in millimeters of the manipulator with two joints throughout its Cartesian workspace obtained with BARON

Table 1 .

 1 Type, problem size and solving times (in seconds).

	#joints Type	Problem Size BARON RealPaver Ecl i ps e #var #ctr #op
	2	T	12	8 28	0.08	0.004 >60
	2	R	6	4 18	0.124	0.004 >60
	3	T	18 12 135	0.124	0.008 t.o.
	3	R	9	6 90	0.952	0.004 t.o.
	4	T	24 16 374	0.144	0.152 t.o.
	4	R	12	8 205	2.584	0.02 t.o.
	5	T	30 20 1073	0.708	3.71 t.o.
	5	R	15 10 480	9.241	0.26 t.o.