Christophe Drouet
email: christophe.drouet@ensiacet.fr

Apatite Formation: Why It May Not Work as Planned, and How to Conclusively Identify Apatite Compounds

Calcium phosphate apatites are inorganic compounds encountered in many di erent mineralized tissues. Bone mineral, for example, is constituted of nanocrystalline nonstoichiometric apatite, and the production of "analogs" through a variety of methods is frequently reported. In another context, the ability of solid surfaces to favor the nucleation and growth of "bone-like" apatite upon immersion in supersaturated uids such as SFB is commonly used as one evaluation index of the "bioactivity" of such surfaces. Yet, the compounds or deposits obtained are not always thoroughly characterized, and their apatitic nature is sometimes not rmly assessed by appropriate physicochemical analyses. Of particular importance are the "actual" conditions in which the precipitation takes place. e precipitation of a white solid does not automatically indicate the formation of a "bone-like carbonate apatite layer" as is sometimes too hastily concluded: "all that glitters is not gold. " e identi cation of an apatite phase should be carefully demonstrated by appropriate characterization, preferably using complementary techniques. is review considers the fundamentals of calcium phosphate apatite characterization discussing several techniques: electron microscopy/EDX, XRD, FTIR/Raman spectroscopies, chemical analyses, and solid state NMR. It also underlines frequent problems that should be kept in mind when making "bone-like apatites. "

Introduction

Synthetic-calcium-phosphate-apatite-based compounds are frequently encountered in the literature dealing with bone tissue engineering. is fact is not surprising though, given the apatitic nature of bone mineral itself [ ]. Detailed investigations on this biomineral (e.g., [ -]) have revealed its nanocrystalline, nonstoichiometric, and hydrated characters, as well as speci c surface features involving labile ions in nonapatitic chemical environments, potentially exchangeable with ions contained in surrounding uids. Similar characteristics were also pointed out for synthetic biomimetic analogs (e.g., [ ]), allowing one to prepare bioactive ceramics [ ].

Such biomimetic nanocrystalline apatites exhibit an overall chemical composition that can generally be described by formulas of the type Ca 10-(PO 4 ) 6-(HPO 4 ) (OH) 2-[ ] or Ca 10--(PO 4 ) 6-(HPO 4 ) (OH) 2--2 [ ], where and depend on conditions of formation and state of ageing (maturation). It should be stressed here that the physico-chemical characteristics of nanocrystalline apatites-whether of biological or synthetic origin-di er signi cantly from those of stoichiometric hydroxyapatite (HA), Ca 10 (PO 4 ) 6 (OH) 2 , especially in terms of stoichiometry, crystal size, crystal disorder, unit cell parameters, surface features, and hydration state [ ].

While the synthesis of apatite materials for the preparation of implantable bioceramics is bound to occupy a privileged place in bone applications thanks to (sub)structural and compositional features close to those of bone mineral, it is not suited-in a self-supported way-for load-bearing applications that require stronger mechanical properties. Metal implants then appear as the most commonly used candidates (e.g., hip or knee prostheses), despite potential drawbacks such as poor bone-bonding abilities or allergy issues [ ]. In this case, the chemical quality of the interface between the implant and the surrounding bone tissue is of prime importance for the success and durability of implantation [ ].

is point has led many researchers and industrial companies to investigate and develop deposition processes of apatite compounds on metal implants (e.g., by plasma spray technologies) in view of favoring bone-bonding properties [ -]. Also, more generally, tests are o en carried out in supersaturated solutions so as to evaluate the ability of a surface to induce the nucleation and growth of a "bone-like" apatite layer [ ]. In these cases, the term "supersaturated" implicitly refers to hydroxyapatite (solubility product sp ( ∘ C) = [ ] for the formula Ca 10 (PO 4 ) 6 (OH) 2 ). Several types of supersaturated solutions have been used, the most famous probably being the so-called SBF solution (Simulated Body Fluid) [ , ] or its multiples (e.g., SBF × . ) [ , ]. Tests are generally performed for several days and o en at ∘ C to approach physiological conditions. Although care should be taken when drawing conclusive statements on a material "bioactivity" on the sole basis of such laboratory tests (run out of the body and in the absence of cells or o en of proteins), such assays may however be informative, for example, for a preliminary "ranking" of implants capabilities [ ], but it has to be kept in mind that additional data are needed (e.g., cell behavior assessments and then in vivo con rmations) to draw a more accurate picture of the actual bioactivity or expected biological behavior of the samples tested.

In all the preceding statements, the formation of an apatite phase is a speci c concern, whether for the production of self-supported bioactive ceramics, for the setup of a coating on implants for improving bone-bonding abilities, or else for drawing relative "bioactivity" information. A great deal of papers and congress communications refer to these aspects, and conclusive statements are commonly drawn. Unfortunately, adequate physico-chemical characterization data are not always attached to such statements, and the establishment of the "apatitic" nature of samples or deposits is sometimes unconvincing or even dubious. e announced conditions of preparation of a compound considered to be apatite sometimes depart from the conventional conditions of stability of apatite, requiring characterization proofs. In other instances, the sole observation of a white deposit a er immersion in supersaturated media is considered as a solid proof for concluding that a "bone-like carbonated apatite layer" has been formed, which is obviously not scienti cally satisfactory.

Various aspects should in contrast be taken into account before making conclusive statements.

e chemistry of calcium phosphates is indeed very rich, with numerous possible phases (beside apatite) able to form depending on experimental conditions [ ]. Also, the "real" conditions of treatments are sometimes slightly di erent from the "intended" ones, due, for example, to experimental mistakes or approximations, and some steps in the synthesis/coating process may not be adequately performed. Furthermore, the substrate itself might be able to react with the immersion solution, for instance, by releasing/capturing some ions or molecular entities into/from the surrounding medium, thus modifying its actual surface state or the external conditions (e.g., e ect on the pH of the medium). Finally, inadequate conditions may lead to variations in supersaturation of pH along the process.

All of these considerations point to a situation that is more complex than may be initially thought. All in all, the conclusion that an apatite phase has been obtained requires, as in every reliable materials science study, adequate physicochemical characterization. is contribution is intended to review the main characteristics to be expected from a calcium phosphate apatite compound, drawn from complementary characterization techniques. e cases of hydroxyapatite and of a nanocrystalline apatite of rather low maturation state are examined for establishing typical "identity cards" for such materials. Characteristics of other minerals that may possibly also be encountered in the above-mentioned contexts are also reviewed, and some frequent issues are commented upon.

Results and Discussion

. . Apatite Formation: Why It May Not Work as Planned. . .. In this rst section, the idea is to provide readers with elements pointing out that apatite formation is not a trivial phenomenon. Whether due to inexact concentration calculations, unexpected alteration/equilibration of a substrate surface upon immersion, or else because of incomplete processing steps, several factors could in fact lead to other outcomes than planned. e following paragraphs describe some of the most likely causes of experimental failure or mistakes to avoid.

. . . e Impact of Incomplete Washing. Most laboratory tests aiming at evaluating the ability of solid surfaces to get covered by an apatite layer upon immersion are carried out in "closeto-physiological" media. Generally, though, only the mineral fraction of body uids is considered (e.g., in SBF solution [ ]), meaning in particular in the absence of proteins or cells. is situation indeed allows a greater ease of use and leads already to some informative results. However, it should be kept in mind at this point that the adsorption of organic substances such as proteins or peptides, or the activity of cells once in actual in vivo environments, could lead to somewhat modi ed outcomes.

In cases when synthetic simulated body uids (only mineral) are prepared, it is customary to use chemical compositions that are more or less close to that of blood plasma (reported in Table , as reported by Krebs [ ], along with speciation data determined in this work using the Visual Minteq . so ware). As can be seen, blood plasma (or synthetic SBF) contains in particular large amounts of sodium and chloride ions, which are predominant ionic species. e observation of a white deposit on the substrate a er withdrawal of the piece is therefore not an absolute proof of the formation of apatite at this stage; indeed, in cases where the washing step has not been thoroughly performed, the formation/residual presence of NaCl aggregates could explain this white coating phenomenon. Figure reports the illustrative example of a titanium alloy disk (TA V) immersed in SBF × for days at room temperature (RT) in the case of either incomplete or more advanced washing. Although SEM observations evidenced the presence of deposited crystals in both cases, EDX analyses pointed out that the poorly washed sample was (as one could indeed expect) mostly covered by NaCl crystals. In the context of immersion tests, the absence of NaCl crystals should then probably be systematically veri ed (by EDX and/or XRD analyses).

. . . Role of "Real" Experimental Conditions. In cases where adequate sample preparation was done prior to analysis, thus including thorough washing steps, the observation of a deposit a er immersion in a hydroxyapatite-supersaturated medium still does not necessarily imply the formation of apatite. Experimental mistakes (e.g., due to the weighing of hygroscopic starting salts for preparing the supersaturated solution) may lead to undesired immersion conditions and then to nucleated phases other than apatite.

is stresses the fact that parameters such as the medium pH should probably be monitored during the immersion step to verify that no signi cant pH change has occurred. is is all more relevant as the precipitation of (hydroxy) apatite implies the incorporation of OH -ions and theoretically implies an acidi cation of the medium: this acidi cation phenomenon may remain of low amplitude if the liquid-to-solid ratio is chosen su ciently high, but this underlines the fact that in some cases the pH may turn out to be inappropriate to form apatite. e incorporation of pH-bu ering agents like Tris can be found e ective, but this implies the addition to the medium of a "foreign" organic compound, which may not always appear appropriate. In cases when the pH follows uncontrolled variations, other phases may stably form instead of apatite (depending on the exact conditions), such as acidic calcium phosphate compounds like brushite CaHPO 4 ⋅ H 2 O (DCPD), monetite CaHPO 4 (DCPA) or octacalcium phosphate (OCP). Indeed, these phases become more stable than hydroxyapatite in acidic pH conditions [ ].

e variation of supersaturation during immersion tests may also lead to a modi cation in deposited phase with time. For information, the saturation index relative to various phases of interest and considering as initial conditions the ionic concentrations of blood plasma (inorganic species, see Table ) is reported in Table . A negative value of this index is indicative of a situation where the medium is supersaturated with respect to the phase concerned. ese data were calculated using the Visual Minteq . so ware, taking into account the solubility products that are generally accepted for these various phases (added in Table ). As may be noted, such conditions (if correctly set up from an experimental point of view) lead to a high supersaturation with respect to HA and to a moderate supersaturation for octacalcium phosphate (OCP), calcite or vaterite, among other phases. In contrast, they are moderately undersaturated with respect to magnesium-containing phases such as MgHPO 4 ⋅ H 2 O and more undersaturated with respect to brushite or monetite. If undesirable (or variations in) ionic concentrations are present during the tests, however, the ionic product is going to evolve and the overall saturation scheme is expected to change. Supersaturation aspects are thus also of prime importance during apatite precipitation plans.

In this regard, the use of constant composition crystal growth (C G) systems [ , ], which keeps the pH, ionic Selected phases sp ( ∘ C) Reference Saturation index (c) for blood plasma conditions (kJ/mol) Hydroxyapatite (HA) Ca (PO ) (OH)

. (a) [ ]

-. Octacalcium phosphate (OCP) Ca (PO ) (HPO ) ⋅ H O . (b) -. Monetite (DCPA) CaHPO . . Brushite (DCPD) CaHPO ⋅ H O . . Calcite c-CaCO . [ ] -.
Vaterite v-CaCO .

-.

Magnesium hydrogenphosphate MgHPO ⋅ H O .

[ ] .

( a) Expressed relative to HA formula in Ca . (b) Expressed relative to OCP formula in Ca . (c) Saturation index (Δ ) [ ] calculated as: Δ = -( / )Ln( / sp ) where is the number of ions involved, IP is the ion product, and sp is the solubility product. A negative value of Δ means that the medium is supersaturated with respect to the concerned phase.

strength, temperature, and relative concentrations constant, can be seen as an advantageous way to control all experimental parameters and derive relevant data on the ability of substrates to provoke the nucleation and growth of apatite. Finally, especially in the case of long immersion tests run in supersaturated uids or for precipitations in media containing organic molecules (glucose, proteins. . .), the development of microorganisms may become signi cant for rather long periods of treatment. In this case, bacterial activity may then contribute to modifying the initial medium in a signi cant way. In order to avoid such issues, the incorporation of additives such as sodium azide NaN 3 in the medium remains an option. Antibacterial agents such as penstrep or fungizones may also be included.

. . . Potential Impact of Substrate Equilibration with the

Medium.

e nature and behavior of the substrate itself, once in solution, may also play a nonnegligible role in the outcome of immersion tests. For instance, substrates exposing ionic or functional molecular gra ed groups (e.g., a er speci c treatments aiming to alter the surface roughness or intending to improve cellular activity) may endure a preliminary surface equilibration with the medium, potentially modifying the initially determined immersion conditions [ ]. If the pH of the medium, or another relevant factor, is altered by the conditioning of the substrate upon contact with the medium, this could in turn lead to uncontrolled or undesirable test conditions. Preliminary immersion tests should thus probably be run for samples for which the inert character has not been clearly established, so as to guarantee the absence of noticeable change in the medium. In this view, pH monitoring could again be seen as relevant during the process.

. . . Incomplete Reactions. Hydroxyapatite is a mineral phase that gathers a large number of ions. As such, its precipitation-especially at moderate temperatures-is bound to remain a rather slow phenomenon [ , ]. is can explain the low degree of crystallinity of precipitated apatites, which subsequently mature in solution (evolution towards stoichiometry) at a pace that depends on experimental conditions [ ]. In this context, the transient formation of precursor phases (metastable but exhibiting a faster formation rate) is also possible despite conditions that are thermodynamically favorable to apatite formation, in line with Ostwald's rule [ ].

Among possible precursors, amorphous calcium phosphate (ACP) and octacalcium phosphate (OCP, triclinic) are two potential candidates [ , ] that have already been suggested as precursors of bone-apatite in vivo [ , ]. A er their formation, both compounds may then be hydrolyzed to apatite in a subsequent step. ACP, although amorphous, is not totally exempt from structuration; it involves building blocks based on Ca 9 (PO 4 ) 6 ⋅nH 2 O clusters (so-called Posner clusters) [ ], which can undergo partial internal hydrolysis through the reaction PO 4

3-+ H 2 O → HPO 4 2-+ OH - [ ],
leading to Ca 9 (PO 4 ) 6-(HPO 4 ) (OH) and allowing in particular the lling by OH -ions of what will become apatitic channels. e hydrolysis of OCP probably involves a rather di erent mechanism. Taking into account the similarities between the structures of OCP and HA (detailed below) [ ], the possibility of undergoing the OCP-to-apatite transformation via a topotactic mechanism appears as a possible scheme [ -]. In this case, the usual plate-like morphology of OCP is expected to be rather conserved upon transformation into apatite. is illustrates then the di culty of assessing phase identi cation uniquely on the basis of morphological features (as discussed below). is fact points again to the necessity to adequately characterize formed phases, as even conditions that are thermodynamically consistent with the formation of an apatite phase may (at least transiently) allow the formation and sometimes stabilization of precursor compounds.

. . On the Characterization of Apatite Compounds. e above statements have unveiled various possible causes for immersion tests failure or mistaken precipitation situations. ey illustrate in particular the possibility of obtaining deposited crystals other than apatitic, due to uncontrolled parameters like pH, supersaturation or substrate reactivity.

is emphasizes the need for appropriate physico-chemical characterization of the phase formed (either on the occasion of immersion tests or more generally when intending to precipitate apatite compounds in any situation), in order to con rm or not the formation of an apatite phase. Several distinctive features (structural, compositional. . .) can be identied by the way of complementary techniques so as to draw conclusive statements. In this context, the following section aims to review the fundamentals of calcium phosphate apatite characterization. Comparisons with other phases that may also be encountered in humid conditions have also been included in the discussion.

. . . Microscopy Observations-TEM, SEM-EDX Techniques. Microscopy observations are o en carried out, for example, for assessing the eventual presence of a deposit on the surface of a substrate immersed in supersaturated uids.

is is also the way to determine the morphological features of a precipitated compound.

e detection of di erent morphologies during the analysis of a single sample should not be considered as insigni cant as it may be indicative of the simultaneous presence of several phases in the specimen. is may, for example, be noticed on the occasion of immersion tests for which the solutionto-solid ratio has been underestimated, leading to strong variations in supersaturation levels along the test.

In cases when a single morphology is observed (the bestcase scenario), care should still be taken before drawing conclusions on the nature of the phase. Figure reports the usual plate-like or petal-like morphology of biomimetic apatite formed on the occasion of immersion tests in supersaturated solutions. Although the observation of this type of morphology is a promising factor in favor of the precipitation of bone-like apatite, this morphology should not be considered as an absolutely conclusive criterion. Indeed, other calcium phosphates may also exhibit anisotropic or platelike morphologies. Platelets are also characteristic, for example, of monetite (CaHPO 4 ) or brushite (CaHPO 4 ⋅ H 2 O), although the particle size is generally larger than for bonelike nanocrystalline apatites. Triclinic OCP also exhibits a petal-like morphology. Moreover, unexpected remaining impurities (e.g., due to poor washing) may also lead to morphological similarities with biomimetic apatite.

EDX elemental analysis can be considered as a precious (although still not su cient) tool for unveiling compositional aspects. It may, for example, rule out the presence of NaCl crystals (Figure ). e observation of calcium and phosphorus lines is then in favor of the formation of a calcium phosphate phase. However, although quantitative Ca/P molar ratios may be drawn from EDX analyses (if appropriately compared with elemental standards), it should be remembered that bone-like apatites are o en largely nonstoichiometric and exhibit a nonapatitic surface layer hosting mostly bivalent ions such as HPO 4 2-instead of PO 4 3-(which may be related to the conditions of formation of such apatites in solution) [ , , -]. erefore, the Ca/P ratio cannot be exploited to unequivocally determine the nature of the calcium phosphate phase (e.g., noncarbonated nanocrystalline apatites with a Ca/P ratio as low as . have been prepared [ ])-this point will be addressed again in the "chemical analyses" subsection. Consequently, while morphological and elemental EDX analyses are valuable for rst characterization statements, they cannot be exclusively used to con rm the apatitic nature of the precipitated phase, and additional characterization is needed.

. . . X-Ray Di raction (XRD). XRD is undeniably a central characterization tool for identifying precipitated crystallized phases. On the contrary, amorphous compounds (sometimes present beside a crystallized phase) will only lead to large halos, which are unfortunately sometimes overlooked. In the case of calcium phosphates, the halos are due to amorphous tricalcium phosphate (ACP) and arise as background deformations under the main di raction lines of apatitic compounds (see Figure ), corresponding to interreticular distances approximately in the ranges . -. and . -. Å. Care should consequently be taken when inspecting these regions so as to verify the absence of broad halos.

e XRD pattern of well-crystallized stoichiometric hydroxyapatite (HA) is also reported in Figure (relative to both cobalt and copper anticathode K 1 wavelengths). In cases when acidic pH values may be reached, other calcium phosphate phases such as monetite or brushite may form, the patterns of which have also been added in Figure . At this point, it should also be remembered that the acquisition of XRD diagrams o en favors preferential crystal orientations.

is is especially true for phases exhibiting highly anisotropic morphologies like monetite or brushite. In such cases, experimental XRD patterns may then signi cantly depart from calculated ones, by variations in peak intensities. A noticeable modi cation of the XRD pattern is, for example, found for brushite, for which the ( ) experimental line becomes noticeably more intense than expected for randomly oriented crystals. Despite such modi cations, the XRD pro les of monetite and brushite distinctively depart from that of apatite compounds, allowing one to rule out or not the presence of such phases.

Preferential orientations and related peak intensity modi cations are also observable for precipitated OCP, but in this case the distinction from an apatite pro le is more subtle (Figure ). e relative similarity of experimental patterns obtained for OCP and a poorly-crystalline apatite may in fact be related to (partial) structural resemblances; the OCP structure can indeed be described as the alternative stacking of "apatitic" layers (with crystallographic positing of ions very close to those in HA) and "hydrated" layers (which enclose in particular all the HPO 4 2-ions contained in OCP) [ ]. In this context, the distinctive presence of the ( ) line of OCP at very low angles (see Figure ) then becomes particularly helpful for distinguishing, on the basis of XRD analyses, precipitated OCP from an apatite phase. Unfortunately, low angle ranges are not always examined or reported in the literature studies (the di ractometer setup must sometimes be speci cally adapted to access this low-angle region).

XRD analysis is thus one major stage in sample characterization. But limitations to this technique also exist as illustrated above (di culty in revealing the presence of amorphous compounds, existence of preferential orientations, and necessity to analyze speci c angle ranges). Also, in the case of immersion tests, the deposit to be expected o en remains of very limited depth, making XRD analyses more di cult to perform (the analysis of thin deposits being more e ectively carried out with glazing angle analyses); complementary analysis may then be required, including in particular spectroscopy techniques that are addressed below.

. . . Vibrational Spectroscopies: FTIR and Raman. In view of the preceding statements, crystallographic data are o en helpful but not always conclusive in terms of phasecation. Vibrational spectroscopies (FTIR, Raman) can then be seen as useful tools for gaining more insight on the constitutive ions of a sample and related ionic environments. Indeed, phosphate and hydroxide groups-which involve covalent bonds-lead to very speci c vibrational features when involved in an apatitic system, which may be exploited for phase identi cations. , namely, stoichiometric HA, a nanocrystalline apatite matured for day (freeze dried), OCP, ACP, monetite, and brushite. is gure may be used to identify some major spectral di erences among these phases. In the case of HA, two bands due to OH -ions in apatitic environments are in particular detectable at ± (as a shoulder to the ] 4 (PO 4 ) band) and ± cm -1 (narrow peak). However, for nonstoichiometric poorly-crystallized apatites obtained by precipitation at moderate temperatures, these OH bands are o en di cult to detect by IR spectroscopy (see Figure ); the low degree of crystallinity tends to enlarge vibrational bands, covering weak OH signals, and the nonstoichiometry disfavors the presence of OH -ions in apatitic channels. Also, such precipitated apatites are generally associated with water molecules, leading to strong absorption in the cm -1 region. e peak at cm -1 is, however, also active in Raman spectroscopy, where water molecules are not signi cantly perceptible. e detection of this band (not to be confused with the strong absorption of brushite in this region, Figure ) is one factor attesting to the presence of an apatite phase.

Besides, although the ] 1 (PO 4 ) band at ± cm -1 is only of minor intensity in IR spectroscopy (Table ), its presence is also of interest. It is indeed observed at this position in apatite phases or OCP but not in ACP (band shi ed to lower wavenumbers, leading to a shoulder to the ] 3 (PO 4 ) band, around cm -1 ), monetite (band shi ed to ca.

cm -1 ), nor brushite (band shi ed to ca. cm -1 ), as shown in Figure . Besides, this ] 1 (PO 4 ) band is very active in Raman spectroscopy, which may allow further identi cation validation.

FTIR spectra of nanocrystalline (biomimetic) apatites show modi ed features compared with well-crystallized stoichiometric HA. First of all, spectra are o en composed of enlarged bands due to rather low degrees of crystallinity (Figure ). In these conditions, bands that appeared isolated in HA may only give rise to merged groups of bands (the decomposition of which requires mathematical treatment). Moreover, as mentioned above, nanocrystalline apatites are generally nonstoichiometric-with calcium and hydroxide vacancies-, and the limited amount of OH -ions in apatitic channels leads to added di culty in detecting such ions. However, in addition to the "regular" apatite spectral features observed in HA, nanocrystalline apatites also exhibit supplementary bands which are due to the presence of nonapatitic ionic environments within a surface layer on the nanocrystals, as mentioned in the introduction section; bands due to nonapatitic (labile) HPO 4 2-and PO 4 3-as well as CO 3 2-ions have in particular been identi ed [ , , ,

, -], and spectral decompositions are then needed to explore in further details each vibrational contribution (see e.g., [ ]). Also, in addition to nonapatitic HPO 4 2-ions, some apatitic HPO 4 2-ions are generally present in the lattice.

From a "practical" viewpoint, some modi cations of IR spectra can be identi ed in the case of nanocrystalline biomimetic apatites-as compared to HA-due to the presence of HPO 4 2-ions associated with the apatite crystals, as indicated by arrows in Figure . is includes ( ) a broad shoulder on the low wavenumbers region of the ] 4 (PO 4 ) band (shoulder typically expanding from to cm -1 that in fact comprises two subcomponents centered around and cm -1 assignable, respectively, to nonapatitic and apatitic HPO 4 2-) and ( ) a shoulder on the high wavenumbers region of the ] 3 (PO 4 ) band (visible on spectra by a burst around cm -1 ). A thin band at cm -1 is also detected for HPO 4 -bearing apatites due to the P-OH stretching. However, if the apatite is partly carbonated (which is noticeable for instance from a broad ] 3 (CO 3 ) absorption band in the region -cm -1 ), then the HPO 4 band at cm -1 becomes almost superimposed with the ] 2 (CO 3 ) band (around cm -1 for B-type carbonate ions), making it di cult to exploit.

e above statements indicate that several parameters may allow one to distinguish apatites from other calcium phosphates. Similarities, however, do remain between the spectral features of nanocrystalline apatites and precipitated OCP, which can be attributed to the above-mentioned structural relationship. Such similarities are even more obvious when analyzing fresh precipitates; indeed, in these conditions the nonapatitic layer on the surface of the nanocrystals is likely to remain rather intact (i.e., not altered by drying processes), which gives rise to a better-de ned IR spectrum than for dried specimens (Figure ). However, a closer look at the spectra, especially in the ] 3 (PO 4 ) region, indicates that two distinct bands at ca. and cm -1 (assignable to HPO 4 2-ions in the OCP lattice con guration) are detectable for OCP but not for apatite. ese di erences may then be considered as other elements for distinguishing between these two phases by IR spectroscopy.

Finally, additional identi cation information can come from the possible presence of carbonate vibration bands, the F : Characteristic FTIR spectra ( -cm -1 range) for a well-crystallized stoichiometric hydroxyapatite (HA), a nanocrystalline apatite (matured day, freeze dried), octacalcium phosphate (OCP), amorphous calcium phosphate (ACP), monetite (DCPA), and brushite (DCPD).

position of which has been cited above. Indeed, while the apatite structure can accommodate carbonate CO 3 2-ions, this is not an option for OCP. e detection of such carbonate bands can then be taken as an additional con rmation of apatite formation. Vibrational spectroscopies, and in particular FTIR, thus represent particularly well-suited tools for phase identi cations in the Ca-PO 4 -H 2 O system, especially for con rming or not the "apatite" nature of a given sample.

. . . Chemical Analyses.

e detection of calcium and phosphorus by EDX analyses has been mentioned previously. e oxidation state of phosphorus in orthophosphate ions used in starting salts or in phosphoric acid is +V; since highly reductive conditions are generally not found for the type of experiments of interest in this paper, this oxidation state is bound to remain unchanged; the observation of phosphorus by EDX may be considered in such cases as a mark of the presence of phosphate ions. e titration of phosphate ions can, however, be performed more accurately using a chemical method, for example, via visible spectrophotometry (using the phosphovanadomolybdenum complex formed in acidic conditions) [ , ]. Other techniques may also be used to this end, such as ICP-AES, but our experience shows that levels of phosphate ions obtained by this technique o en tend to be underestimated and the analysis of a good reference sample such as calcined stoichiometric HA should probably always be concomitantly done for cross-checking and potentially applying data correction.

e calcium content of the formed compound can be reached without signi cant issues, for example, either by complexometry (e.g., with EDTA), via atomic absorption, or ICP techniques. If XRD, SEM, and FTIR data converge to show that the sample is single phased and apatitic in nature, then these titrations may allow the Ca/P molar ratio of the specimen to be determined. However, the apatite structure is well known for its ability to host a great number of di erent substituents that are also o en present in precipitating or immersion media such as sodium (Na + ) or magnesium (Mg 2+ ) ions. us, the Ca/P ratio may not be the best parameter to evaluate the overall nonstoichiometry of the apatite phase. Indeed, if other cations are also present in the structure, then the "cations/P" ratio should be evaluated instead. And this also holds if anionic species are incorporated in phosphate sites (such as B-type carbonate ions whose presence may be detected by FTIR analyses, as mentioned above). erefore, apatite's overall nonstoichiometry should be evaluated from a more general consideration of cations in calcium sites over anions in phosphate sites, which I propose to denote as Ca * /P * . is nonstoichiometry parameter should, however, only be taken as an informative datum, since (as we mentioned previously) nanocrystalline apatites may exhibit Ca * /P * ratios in a wide range, typically from ca. . up to . . erefore, chemical titrations appear as an interesting additional tool for the detailed characterization of the apatite phase obtained, but its apatitic nature should be previously con rmed by other techniques, as illustrated above (XRD, SEM, vibrational spectroscopy, and/or solid state NMR as is discussed below).

. . . Solid State NMR. Although it is not always available on site, solid state NMR remains another helpful technique for the exploration of physico-chemical characteristics of calcium phosphate phases. 1 H and 31 P MAS-NMR spectra were found to be particularly attractive for gaining information on the local chemical environment of phosphates ions in such compounds (see, e.g., [ , , ]).

In the case of stoichiometric HA, 31 P MAS-NMR data show a unique peak around . ppm (with respect to H 3 PO 4 ), which corresponds to apatitic PO 4 3-ions [ ]. In contrast, for nanocrystalline apatites, additional contributions were observed, taking the form of a dissymmetric peak with a maximum (for dried samples) around . ppm (wet samples exhibit a peak maximum at greater elds, around . ppm [ ]) and a shoulder at . ppm, which was related to HPO 4 2ions associated with the apatite phase (Figure ). e position of this shoulder is indeed similar to that of HPO 4 2-ions in brushite [ ]. However, in contrast to FTIR data for which some similarities were found between nanocrystalline apatite and OCP (see FTIR subsection above), it may be noted that there are no direct similarities between the NMR characteristics of OCP and apatite. Indeed, OCP is instead characterized by a very distinct 31 P pro le, with peaks at -. , , . , and . ppm [ ]. ese signi cant di erences can therefore be exploited for phase identi cation purposes.

Finally, proton 1 H NMR spectra can also be useful for apatite phase identi cation due to the presence of a characteristic peak at . -. ppm (relative to TMS) attributable to apatitic OH -ions [ , , ].

e solid state NMR technique therefore appears as another interesting tool, especially for distinguishing between OCP and apatite.

Experimental Section

. . Compounds Synthesis. All reagents used in this study were reagent grade.

Stoichiometric hydroxyapatite (HA), Ca 10 (PO 4 ) 6 (OH) 2 , was prepared by adding dropwise and underre ux a solution of diammonium hydrogenphosphate into a boiling solution of calcium nitrate, in stoichiometric proportions and in ammoniated conditions. Note that for this preparation, decarbonated water (previously boiled up) was used throughout the process. A er hours (at pH∼ ), the precipitate was ltered, thoroughly washed with deionized water, and dried at ∘ C prior to calcination at ∘ C for h.

e nanocrystalline apatite samples studied in this work were prepared by precipitation from mixing aqueous solutions of diammonium hydrogenphosphate in excess ( . M) and calcium nitrate ( . M), at ∘ C and at pH = . close to the physiological value. e excess of phosphate ions was used to provide an internal pH bu er without any addition of external bu er agent. A er rapid mixing ( min), the precipitates were le to mature in solution for day and then ltered, washed, and freeze dried. Brushite, CaHPO 4 ⋅ H 2 O, was synthesized via precipitation by stoichiometric addition of an ammoniated calcium nitrate solution into a solution of ammonium dihydrogenphosphate, followed by a -hour maturation stage. e precipitate was then ltered, washed, and dried at ∘ C overnight.

Monetite, CaHPO 4 , was obtained by heat treatment of brushite (at ∘ C) in air.

Amorphous calcium phosphate (ACP), am-Ca 3 (PO 4 ) 2 , was prepared by rapidly adding an ammoniated solution of calcium nitrate into an ammoniated solution of diammonium hydrogenphosphate, followed by a direct ltration, washing with ammoniated water, and freeze drying for hours.

Octacalcium phosphates (OCP, triclinic), Ca 8 (PO 4 ) 4 (HPO 4 ) 2 ⋅ H 2 O, was obtained by hydrolysis, under stirring, of brushite crystals in the presence of an identical weight of ammonium hydrogenphosphate, at ∘ C for hours. is was followed by a step of ltration, washing, and freeze drying. e sample was then conserved at low temperature (-∘ C) to avoid any evolution.

When appropriate (mentioned in the text), simulated body uid (SBF) and -times concentrated SBF ( × SBF) were prepared by following Kokubo's protocol [ ] or by multiplying all concentrations by a factor of , respectively.

. . Physicochemical Characterization. X-ray di raction (XRD) analyses were carried out on an INEL CPS curved counter di ractometer using the CoK 1 radiation (with = . Å). Fourier transform infrared (FTIR) spectra were obtained on a Nicolet spectrometer, using the KBr pellet method, in the range -cm -1 ( scans, resolution cm -1 ). SEM-EDX analyses were performed on a LEO VP microscope operated at -kV.

Conclusions

is review was aimed at remembering the necessity and the means to carry out adequate physico-chemical characterization prior to drawing conclusive results on the apatitic nature of a given sample or deposit.

Several possible reasons for experimental failure were brought up in the rst part of this paper, where various aspects that are not always taken into account were underlined (possible experimental mistakes, poor washing step, evolution of a substrate in solution, contamination of a medium by microorganisms, residual presence of precursor phases, etc.).

e second part of this review considered the fundamentals of apatite characterization based on various complementary techniques. Even in thermodynamically adequate conditions, some kinetic aspects or other factors-such as the presence of a stabilizing impurity in the medium-could mislead the experimentalist and give rise to unexpected results. In the case of immersion tests in supersaturated solutions, for example, the formation of a white deposit, even characterized by a plate-like morphology, does not necessarily indicate the formation of apatite. Characterization tools like XRD, FTIR/Raman, or solid state NMR prove to be especially well suited to reliably identify apatite compounds, including distinguishing between OCP and nanocrystalline apatite.

All that glitters is not gold. . . all that is white is not apatite either.

  EDX observations for a TA V disk immersed in SBF × for days at RT-role of washing. T : Saturation index relative to various phases of interest and considering as initial conditions the ionic concentrations of blood plasma reported in Table .

  -like morphology (SEM) for biomimetic apatite (example obtained a er immersion of a titanium-based disk in Kim's supersaturated uid [ ] at ∘ C for days then ∘ C for days) and corresponding EDX spectrum. relative to CoK 1 wavelength) 2 ∘ (relative to CuK 1 wavelength)F: Characteristic XRD patterns for a well-crystallized stoichiometric hydroxyapatite (HA), a nanocrystalline apatite (matured day), octacalcium phosphate (OCP), amorphous calcium phosphate (ACP), monetite (DCPA), and brushite (DCPD) (relative to either CoK 1 or CuK 1 ).

  FTIR features for nanocrystalline apatite (dried and undried) and OCP in the ] 3 (PO 4 ) vibration range cm -1 . 31 P MAS-NMR spectra characteristic of nanocrystalline apatites (wet or lyophilized), OCP, and DCPD (reprinted with permission from [ ]).

  Chemical composition of blood plasma (inorganic species) from Krebs [ ], at pH . and ∘ C, and speciation data (calculated using the Visual Minteq . so ware). Using Davies' equation for calculation of activity coe cients (total ionic strength ∼ . M).

		Main ion types	Average global concentration (mM)	Main species (and % of total concentration * )	Corresponding activities * * (mM)
		Sodium	.	Na + ( %)		.
		Chloride		Cl -( %)		.
		Carbonate		HCO -( %) H CO (aq) ( %)	.	.
		Potassium	.	K + ( %)	.
				Ca + ( %)	.
		Calcium	.	CaHCO + ( %)	.
				CaCl + ( %)	.
				HPO -( %)	.
		Phosphate (inorganic)		NaHPO -( %) H PO -( %)	. .
				CaHPO (aq) ( %)	.
				Mg + ( %)	.
		Magnesium	.	MgCl + ( %)	.
				MgHCO + ( %)	.
		Sulfate (inorganic)	.	SO 2-( %) NaSO 4 -( %)	. .

T : *

For species in amount ≥ %. * *

Table reports

 reports

	the typical vibrational features of PO 4	3-

and OH -ions in well-crystallized stoichiometric HA, and Figure reports characteristic FTIR spectra for various calcium phosphates of interest in this study (obtainable in wet conditions)

  FTIR characteristic bands (PO 3-and OH -) for stoichiometric HA.

	Band position (cm -) (± cm -)	Hydroxyapatite (HA) Ca (PO ) (OH) Relative intensity	Attributions
			PO 3-groups (apatitic)
	(isolated)	very weak	] (PO ): OPO bending
	and	(merged together)	moderate	] (PO ): OPO bending
	(isolated)	moderate	] (PO ): OPO bending
	(isolated)	weak	] 1 (PO ): PO stretching
	,	(both isolated)	intense	] (PO ): PO stretching
			OH -groups (apatitic)
	(shouldered to ] (PO ))	moderate/weak	] (OH): OH libration
	(shouldered to water O-H)	moderate/weak	] (OH): OH stretching

T :

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-, pp. -, .


Le Cui

Eric Marchand

Sinan Haliyo

Stéphane Régnier

Hybrid Automatic Visual Servoing Scheme using Defocus Information for 6-DoF Micropositioning

Direct photometric visual servoing uses only the pure image information as a visual feature, instead of using classic geometric features such as points or lines. It was demonstrated efficiently in 6 degrees of freedom (DoF) positioning. However, in micro-scale, using only image intensity as a visual feature performs unsatisfactorily in cases where the photometric variation is low, such as motions along vision sensor's focal axis under a high magnification. In order to improve the performance and accuracy in those cases, an approach using hybrid visual features is proposed in this paper. Image gradient is employed as a visual feature on z axis while image intensity is used on the other 5 DoFs to control the motion. A 6-DoF micro-positioning task is accomplished by this hybrid visual servoing scheme. The experimental results obtained on a parallel positioning micro-stage under a digital microscope show the robustness and efficiency of the proposed method.

I. INTRODUCTION

Automatic and reliable positioning, handling and assembly of the micro-structures have shown their potential in fabrication of micro-structures and microelectromechanical systems (MEMS) over the past few decades. In the micro-scale, the prevailing spatial sensing technique is vision, through optical or electron microscopy. Visual servoing is hence an unavoidable tool to automate manipulation tasks [START_REF] Pawashe | Two-dimensional vision-based autonomous microparticle manipulation using a nanoprobe[END_REF], [START_REF] Fatikow | Visual feedback methods for nanohandling automation[END_REF]. The accuracy in position and orientation of a sample in the camera reference frame is one important issue [START_REF] Devasia | A survey of control issues in nanopositioning[END_REF], [START_REF] Ouarti | Robposit, a robust pose estimator for operator controlled nanomanipulation[END_REF]. Many approaches in this domain rely the observation of object features [START_REF] Sun | A visually servoed mems manipulator[END_REF], [START_REF] Ogawa | Microrobotic visual control of motile cells using high-speed tracking system[END_REF]. The pose (i.e., position and orientation) of samples is computed by estimating the position and orientation of these features. Furthermore, available imaging techniques at the micro-scale by an optical or electronic microscope exhibit several limitations such as high signalto-noise ratio, low refresh rate, contrast, etc., leading to the bottleneck of pose detection.

Recently, a novel visual servoing technique has been proposed to overcome these different issues. This approach uses the photometric information [START_REF] Collewet | Visual servoing set free from image processing[END_REF], [START_REF] Collewet | Photometric visual servoing[END_REF] as a visual feature in the image based control law. In this case, only the image intensity is needed while visual tracking, including feature extraction and motion prediction is no longer necessary. This approach is demonstrated in micro-positioning on 3-DoF (translation along x and y axes, and rotation around z 1 Le Cui and Eric Marchand are with Université de Rennes 1, Lagadic group, IRISA, Rennes, France le.cui@univ-rennes1.fr, eric.marchand@irisa.fr sinan.haliyo@upmc.fr, stephane.regnier@upmc.fr axis) [START_REF] Tamadazte | A direct visual servoing scheme for automatic nanopositioning[END_REF], and on 6-DoF with some limitations on translation along z axis [START_REF] Cui | 6-dof automatic micropositioning using photometric information[END_REF].

Particularly in micro-nano scale, the motion along z axis is difficult to observe due to microscope projection model [START_REF] Cui | Calibration of scanning electron microscope using a multi-images non-linear minimization process[END_REF].

As a microscope provides a long focal length, a motion along z axis causes only a tiny variation on the observed image. Using only image intensity information as a visual feature, the accuracy of the translation on z axis is inferior to other DoFs. Accurate depth information is an important issue for correct positioning. Many approaches in depth detection have been proposed in computer vision. One idea is to employ stereo vision and reconstruct the tree-dimensional (3D) image from several two-dimensional (2D) images [START_REF] Pouchou | 3d reconstruction of rough surfaces by sem stereo imaging[END_REF], in which the reliable extraction of features is necessary. Both extraction and feature matching could be computationally expensive. Considering focus information, depth from focus (DFF) [START_REF] Nayar | Pattern analysis and machine intelligence[END_REF] has been proposed. The basic of this method is to obtain different focus levels by adjusting the camera parameters (i.e., the distance between the lens and image plane, the focal length and the aperture radius). It involves obtaining many observations for the various camera parameters and estimating the focus using a criterion function. Since many camera parameters and observations should be considered, it is not practical in positioning. Alternatively, the depth from defocus (DFD) based approaches have been also widely discussed [START_REF] Pentland | A new sense for depth of field[END_REF], [START_REF] Favaro | Observing shape from defocused images[END_REF]. The main idea of these methods is objects at a particular distance from the lens will be focused in an optical system, whereas objects at other distances will be blurred. By measuring the amount of defocus of the object on the observed image, the depth of the object with respect to the lens can be recovered from the geometric optics. In these methods, the defocus parameters can by estimated from the image by an inverse filter in the Fourier domain [START_REF] Pentland | A new sense for depth of field[END_REF], by modeling the depth and the image as separate Markov random fields [START_REF] Bhasin | Depth from defocus in presence of partial self occlusion[END_REF], or by some spatial domain-based techniques [START_REF] Ziou | Depth from defocus estimation in spatial domain[END_REF], etc. Inspired from the DFD, considering that the image sharpness and focus vary significantly when using an optical path which provides a small depth of field, the image gradient information can be hence used for controlling z motion in a visual servoing scheme.

This study addresses a closed-loop control scheme for visual servoing using image intensity and image gradient for a micro-positioning task in 6-DoF. The proposed method is demonstrated experimentally on a 6-DoF parallel-kinematics positioning stage and a digital microscope. The manuscript is organized as follows: Section II briefly recalls the classic visual servoing and introduces the general approach. Section III describes selected visual features, image intensity and gradient. The hybrid control laws constructed with these features are presented in Section III C. Experimental results obtained on a 6-DoF parallel micro-assembly workcell are shown in Section IV.

II. CLASSICAL VISUAL SERVOING

Two distinct robot-camera relation cases exist in classical visual servoing [START_REF] Chaumette | Visual servo control, Part I: Basic approaches[END_REF]: eye-in-hand case, in which the camera is installed in the end-effector, and alternatively, the eye-tohand case, where the camera is fixed and look toward the end-effector. In micro/nano-robotics, the eye-to-hand case is generally considered since the sensor (microscope) is usually motionless.

In classical visual servoing, the objective is to minimize the error e between the current visual feature s(q) and the desired one s ⇤ : e(q) = s(q) s ⇤

The relation between the time derivative ṡ and the robot joint velocity q is given by:

ṡ = J s q (2)
where J s represents the visual feature Jacobian. With an exponential decrease of the error ė = e, with (1) and ( 2), the control law can be expressed as:

q = J + s e (3)
where is the proportional coefficient and J + s is the pseudoinverse of J s . Considering the eye-to-hand visual servoing context, the Jacobian J s can be expressed as:

J s = L s c V F F J n (q) (4)
where L s represents the interaction matrix, which links the relative camera instantaneous velocity v and the feature motion ṡ, c V F is the motion transform matrix which transforms velocity expressed in camera reference frame onto the robot frame, F J n (q) is the robot Jacobian in the robot reference frame.

To improve the robustness of algorithm, a Levenberg-Marquardt-like method is considered:

q = (H + µ • diag(H)) 1 J > s e(q) (5)
where µ is a coefficient whose typical value ranges from 0.001 to 0.0001. diag(H) represents a diagonal matrix of the matrix H = J > s J s .

III. PURE IMAGE INFORMATION BASED VISUAL

FEATURES

In visual servoing, one or more feature information, such as geometric measurements (e.g. position and orientation of interesting points) or direct image information including image gradient [START_REF] Marchand | Using image gradient as a visual feature for visual servoing[END_REF], [START_REF] Marchand | Controlling a camera in a virtual environment: Visual servoing in computer animation[END_REF] and image entropy [START_REF] Dame | Mutual information-based visual servoing[END_REF] can be extracted as visual features. In [START_REF] Tamadazte | A direct visual servoing scheme for automatic nanopositioning[END_REF], [START_REF] Cui | 6-dof automatic micropositioning using photometric information[END_REF], the photometric information is shown to be efficient for 6-DoF visual servoing. However, the accuracy of translation along z axis is much inferior to others due to the lack of observation on z axis in micro/nano imaging: At high magnifications, existing models assume an orthogonal projection hence the output image varies slowly over the object/camera distance. In this paper a hybrid approach with features extracted from the pure image information is proposed for 6-DoF visual servoing. The general idea is to use image gradient information for controlling the motion on z axis and to use image intensity information to control the other degrees of freedom.

A. Image intensity as a visual feature

First considering the intensity of all the pixels I from the pure image as the main visual feature s, the cost function is defined as: e I (q) = I(q) I ⇤

For a pixel x = (x, y) on image plane, the time deviation of x can be expressed by

ẋ = L x v. (7)
where v = (v, w) contains the relative camera instantaneous linear velocity v and angular velocity w, L x is the interaction matrix:

L x =  1 Z 0 xy (1 + x 2 ) y 0 1 Z 1 + y 2 xy x . (8)
Let I(x, t) be the intensity of the pixel x at time t, then

rI =  @I @x 0 0 @I @y , (9)
the total deviation of the intensity I(x, t) can be written as

İ(x, t) = rI ẋ + İ, (10)
where İ = @I @t represents the time variation of I. According to [START_REF] Horn | Determining optical flow[END_REF] based on the temporal luminance constancy hypothesis, İ(x, t) = 0. In this case,

İ = rIL x v = L I v. (11)
Considering the entire image, I = (I 00 , I 01 , • • • , I MN ), where M, N represent the image size:

İ = 0 B @ L I00 . . . L I MN 1 C A v = L I v ( 12
)
where İ is the variation of the whole image intensity. L I is a MN ⇥ 5 matrix that theoretically allows a control law able to compute the 5 DoFs.

B. Image gradient as a visual feature

As mentioned before, one difficulty in micro/nano vision at a high magnification is the lack of observation on z axis. Actually, for a sensor with small depth of field, it is evident that the image sharpness changes when z position changes. That leads the motion on z can be controlled according to the image sharpness. Hence, the image gradient is chosen as a visual feature to control the motion along z axis. An approach employing image gradient information in visual servoing is proposed in this part. The general idea is to consider that the image gradient varies when object position on z axis changes, where the focal length of the sensor is always a constant. Practically, for a sensor with a small depth of field, the focal length of sensor can be adjusted in order to acquire a sharp image at the desired pose. In this case, the aim is to minimize the error of image gradient between the image at current pose and the image at desired pose to move the object to the target pose.

The linear image formation model, which is commonly used in the case of optical sensors [START_REF] Nayar | Pattern analysis and machine intelligence[END_REF] is employed here. Let Z be the current position on z axis, the defocus image I(x, y, Z) at the position Z can be expressed as the convolution of a sharp image I ⇤ (x, y, Z ⇤ ) at the desired pose Z ⇤ and a defocus kernel f (x, y):

I(x, y, Z) = I ⇤ (x, y, Z ⇤ ) ⇤ f (x, y) (13)
In previous studies, such as [START_REF] Ens | An investigation of methods for determining depth from focus[END_REF], the Gaussian kernel is widely used as an approximation of defocus model by many authors. Its probability density function (PDF) can be expressed by

f (x, y) = 1 2⇡ 2 e x 2 +y 2 2 2 . ( 14
)
For a small displacement dZ on z axis, consider a proportional relation between d and dZ:

d = mdZ. ( 15
)
where m is a constant coefficient.

For an image I(x, y, Z) at Z on z axis, the square of the norm of the image gradient at a point (x, y) on the image plane is

g(x, y, Z) = krI(x, y, Z)k 2 = rI 2 x (x, y, Z) + rI 2 y (x, y, Z) (16)
Considering the square of the norm of the image gradient for the whole image as the visual feature:

G(Z) = M X x=0 N X y=0 g(x, y, Z) = M X x=0 N X y=0 (rI x 2 (x, y, Z) + rI y 2 (x, y, Z)) (17)
One goal is then to minimize the error of the current image gradient G(Z) and the desired image gradient G ⇤ (Z ⇤ ). In this case, the cost function is defined as:

e G (Z) = G(Z) G ⇤ (Z ⇤ ) (18)
The relation between the relative camera instantaneous linear velocity v z along z axis and the time variation of image gradient G is

Ġ = L G v z ( 19
)
where L G is the Jacobian (hence a scalar) which can be expressed by:

L G = @G @ @ @Z (20)
From [START_REF] Favaro | Observing shape from defocused images[END_REF], it leads to

L G = m @G @ . ( 21
)
where @G @ can be expressed by

@G @ = M X x=0 N X y=0 2(rI x (x, y) @rI x (x, y) @ + rI y (x, y) @rI y (x, y) @ ) (22)
In ( 13), the convolution can also be written as:

I(x, y) = X u X v I ⇤ (x u, y v)f (u, v). (23)
From ( 14), compute the derivative

@f (u, v) @ = 1 2⇡ (u 2 + v 2 2 2 ) 5 e u 2 +v 2 2 2 . ( 24
)
According to ( 23) and ( 24):

@rI x (x, y) @ = X u X v r(I ⇤ x (x u, y v) • 1 2⇡ (u 2 + v 2 2 2 ) 5 e u 2 +v 2 2 2 ) (25)
and

@rI y (x, y) @ = X u X v r(I ⇤ y (x u, y v) • 1 2⇡ (u 2 + v 2 2 2 ) 5 e u 2 +v 2 2 2 ) (26)
Considering ( 25) and ( 26) in [START_REF] Horn | Determining optical flow[END_REF], L G can be finally computed.

C. Control law for hybrid visual servoing

For the hybrid visual servoing, both image intensity and image gradient are considered as visual features s = (I(q), G(Z)) > . Using image intensity as a visual feature, the velocities (the linear and angular velocities on x, y axes, and the angular velocity around z axis) of the end-effector can be computed from [START_REF] Ouarti | Robposit, a robust pose estimator for operator controlled nanomanipulation[END_REF], where the Jacobian is

J I = L I c V F F J n (q). (27)
Similarly, using image gradient as a visual feature, the linear velocity along z axis is:

Ż = z L 1 G e G (Z) (28)
where z is an exponential coefficient. During the visual servoing process, the control laws for motion along z axis and the other 5 DoFs are computed respectively and simultaneously.

IV. EXPERIMENTAL RESULTS

A. Experimental setup

Experiments are accomplished on a micropositioning workcell installed on an anti-vibration table shown in Fig. 1. It contains a 6-DoF positioning-kinematics micro-stage 1 as well as its modular control system and a digital microscope 2 with an aperture-adjustable lens towards the top-plate of positioning stage. Experiments are realized on an optical magnification of 60⇥.

✓ X +/-10°1 µrad ✓ Y +/-10°1 µrad ✓ Z +/-20°1 µrad
The SmarPod positioning stage is a parallel robot (hexapod) that provides three positioners supporting a top-plate. By the motion of its positioners, the top-plate can be moved in three directions and rotated around three axes. The hexapod and the reference frame are shown in Fig. 2. Table I describes its specifications.

The specimen is a microchip which measures 10 mm⇥5 mm, with 0.5 mm in thickness. The resolution of acquired 1 SmarPod 70.42-S-HV made by SmarAct and its positioner SLC 17.20-S-HV 2 Basler acA1600-60gm

B. Validation of the method

First, the positioning stage is moved from -2.3 mm to 2.1 mm along the z axis to evaluate the variation of image gradient with respect to z position. Images are acquired at each 40 µm step. By computing the image gradient per pixel for each image, the relation between the image gradient and z position is shown in Fig. 3. It can be seen from the figure that the depth of field is small enough for an accurate positioning and a single optimum is found in image gradient.

In positioning experiments, the stage is first set to an initial pose then moved to a predefined desired pose iteratively by comparing the image at the desired pose with the image at the current pose. The focus of microscope is adjusted so that the image is focused at the desired pose.

To validate the method, the initial pose of positioning stage is set to 500 µm in x and 1 mm in y axes, 2 mm in z axis; 0.1°around x axis, 2°around z axes away from the desired pose to test the performance of the proposed method. The initial image and desired image after image processing is shown in Fig. 5(a) and Fig. 5(b), respectively. Fig. 5(c) to Fig. 5(d) shows the evolution of image intensity error e I (q) = I(q) I ⇤ until the end of visual servoing procedure. The velocities converge fast to 0. As a consequence of optimization, the error image is almost null. The experimental results are shown in Fig. 4. Since visual servoing is robust to calibration errors [START_REF] Espiau | Effect of camera calibration errors on visual servoing in robotics[END_REF], the positioning task without explicit calibration performs also quite well. The image intensity error as well as image gradient error per pixel decrease to negligible values when the velocities converge. The object pose errors between the final pose and the desired pose reach 0.65 µm, 0.47 µm and 0.17 µm in translation along x, y, z axes; 0.027°, 0.036°and 0.003°in rotation around x, y, z axes, respectively. It can be mentioned that in Fig. 4(c), the image gradient error increases around the 70th iteration. It is mainly because the sample is too large that cannot be presented in the whole image. When the positioning stage is moving, details of the sample on the image vary, which causes the computation of image gradient to be slightly disturbed. In experiments, the proposed visual servoing scheme shows robustness to such situations.

C. Hybrid visual servoing vs visual servoing using image intensity

Experiments have been achieved to evaluate the proposed hybrid visual servoing by comparing to an approach using only image intensity [START_REF] Cui | 6-dof automatic micropositioning using photometric information[END_REF] in the same conditions. The initial pose of the positioning stage is set to be 2 mm on z axis and 2°around z axis away from the desired pose. The positioning task based on the proposed hybrid visual servoing and image intensity based visual servoing are accomplished respectively. The evolution of joint velocity are illustrated in Fig. 6 and Fig. 7. The positioning error on translation along z axis with hybrid visual servoing is 0.26 µm, which is smaller than with the image intensity based visual servoing (1.66 µm), where also more iterations are needed to converge. On other DoFs, both these two methods perform equivalent, where the pose errors are less than 0.6 µm in translation along x and y axes, less than 0.01°in rotation around x, y and z axes. Furthermore, because of the limited travel range of the positioning stage on z axis, the initial pose on z cannot be extremely far away from the desired pose. Indeed, in that case, the image-intensity-only method fails to converge because little details can be extracted from the initial blur image. However, the hybrid visual servoing performs well since the motion on z axis can be conducted even the image is heavily blurred.

D. Robustness to light variation

As the proposed method uses photometric information, the sensibility to variable light conditions is indeed an important issue. Therefore, the robustness to light variation of the proposed method is tested. The initial pose of the stage is also set to be 2 mm on z axis and 2°around z axis. Fig. 8 shows the evolution of joint velocity. The luminance of the environment light is changed suddenly at the 8th iteration. Oscillations in velocities appear, caused by the lighting changes. However, the convergence and the accuracy are unaffected in spite of the changing light. The system keeps stable for a small perturbation occurring during the positioning task.

V. CONCLUSION In this paper, a hybrid visual servoing scheme is proposed for an accurate and robust 6-DoF micropositioning. Differently from traditional visual tracking and object localization approach, only pure image photometric information is needed in this method. The image intensity information is employed to control the linear motion along x, y axes and the angular motion around x, y, z axes. To improve the precision of positioning on z axis, image gradient is introduced as a new visual feature according to the variation of image sharpness due to the motion along z axis. This method is validated by experiments on a 6-DoF parallel positioning stage and a digital microscope. Although the initial pose is far away from the desired pose (2 mm in translation and 2°i n rotation), the results are accurate (pose errors are below 1 µm in translation, below 0.04°in rotation around x, y and 0.003°in rotation around z). Future work will be to modify the sensor projection model and to validate this approach with a scanning electron microscope.

Fig. 1 .Fig. 2 .

 12 Fig. 1. The micropositioning workcell

Fig. 3 .

 3 Fig. 3. Image gradient per pixel with respect to z position

Fig. 4 . 6 -

 46 Fig. 4. 6-DoF positioning using hybrid visual servoing (a) Evolution of joint velocity (in mm/s and rad/s). (b) Evolution of object pose error (in µm/s and degree). (c) Evolution of image intensity error and image gradient error per pixel. (d) Object trajectory in camera frame

Fig. 6 .Fig. 7 .

 67 Fig.6.Positioning using only image intensity based visual servoing: Evolution of joint velocity (in mm/s and rad/s)

Fig. 8 .

 8 Fig. 8. Positioning using hybrid visual servoing with lightning perturbation: Evolution of joint velocities (in mm/s and rad/s)

ACKNOWLEDGMENT

The authors would like to thank Soukeyna Bouchebout, Mokrane Boudaoud and Jean Ochin Abrahamians for their help during the experiments at ISIR, Université Pierre et Marie Curie. The positioning stage is supported by French Robotex platform. This work was realized in the context of the French ANR P2N Nanorobust project (ANR-11-NANO-0006).


Direct visual servoing based on multiple intensity histograms

Quentin Bateux, Eric Marchand Abstract-Classically Visual servoing considered the regulation in the image of a set of visual features (usually geometric features). Recently direct visual servoing scheme, such as photometric visual servoing, have been introduced in order to consider every pixel of the image as a primary source of information and thus avoid the extraction and the tracking of such geometric features. Previous works proposed methods to use directly the image intensities in the definition of the control law, by using for example mutual information. In this paper, we propose a method to extend these works by using a global descriptor, namely intensity histograms, on the whole or multiple sub-sets of the images in order to achieve control of a 6 degrees of freedom (DoF) robot. The results are then demonstrated through experimental validations.

I. INTRODUCTION

The goal of visual servoing is the control of the dynamic of a system by using visual information provided by one or multiple cameras [START_REF] Chaumette | Visual servo control, Part I: Basic approaches[END_REF]. The goal is to regulate an error defined in the image space to zero. Usually, as presented in [START_REF] Hutchinson | A tutorial on visual servo control[END_REF][2] [START_REF] Chaumette | Visual servo control, Part II: Advanced approaches[END_REF] this error, to be minimized, is based on visual features that correspond to geometric features (points, lines,...) and requires extraction and matching of those features in the current and desired image. This real-time spatio-temporal tracking of these visual features [START_REF] Marchand | Feature tracking for visual servoing purposes[END_REF] is a non trivial task and also one of the bottlenecks of the expansion of visual servoing.

Recently, to avoid this tracking and matching process, direct approaches have been proposed [START_REF] Collewet | Photometric visual servoing[END_REF], where the feature matching and extraction step has been removed. Within this method, the control is built by minimizing the error between the current and desired images. Only pure photometric information (i.e., image intensity) is considered here. Although photometric visual servoing [START_REF] Collewet | Photometric visual servoing[END_REF] is very efficient and very precise it may suffer from several drawbacks in non controlled environment. Since this is a pixel-wise comparison between the images it may be sensitive to illuminations variations and occlusions... leading to less precise positioning results. A way to improve this method is to use a global approach in order to work with a more compact representation of the visual information. To solve these problems, [START_REF] Delabarre | Visual servoing using the sum of conditional variance[END_REF] proposes to dynamically adapt the reference image to the illumination conditions of the current image acquired by the camera. The reference image is then replaced by an expected image computed using a probabilistic expectation operator (that take into account illumination condition in both current and learned desired image). Another direct method [START_REF] Dame | Mutual information-based visual servoing[END_REF] considered the mutual information as the similarity measure between Quentin Bateux and Eric Marchand are with Université de Rennes 1, IRISA, Inria Rennes, France, Quentin.Bateux@irisa.fr, Eric.Marchand@irisa.fr desired and current images. This information theoretic approach allows to servo the camera in order to maximize the quantity of information shared by the current and desired image (that is the mutual information which is built by a measure of image entropy).

In this paper we propose to represent the image by intensity histograms and to build a control based on such visual features. Histograms have been widely considered in computer vision for tracking, matching, detection or images retrieval. In tracking, the Mean Shift tracking algorithm [START_REF] Comaniciu | Kernel-based object tracking[END_REF] and its sequel proved the feasibility of an histogram-based tracking scheme. A color histogram is used to describe the region to track. Based on the Bhattacharyya coefficient, tracking is accomplished by iteratively finding the local minima of the distance measure functions using the mean shift algorithm. Only histogram computation is considered. An extended version of the mean-shift, considering histogram and based on non-linear optimization, has also been proposed in [START_REF] Hager | Multiple kernel tracking with ssd[END_REF]. In shape recognition or detection and image indexing, the use of methods that consider histogram based descriptors have benefited greatly to some applications. Histograms of gradients have been widely considered in key-points detection and matching method such as Sift [START_REF] Lowe | Distinctive image features from scale-invariant keypoints[END_REF] of pedestrian detection such as in [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF]. Image indexation or retrieval algorithms, such as "Gist" [START_REF] Oliva | Building the gist of a scene: The role of global image features in recognition[END_REF], also consider histogram to globally describe the image.

In this paper, we extend the direct visual servoing approach by using gray-level intensity histograms. The basic idea is to compute the histogram of the the images acquired for the current and desired pose and control the camera in order to minimize an adequate distance between these two histograms. The idea is to use for our features a distance measure between histograms computed on both the images taken on initial and desired poses. This approach allows us to benefit from the properties of histograms, such as increased robustness to noise, and flexibility by choosing the number of classes considered in the computation of the histogram.

This paper is structured as follows. The next section provides elements in visual servoing, Section 3 develops the structure of the minimization algorithm involved in the presented method. Then Section 4 and 5 will present respectively the theoretical basis of the presented method, and then the experimental developments that validate the method.

II. DIRECT VISUAL SERVOING

A. Positioning Task

The aim of a positioning task is to reach a desired pose of the camera r * , starting from an arbitrary initial pose. To achieve that goal, one needs to define a cost function that reflects, in the image space, this error. Most of the time this cost function f is an error measure which needs to be minimized. Considering the actual pose of the camera r the problem can therefore be written as an optimization process:

r = arg min r ρ(r, r * ) (1)
where r, the pose reached after the optimization process (servoing process), is the closest possible to r * (optimally r = r * ). For example, considering a set of geometrical features s, the task will typically have to minimize the difference between s(r) and the desired configuration s* which leads to:

r = arg min r (s(r) -s * ). (2)
This visual servoing task is achieved by iteratively applying a velocity to the camera. This requires the knowledge of the interaction matrix L s of s(r) that links the variation of ṡ to the camera velocity and which is defined as:

ṡ(r) = L s v ( 3
)
where v is the camera velocity. This equation leads to the expression of the velocity that needs to be applied to the robot. The control law is classically given by:

v = -λL + s (s(r) -s * ) ( 4
)
where λ is a positive scalar and L + s is the pseudo-inverse of the interaction matrix.

As pointed out in [START_REF] Collewet | Visual servoing set free from image processing[END_REF], such control is very similar to a Gauss-Newton minimization method which is widely considered to solve non-linear optimization problems. In this approach, the determination of the direction of descent will affect widely the performances of the servoing, as it will affect the radius of convergence of this method. In order to keep as much flexibility as we can, we will alternatively consider a control inspired from the Levenberg-Marquardt method, which can be expressed as:

v = -(H + µ diag(H)) -1 L s (s(r) -s(r * )) (5)
with H = L s L s where µ is a gain determining the behavior of the control law, from a steepest gradient behavior (µ small) to a Newton's behavior (µ large).

III. HISTOGRAM-BASED VISUAL SERVOING

Histograms have been considered as a powerful way to provide a global description of a set of values, especially for tracking applications or image indexing where they have been used to perform comparison between two images (or part of images) in a global way. Since pixel-wise comparison (SSD) may not be robust enough in case of illumination offset or noise, such global descriptor appears to be an interesting choice.

A. Histogram: definition and overview

The main idea behind histogram computation is to classify each pixel of the image into a corresponding bin, according (in the case considered in this paper) to the intensity value of this pixel. Each bin consists of a scalar value that is incremented each time a corresponding element is found, and once each pixel of the image has been tested, the bins values will be normalized according to the number of pixels in the image. This leads to the following classical expression of an intensity histogram:

p I (i) = 1 N x Nx x δ(I(x) -i) (6)
where x is a 2D pixel position in the image plane, the pixel intensity i ∈ [0, 255] if we use images with 256 gray-levels, N x the number of pixels in the image I(x), and δ(I(x) -i) the Kronecker's function defined such as:

δ(x) = 1 if x = 0 0 otherwise (7)
p I (i) is nothing but the probability for a pixel of the image I to have the intensity i.

B. Distance between histograms

As stated, our goal here is to find a way to use such histogram as a visual feature within a visual servoing control law. In order to use intensity histograms as our cues, we need to be able to compare them with each other. As defined in equation ( 1) a correlation function ρ(.) has to be defined. The traditional way of doing this is to compare the histograms bin-wise, by using for example the Matusita distance, which can be expressed as [START_REF] Cha | On measuring the distance between histograms[END_REF]:

ρ(I, I * ) = Nc i p I (i) -p I * (i) 2 (8)
where N c is the number of bins considered in the histograms.

C. Computation of the interaction matrix

The main problem that arises with this formulation of the histogram is that the Kronecker's function (δ(.) used in equation ( 6)) is non-differentiable, which will prove problematic for our purposes, since we need to compute the interaction matrix [START_REF] Chaumette | Visual servo control, Part I: Basic approaches[END_REF] (or image Jacobian [START_REF] Hutchinson | A tutorial on visual servo control[END_REF]) for this new error function (the Matusita distance given in equation ( 8)). To solve this problem, as in [START_REF] Dame | Mutual information-based visual servoing[END_REF], we choose here to use Bsplines of at least second order (which can be differentiated once) to approximate and smooth our bins , as illustrated in Fig. 1 . This technique yields the following formulation of an histogram:

p I (i) = 1 N x Nx x φ( Ī(x) -i) (9)
where N x is the number of pixels, x = {x, y} is a single image pixel, φ(.) is a B-spline differentiable once, Ī is the image reduced to N c intensity values (each histogram will then contain N c bins). For 8 bits images, we have

Ī(x) = N c 255 I(x).
A detailed description of B-Spline functions is given by Unser et al. in [START_REF] Unser | B-spline signal processing: Part i-theory[END_REF] but here are interesting properties of B-Spline in this case: the integral of the function is 1, so the result does not have to be re-normalized and the computation of the derivatives is easily obtained. In the following experiments a B-Spline of order 2 as been selected:

φ(t) =      t + 1 if t ∈ [-1, 0] -t + 1 if t ∈ [0, 1] 0 else (10)
As shown in the previous part, the interaction matrix can be seen as the Jacobian of the feature vector according to each pose parameter r = {t x , t y , t z , r x , r y , r z }. This definition leads to the following expression: L ρ = ∂ρ(I,I * ) ∂r where I * and I are the images recorded at the desired pose and at the current pose respectively, and ρ(I, I * ) the Matusita distance between the two histograms.

Using chain rule derivation, we can obtain the following expressions:

L ρ = ∂ ∂r Nc i pĪ(i) -pĪ * (i) 2 (11) = 2 Nc i ∂ ∂r pĪ(i) pĪ(i) -pĪ * (i) with ∂ ∂r pĪ(i) = 1 2 pĪ(i) ∂pĪ(i) ∂r (12)
where

∂pĪ(i) ∂r = ∂ ∂r Nx x φ( Ī(r, x) -i) = Nx x ∂ ∂r φ( Ī(r, x) -i) (13)
and from [START_REF] Dame | Mutual information-based visual servoing[END_REF],

∂ ∂r φ Ī(r, x) -i = ∂ ∂i φ Ī (r, x) -i ∂ Ī (r, x) ∂r = ∂ ∂i φ Ī (r, x) -i LĪ (14)
with, from [START_REF] Collewet | Visual servoing set free from image processing[END_REF], the interaction matrix that links the variation of the image intensity to the camera velocity is given by: LĪ = -∇ x ĪL x + ∇ y ĪL y [START_REF] Unser | B-spline signal processing: Part i-theory[END_REF] where L x and L y are the lines corresponding to the x and y-coordinates of L x is the interaction matrix corresponding to a single 2D point x = (x, y) in the image, and is defined in [START_REF] Chaumette | Visual servo control, Part I: Basic approaches[END_REF] as:

L x = -1 Z 0 x Z xy -(1 + x 2 ) y 0 -1 Z y Z 1 + y 2 -xy -x
Finally, after simplification, we get the following L ρ interaction matrix for our control law:

Lρ = Nc i   1 Nx 1 - pĪ * (i) pĪ(i) Nx x ∂ ∂i φ( Ī(r, x) -i) LĪ  

D. Using Kernels in histogram

In works such as tracking problems [START_REF] Comaniciu | Kernel-based object tracking[END_REF], [START_REF] Hager | Multiple kernel tracking with ssd[END_REF], that are related to our current problematic, a weighting kernel function as been successfully introduced in the computation of the histogram, in order to give more weight to the pixels in the center of the image and less to those on the borders, as the latter are less prone to contain relevant information. The use of this kernel can be here justified since some pixels will enter the camera field of whereas other will disappears. The kernel will then allow to give more importance to the pixels at the center of image. The histogram computation expression becomes then:

pĪ(i) = C Nx x K( x -c h )δ(i -Ī(x)) (16)
with C a normalization constant such as C =

1 Nx Nx x K( x-c
h ) , and K an isotropic kernel of bandwidth h, centered on c, the center of the image. In our application, we will take K as a gaussian kernel, expressed such as:

K( x -c h ) = e -||x-c|| 2 2 h . (17)

E. Multiple kernels extension

As expected from previous work using intensity histograms distances [START_REF] Comaniciu | Kernel-based object tracking[END_REF], processing a single histogram on the whole image fails to control more than translations on the x/y axis. A solution that as been proposed in tracking to extend the sensibility of histogram-based method to more degrees of freedom has been to use multiple kernels throughout the image, as in [START_REF] Hager | Multiple kernel tracking with ssd[END_REF]. Here we adapt this idea by dividing the image in multiple areas and associating a histogram to each area, then stacking the resulting error vectors and interaction matrix. The Fig. 2 shows an example of the application of this method using 9 kernels on the image : each sub-part will be used to compute its own histogram, and thus giving us a new distance to use as a distinct feature when we compare to the 9 histograms computed on the desired pose image. The global interaction matrix then becomes:

L ρ = L ρ1 L ρ2 . . . L ρn T (18)
where L ρ i is the interaction matrix given by the i-th histogram distance, using the control law defined in the previous section. By using enough kernels on the image, it becomes them possible to control every six degrees of freedom of the robot, as it will be shown in the next section.

After several experiments, the separation of the image into 5 horizontally and vertically (25 sub-parts) proved to be an adequate choice, as less areas can prove problematic to a good decoupling of the DoF, and more can leads to increasing the sensibility to small changes in the scene and thus to an harder determination of the parameters to be tuned (mainly regarding the determination of the gain and the number of bins in the histograms). Regarding the effect of the multiple kernel approach on the cost function, Fig. 3 shows the mapping of the cost function in the case where we use only one kernel on the whole image to calculate our histogram. It can be seen that even for the two DoF t x and t y , there is no clear minimum, which can only prevent our system to reach the desired pose. On the other hand, if we separate the image into 5 along x and y, and if we create a distance feature from each of the generated sub-parts, the cost function show a clear minimum, with a rather large valley, as seen on the Fig. 4. IV. EXPERIMENTAL RESULTS To validate the proposed approach, several experiments have been performed using a camera mounted on a 6 DOF gantry robot. Independently from the experiment, the computation time remains low. The control law is computed at video rate. A velocity is computed and sent to the robot every 40 ms for a 320 × 240 input image using a 2.4-GHz computer. All implementation are done using the ViSP library [START_REF] Marchand | Feature tracking for visual servoing purposes[END_REF]. In order to validate our approach, we have done several experiments to test the overall efficiency of this method. The goal of these experiments is always the same, we indicate a desired pose to the robot by taking the associated image, then we move the robot to another pose, and the camera velocities are computed at nearly video rate according to the proposed control law in order to reach the desired pose. Four main experiments have been performed, testing this method in nominal conditions (with 4 DoF in the preliminary simulation and 6 DoF on a real robot), in the case of nonplanar scene and in the presence of occlusions added during the experiment.

1) Simulation for a 4 DoF positioning task: For the first experiment, in order to validate the proposed control law, we consider a 4 DoF positioning task in simulation. Simulated image of a planar environment is provided by the rendering engine provided in ViSP [START_REF] Marchand | Feature tracking for visual servoing purposes[END_REF]. In this test case, the camera is set at a distance of 0.2m of the image plane, and the initial offset in the camera position was r init = (1cm, -1cm, -2.5cm, 0 • , 0 • , 10 • ). As shown in the Fig. 5, the control law succeeds to converge, allowing further testing on the more complex 6DoF test case, applying the control law to the servoing of an actual robot. 2) 6 DoF positioning task Nominal conditions: For the second experiment, we used a planar scene. The depth has been considered as a constant during all the experiments with Z = 0.80m in equation ( 16). The initial displacement with respect to the desired position (which is parallel to the scene plane) was chosen in this experiment was r init = (-2cm, 16cm, 6cm, 16 • , 1 • , 0 • ). Fig. 6 shows the results of this experiment. It is interesting to note that with this initial pose, the plane of the picture and the one of the camera are not parallel, which leads to non-uniform relative depth for the pixel of the image acquired. Although the depth is assumed to be constant, our system exhibits a correct behavior, despite this strong hypothesis in the computation of the interaction matrix. The Fig. 6 The convergence shows good properties regarding the evolution of the Matusita distance, with a satisfying exponential decrease, which can be found also in the evolution of the pose errors.

3) Non-planar scene: In this experiment, we test the sensibility that can be induced by our hypothesis of planar scene by introducing a 3D object inside the scene. The object used can be seen in Fig 7(e) and 7(f). The initial pose that was chosen in this experiment was r init = (-1cm, 0cm, -10cm, 0 • , 0 • , -19 • ), which is also a way the test the sensibility of the constant depth that is used in our interaction matrix. Fig. 7 show the result of this experiment, and we can see that all the desirable properties of the first experiment are kept, proving the robustness of our method regarding the depths approximations, In this experiment we also considered only 16 bins in the histograms, which enlarges the converging area.

4) Occlusions during positioning: In this last experiment, we introduce an object that is external to the scene during the positioning process. This introduces an significant dissimilarity between the desired image and the one that has actually to be reached. Since it introduces a significant change in the intensity levels of many pixels, it also show that the proposed method is robust to local changes. The initial pose that was chosen in this experiment was r init = (-2cm, 16cm, 6cm, 16 • , 1 • , 0 • ), the same one as in the second experiment, in order to allow comparison in the behavior. The Fig. 8(b) displays the moment when we add an object in the scene. This object can clearly be seen in the Fig. 8(f) and 8(h). We can see that the system still converges, despite noisier computed velocities and trajectories.

5) Increasing the convergence range: In order to increase the convergence area in the case of larger noises such as changes in illumination or wider occlusions, experiments has shown that using a Gaussian filtering as preprocessing step provides significant stabilization of this method in most cases, but at the cost of the final precision. Still, this could prove a valuable solution if it were to be coupled with our current technique, based on unfiltered images, inside a two- step method, where the first step could bring the system in the converging area, and then switching in order to get good final precision.

V. CONCLUSION AND PERSPECTIVES

In this paper, we have shown that it was possible to extend the current direct photometric visual servoing method by using more compact global descriptors: intensity histograms. We proposed a new control law that minimizes the Matusita distance between histograms. Interaction matrix related to this error function has been exhibited. We validated this approach for 6 DoF positioning tasks on various scene with planar and non-planar scenes. Future works will concern the elaboration of control laws based on more complex and more resilient global descriptors such as color histograms or histograms of oriented gradients [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF].

Fig. 1 .

 1 Fig. 1. Smoothing and approximating an histogram by using a B-spline

Fig. 2 .

 2 Fig. 2. Illustration of multi-kernels approach with 2 horizontal and vertical separations (9 areas)

Fig. 3 .

 3 Fig. 3. Cost function map for single kernel approach

Fig. 4 .

 4 Fig. 4. Cost function map for 25 kernels approach

Fig. 5 .

 5 Fig. 5. Simulation with 4 DoF. Camera velocities in m/s and rad/s in (a). (b) Matusita distance. (c) Positioning error: translational part of r, (d) rotational part of r.

  (a) depicts the velocities of the camera ; Fig. 6(b) the evolution of the Matusita distance ; Fig. 6(c) and 6(d) the pose errors (resp. the translational and rotational parts) between r and r * ; Fig. 6(e), 6(f) show the initial and final image acquired by the camera ; Fig. 6(g), 6(h) depicts the initial and final values of I -I * .

Fig. 6 .

 6 Fig. 6. Nominal conditions. Camera velocities in m/s and rad/s in (a). (b) Matusita distance. (c) Positioning error: translational part of r, (d) rotational part of r. (e) Initial image (128 gray levels used). (f) Desired image (128 gray levels used). (g) I -I* at initial position. (h) I -I* at the end of the motion

Fig. 7 .Fig. 8 .

 78 Fig. 7. Positioning task with respect to non-planar scene. Camera velocities in m/s and rad/s in (a). (b) Matusita distance. (c) Positioning error: translational part of r, (d) rotational part of r. (e) Initial image (16 gray levels used). (f) Desired image (16 gray levels used). (g) I -I* at initial position. (h) I -I* at the end of the motion