Sean P Ogilvie
email: s.ogilvie@sussex.ac.uk

Matthew J Large

Marcus A O'mara

Anne C Sehnal

Aline Amorim Graf

Peter J Lynch

Adam J Cass

Jonathan P Salvage

Marco Alfonso

Philippe Poulin

Aline Amorim Graf

Alice A K King

Alan B Dalton
email: a.b.dalton@sussex.ac.uk

Nanosheet-Stabilized Emulsions: Near-Minimum Loading and Surface Energy Design of Conductive Networks

Keywords: emulsions, graphene, molybdenum disulfide, liquid phase exfoliation, surface energy

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Introduction

Liquid phase exfoliation of pristine graphene and related two-dimensional (2D) materials has enabled assembly of solution-processed nanosheet networks with a broad range of electronic, electrochemical, thermal and mechanical properties. [1] While thin film applications such as printed electronics [2] and energy storage materials [3,4] are well-developed, macroscopic structures are typically limited to random networks in polymer matrices. [5] This limits the range of accessible applications and often requires high loadings to achieve the desired properties. As such, more controlled assembly techniques are required to broaden the range of possible structures and realize enhanced functional properties at low loading level.

Pickering emulsification is a versatile technique for assembling solid particles at the interface between two immiscible liquids, resulting in solid-stablized droplets of one phase in the other [6] . Clearly, for 2D nanosheets, this presents a route towards assembly of structures where the degree of exfoliation is maintained in situ, preserving high number densities of nanosheets, which act as both emulsion stablizer and functional filler. In addition, their atomically-thin nature and correspondingly high specific surface area could allow stabilisation of microscale droplets with nanoscale film thickness, potentially enabling macroscopic functionality at low nanosheet loadings. This is analogous to the segregated network approach demonstrated for carbon nanotube [7] and graphene composites [8] , recently extended to enable state-of-the-art battery electrodes [9] .

While Pickering emulsification has been studied for clays [10,11] , graphene oxide (GO) [12] , reduced GO [13] and graphitic multilayers [14][15][16] , emulsions stablized by pristine few-layer nanosheets have not yet been realized. This is likely because of the difficulty in exfoliating these materials in appropriate liquids to allow emulsification. Here, we develop a framework for understanding and design of emulsion stablized by pristine few-layer nanosheets to enable their applications including ultra-low loading functional composites and energy storage materials.

Exfoliation and emulsification

The mechanism of Pickering emulsification is that two immiscible liquids partially wet the solid stablizer such that the total interfacial energy is less than that of the oil-water interface. [6,16] This is typically achieved with a high surface tension "water" phase, most often water, and a low surface tension "oil" phase which can be any water-immiscible organic. Given the poor dispersability of pristine 2D materials in water (without surfactant, which acts to stablize the emulsion itself), the most obvious route to formation of these emulsions is exfoliation into the oil phase followed by emulsification with water as shown in Figure 1a, to produce emulsions whose orientation (whether o/w or w/o) is determined by the relative interfacial energies at the three-phase boundary shown in Figure 1b.

In order to realize emulsions stablized by few-layer nanosheets, they must be well-exfoliated before emulsification. This requires solvents which are well-matched in surface energy and Hansen parameters to the nanosheets [17] , which precludes water-immiscible organics such as chloroform, ethyl acetate and common monomers such as methyl methacrylate, and also immiscible with water, which precludes common exfoliating solvent such as N-methyl-2-pyrrolidone, dimethylformamide, acetone and most alcohols. Using this solvent selection approach, illustrated in Figure 1c, cyclopentanone (CPO) and cyclohexanone (CHO) were identified as water-immiscible solvents for direct exfoliation and emulsification, which also have relatively low boiling point to facilitate subsequent evaporation.

In practise, we find higher concentrations and stability for graphene/CHO and MoS2/CPO and hereafter use these as standard exfoliating solvents for these materials.

Nanosheet-stablized emulsions of water droplets in a continuous oil phase can be formed by addition of water to these cycloketone dispersions of few-layer nanosheets followed by simply shaking by hand. These droplets, shown in Figure 1d and 1e, are typically between 10 and 500 µm in diameter and are optically semi-transparent, indicating that the nanosheets form disordered films of <20 monolayers, confirming stabilisation of emulsion by few-layer nanosheets.

Conductive segregated networks

These nanosheet-stablized emulsions droplets represent potential building blocks of segregated networks where the templated self-assembly ensures system-scale conductivity with all nanosheets contributing to the network conductivity. As such, the relationship between droplet size and nanosheet volume fraction will inform the resultant properties of the conductive networks.

To characterise this the water-in-cycloketone emulsions were formed with fixed ratio of liquids but varying nanosheet volume fraction and average droplet diameter measured by statistical optical microscopy. Figure 2a shows the average droplet diameter as a function of volume fraction with values between 10 and 500 µm for nanosheet loadings across three order of magnitude below 1 vol.%. This can be understood in terms of a simple geometric relation equating the surface area of the nanosheets to the surface area of the droplets to give an expression relating droplet diameter and nanosheet volume fraction

〈𝑑〉 = 6𝑐 2𝐷 〈𝑁〉 𝜙 (1)
where 𝜙 is the volume fraction of the nanosheets relative to the droplet phase and 〈𝑁〉 here denotes the area-averaged film interfacial thickness as a number of monolayers, rather than the thickness of the individual nanosheets, and 𝑐 2𝐷 is the interlayer spacing in the bulk material (full derivation in Supporting Information).

It is worth noting that this model assumes constant interfacial film thickness in order for the 𝜙 -1

scaling to be realized. In practise, free exponent fitting of the data gives a value of 𝜙 -0.5 rather than 𝜙 -1 , for both graphene and MoS2, as shown in Figure 2a. This is equivalent to the interfacial film thickness increasing with loading level as 〈𝑁〉 ~ 𝜙 0.5 , as shown in Figure 2b, consistent with the interpretation that the droplets are being overcoated to multiple nanosheets' thickness. This likely arises from droplets of one liquid in another being formed at smaller droplet size than can be stablized, due to unavailability of nanosheets and/or nanosheet rigidity preventing stabilising of submicron droplets. The inferred interfacial film thicknesses take values between 5 and 50 for graphene and 0.3 and 5 for MoS2, perhaps suggesting some material or solvent influence of interfacial film formation.

Nevertheless, the robustness of the 𝜙 -0.5 scaling across both materials suggests some well-defined physics governs droplet formation and allow realisation of emulsions stablized by pristine few-layer nanosheets.

It is intuitive that network conductivity will increase with both reduction of droplet size, due to increased parallelisation of the network, and increasing nanosheet volume fraction. Interestingly, these interfacial films do not exhibit percolative behaviour typically associated with nanocomposites; there is no clear percolation threshold because reducing the volume fraction simply increases the droplet size until there is a single large droplet whose diameter is approaching the size of the container. It is interesting to note that emulsions are essentially films in low loading level limit and random composites in the high loading level limit. As such, their conductivity-volume fraction relationship can be fitted to power law scalings, as shown in Figure 2c, which are simply percolation curves, accounting for the scaling of paths in the network, but with an ultra-low near-negligible threshold.

These networks have conductivities approaching those of typical graphene-polymer composites [5] and, to the best of our knowledge, are the lowest loading levels ever reported for graphene-based conductive networks, as shown in Figure 2d.

Emulsion inks

The formation of disordered interfacial films with controllable thickness presents the possibility of dispersing nanosheets at high concentration with energetic, rather than solely kinetic, stability, highlighting their suitability as inks for deposition of thin films. As illustrated in Figure 1D, the ideal combination of properties for an emulsion ink are realized in water-in-cycloketone emulsions. In addition, nanosheet-coated water droplet sediment onto and are stable in contact with hydrophobic polymeric substrates such as PET, in contrast to on glass where they wet and spread or buoyant oil droplets with rise and burst at the air interface.

This stability of deposited water droplets on polymeric substrates confers a degree of spatial control to deposition of emulsion inks even for drop-wise manual deposition. As shown in Figure 3a, water droplets are stable on substrate until spreading and evaporation of the capping layer of solvent. The exposed graphene-coated water droplet then forms an unstable three-phase interface with the air (only stable for air-in-water), resulting in deformation, drying and collapse of the droplet onto the substrate.

By contrast with dispersions, where wetting of the substrate by the liquid results in loss of any spatial control and drying results in some degree of coffee-ring effect, the collapse of these droplets appears to minimise this effect in emulsion inks as shown in Figure 3b.

This uniform drying and spatial control of emulsions, along with the ability to prepare at higher concentrations than dispersions, facilitates drop-wise deposition of nanosheet networks by hand with greater control than drop casting or spray coating from dispersions. The deposited nanosheets form dense packed networks with macroscopic electrical conductivity as shown in Figure 3c. Interestingly, the conductivities exhibit thickness-dependent scaling as observed previously but the macroscopic non-uniformity introduced by manual depositing result in critical thicknesses of ~1 μm, cf. 50-200 nm in previous studies of vacuum filtration or inkjet printing. [18,19] Nevertheless, the measured conductivities of graphene films reach bulk-like thickness-independent values of ~3000 S/m as shown in Figure 3c. This can be attributed to the formation of dense packed networks of nanosheets illustrated by the inset SEM image in Figure 3c and AFM height image in Figure 3d. In addition, Raman mapping of individual deposited droplets shows spectra characteristic of liquid-exfoliated fewlayer graphene with 2D/G ratio associated with layer number uniform across the droplet (Figure 3e).

Furthermore, the G peak intensity is similarly uniform as shown in Figure 3f, indicating that individual emulsion droplets can be deposited without coffee ring effects which may enable inkjet printing of emulsion droplets. Indeed, these emulsions exhibit the expected non-Newtonian properties; they are found to be shear thinning with viscosity given by characteristic power law of the form 𝜂 ≈ 0.1 𝛾̇-0.58 (as shown in Supporting Information Figure S2), highlighting the possibility of tuning emulsion droplet size, oil-to-water ratio and composition to give the required viscosity and shear rate combination to allow inkjet printing of individual emulsion droplets.

Nanosheet surface energy

In order to realize the full range of applications envisaged, it will be necessary to form nanosheetstablized emulsions with liquids other than water and cycloketones. However, for the reasons illustrated in Figure 1d, it is quite challenging to use alternative solvents while retaining the high degree of exfoliation required for ultra-low loading applications. In practice, this can be achieved using a solvent transfer step based on liquid cascade centrifugation. [20] Dispersions are prepared in cycloketones as normal and subjected to further centrifugation to sediment the majority of the nanosheets, the supernatant is discarded and the sediment is redispersed into an alternative solvent of choice before immediate emulsification. This allows for production of well-exfoliated materials in solvents where this would not be possible by direct exfoliation such that few-layer nanosheet-stablized emulsions can be produced with relatively arbitrary oil and water phases.

This approach allows us to investigate emulsification of liquids with different surface tensions to modify the three-phase boundary shown in Figure 1b. Having established that graphene, MoS2 and boron nitride (BN) are capable of stabilising water-in-cycloketone emulsions, suggesting preferential wetting of the nanosheets by the cycloketone compared with the water, it was noted that less polar oil and/or water phases would be required to produce oil-in-water emulsions.

The stability and orientation (whether oil-in-water or water-in-oil) of these emulsions is determined by the three-phase boundary and associated interfacial energies and spreading coefficients. These are defined as

𝑆 𝑠𝑜 = 𝛾 𝑠𝑜 -𝛾 𝑠𝑤 -𝛾 𝑜𝑤 (2)
𝑆 𝑠𝑤 = 𝛾 𝑠𝑤 -𝛾 𝑠𝑜 -𝛾 𝑜𝑤 (3)
where 𝑆 𝑠𝑜 and 𝑆 𝑠𝑤 are the spreading coefficients for solid/oil and solid/water interfaces respectively and the subscripts of the surface energies denote the contributions as shown in Figure 1c. The criterion is typically that they must both have the same sign (positive or negative) for an emulsion to be stable, where one phase preferentially wets the solid stablizer and therefore forms the continuous phase while the other forms the droplet phase, as illustrated in Figure 4a. [6] From the definitions of the spreading coefficients, and in line with intuition, it can be shown (see Supporting Information) that phase inversion occurs at

𝛾 𝑠𝑜 = 𝛾 𝑠𝑤 (4)
where the phase which has the lowest interfacial tension with the solid will form the droplets, independently of the interfacial tension of the two phases.

While interfacial tensions between liquids can be measured, it would be preferable to understand the spreading coefficients as a continuous function of the individual and well-known surface tensions of the liquids. To facilitate this, well-established simple models for interfacial tension as a function of surface tension [16,21,22] can be employed, such as the following geometric mean model;

𝛾 𝑎𝑏 = 𝛾 𝑎 + 𝛾 𝑏 -2√𝛾 𝑎 𝛾 𝑏 (5)
For graphene and related materials, as solids, the surface entropy (and therefore surface tension) is poorly-defined and therefore it is more correct to infer the surface energy from its interaction with liquids of known surface energy [23] or by inverse gas chromatography. [24,25] As such, liquid-exfoliated graphene is understood to have a surface energy close to 70 mJ/m 2 based on good exfoliation and dispersion into solvents with surface tensions close to 40 mN/m.

As such, this inversion threshold can be further simplified, by substituting equation (5) into equation ( 4) to be given in terms of surface energies

√𝛾 𝑜 + √𝛾 𝑤 = 2√𝛾 𝑠 (6)
where lower surface energies of the liquid phases give o/w and higher surface energies give w/o, and the threshold itself is determined by the surface energy of the solid stablizer; in this case, the layered nanosheets.

In practice, this equation describes all experimental observations in terms of stability and orientation for all combinations of liquids, air and substrate interfaces and nanosheet type (graphene, MoS2 and BN), confirming the nanosheet surface energies to be close to 70 mJ/m 2 as shown in Figure 4b.

Importantly, this equation only describes all experimental results when considering surface energies (rather than tensions) as the interfacial properties are non-linearly related to individual surface properties. In addition, the same emulsion orientations are observed for graphene, MoS2 and BN suggesting they have little difference in their effective surface energies.

Importantly, using this equation, it is possible to measure the surface energy of layered nanosheets based on inversion of an emulsion by changing its composition. To perform this measurement on wellexfoliated few-layer nanosheets, cycloketone dispersions were diluted with pentane and immediately emulsified with water to determine their orientation as a function of pentane volume fraction. The surface tension of the cycloketone/pentane dispersions was measured and used to calculate bounds of the surface energy of the nanosheets based on the emulsion orientation. Inversion of these emulsions was observed to occur at a pentane volume fraction between 0.90 and 0.95, with a surface tension of ~17 mN/m, as shown in Figure 4c, corresponding to a nanosheet surface energy of 71 ± 0.5 mJ/m 2 .

These measurements illustrate that the formation of oil-in-water emulsions requires the use of either a very low surface energy oil phase and/or water phase with lower surface energy than water such as ethylene glycol. As a result, oil and water phases which yield oil-in-water emulsions tend to be poor solvents for dispersion of liquid-exfoliated nanosheets. Consequently, an alternative approach is required to allow few-layer nanosheets to be emulsified with liquid phases chosen for the given application. In practice, this can be achieved using surfactant-exfoliated nanosheets where a sufficient amount of the free surfactant has been removed by, for instance, liquid cascade centrifugation followed by redispersion into pristine deionised water, such that only surfactant bound to the nanosheets remains. This allows formation of stable emulsions and provides a route to using water as a universal carrier for nanosheet-stablized emulsions. In addition, by adding ethylene glycol to the water phase, the surface energy can be determined for a range of oil phases and fitted to yield the surface energy of the surfactant-exfoliated graphene (Figure 4d), which is found to be consistent with the solvent-exfoliated materials and indicates that emulsion formation is indeed dictated by the nanosheet surface energy.

Given the robustness of emulsification to residual surfactant, the influence of pH on emulsification was investigated. Functional groups present at the edges of pristine nanosheets would not be dissociated in disperse (to reduce polarity and improve surface energy matching) but these could be deprotonated at elevated pH, such as in emulsification with a basic solution. In practice, using a standard cycloketone dispersion and KOH solution, emulsions are found to form as oil-in-water as shown in Figure 4e indicating that the deprotonation induced between pH 9 and 10 is sufficient to increase the surface energy of the nanosheets above the threshold required to invert these emulsions, around 80 mJ/m 2 . As shown this approach can be applied for graphene, MoS2 and BN, suggesting some similarity in their edge functionalities, likely S-H and N-H groups respectively. This basic inversion can be performed with solvent-or surfactant-exfoliated nanosheets, presenting an all-waterbased approach for emulsifying few-layer nanosheets and controlling the orientation of subsequent emulsions.

Conclusions

Nanosheet-stablized emulsions represent an unexplored approach for assembly of layered materials where the combination of high surface area and functional properties have much promise for applications. Here, we have developed a framework for preparation of emulsions stablized by pristine few-layer nanosheets. Graphene-or MoS2-stablized water-in-cycloketone emulsions have been shown

to exhibit system-scale conductivity in their as-produced liquid form. Conductivities of ~10 -4 S/m at nanosheet volume fractions of ~10 -5 have been obtained, which represent the lowest loading level nanosheet-containing conductive composites ever reported. Their potential as emulsion inks is highlighted by the ability to drop-cast by hand into films with conductivities equivalent to other deposition techniques, facilitated by their high concentration and drying dynamics, providing spatial control, which would not be possible with standard dispersions.

To exploit the full potential of these emulsion structures, other compositions will be required to form polymer composites, charge separation interfaces, phase change materials, etc. For such applications, it will often be necessary to form oil-in-water emulsions where the water phase can be removed to form dry or solid structures. The orientation and stability of nanosheet-stablized emulsions can be understood in terms of the surface energies of the constituent phases and the inversion used to measure nanosheet surface energy to allow subsequent emulsion design. The use of basic conditions to promote deprotonation of nanosheet edge functionalities has been identified as an alternative approach to increase the surface energy of pristine nanosheets sufficiently to yield o/w emulsions. These results emphasise the robustness of the framework developed here to understand and design functional nanosheet-stablized emulsions and highlight their potential for a wide range of applications.

Experimental Section

Exfoliation and emulsification: MoS2 and BN powders were purchased from Sigma Aldrich. Graphite powder was provided by Zenyatta Ventures Ltd. MoS2 was subjected to an initial sonicationcentrifugation step to remove impurities and very small nanosheets; the bulk powder was added to 30 mL of cyclopentanone (CPO) at an initial concentration of 25 g/L and sonicated using a Sonic Vibracell VCX130 at 60% amplitude for 1 hour under ice bath cooling. The dispersion was centrifuged (Thermo Scientific Sorvall Legend X1 with High Conic II rotor) at 5000 g for 5 mins, the supernatant containing the impurities and very small nanosheets was the discarded and the sediment was redispersed into 30 mL of fresh CPO. Graphite and BN powders were added to 30 mL of cyclohexanone at an initial concentration of 25 g/L. The subsequent sonication step used was the same for MoS2, graphite and BN; sonication using a Sonic Vibra-cell VCX130 at 60% amplitude for 3 hours under ice bath cooling. MoS2 dispersions were centrifuged at 5000 g for 5 mins and graphene and BN dispersions were centrifuged at 5000 g for 30 mins. This typically yields dispersions of nanosheets with N<10 for all materials, as confirmed with spectroscopic metrics by UV-visible extinction spectroscopy (Shimadzu UV3600Plus spectrometer). Extinction spectroscopy was also used in conjunction with previously measured extinction coefficients to determine concentration of these dispersions. Concentrations for these processing conditions are typically ~0.1 g/L. These cycloketone dispersions can be emulsified with deionised water by transferring to silanised vials and adding water at ~1:10 by volume followed by vigourously shaking by hand to homogenise. This gives nanosheetstablized water droplets which sediment through the cycloketone continuous phase. These droplets were collected and deposited on PET to perform statistical measurements of average droplet diameter by optical microscopy (Olympus BX53-M optical microscope). In order to measure droplet size as a function of nanosheet volume fraction, the stock dispersion were diluted with cycloketone and fixed volume was emulsified with fixed volume of water to control droplet size while maintaining a fixed volume of droplets. These samples were transferred into channels milled into PTFE with copper tape contacts to allow electrical measurements using a Keithley 2600 sourcemeter. I-V characteristics were obtained and resistances normalised to channel dimensions to calculate conductivity.

Emulsion inks: Water-in-cycloketone emulsions of graphene and MoS2 were prepared as described above. Samples were deposited by onto PET substrate heated to 80 °C by manual drop casting of 0.1 mL (per pass) of densely-packed emulsion over an area of 1 cm 2 . The sheet resistance was measured using a Keithley 2600 sourcemeter after every deposition pass. Once dry, another 0.1 mL was deposited and this was repeated until optical microscopy showed the films to have nearly complete area coverage, around 5 passes. At this stage, AFM was performed using a Bruker Dimension Icon with ScanAsyst-Air probes to measure topography and determine approximate thickness per pass. For Raman mapping of deposited droplets, samples were deposited onto silicon wafers and their Raman spectra were mapped using a Renishaw inVia Raman microscope with 660 nm excitation using a x50

objective. The deposition process was repeated until the sheet resistance began to decrease with the reciprocal of pass number, indicating that the thickness-independent bulk-like conductivity regime had been reached.

Solvent transfer and emulsion inversion:

In order to prepare emulsions stablized by well-exfoliated nanosheets in solvents which are conventionally considered poor for LPE, cycloketone dispersions were subjected to further centrifugation of 10000 g for 16 hours to result in sedimentation of almost all of the dispersed nanosheets. The cycloketone supernatant was discarded and the sediment redispersed into a new oil phase such as pentane, hexane, ethyl acetate, methyl methacrylate, dichloromethane or styrene. These oil phases span the range of surface energies of water-immiscible organic solvents and are immiscible with alternative high surface energy water phases; ethylene glycol and formamide (with the exception of ethyl acetate-formamide). As such, these combinations were used to identify emulsion orientation and stability. The solvent-transferred dispersions were emulsified with ethylene glycol, formamide and water at 1:1 by volume (to ensure sufficient oil and water phase to stablize either orientation of emulsion) and their orientation determined by identifying buoyancy and/or stability on glass or silanised vials or at the air interface. These orientations were used to verify the surface energy model presented and found to be identical for graphene, MoS2 and BN emulsions whether exfoliated or bulk material was used. In order to perform the inversion experiment, a CHO dispersion was diluted to varying volume fractions of pentane and the mixed solvent dispersion emulsified with water and orientation determined. Samples between which the emulsion orientation inverted were used to calculate a range for the surface energy of the nanosheet films.

Emulsification by surfactant-exfoliated nanosheets and basic inversion. For the emulsification of surfactant-exfoliated nanosheets, dispersions were prepared using the exfoliation parameters described above on dispersions of graphene, MoS2 or BN in 0.25 g/L aqueous Triton X-100 solution, which yields a dispersion with the minimal amount of surfactant, likely bound to the sheets rather than free in dispersion. Surfactant concentration of 0.1 g/L was found to result in significantly reduced concentration, while dispersions produced by exfoliation at higher surfactant concentration required washing by vacuum filtration and redispersion in order to allow stable emulsification. For the emulsion inversion by basic deprotonation, cycloketone dispersions were prepared and emulsified with pH 13 KOH solution, diluted to yield water phases with controlled pH, resulting in formation of buoyant oil droplets in a continuous phase of the basic solution above pH 9. Surfactant exfoliation and basic inversion can also be achieved by blending aqueous surfactant dispersions of nanosheets with KOH solution followed by emulsification with an arbitrary oil phase. 𝑅 𝑡 = 𝑅 𝑗 + 1

( 1 𝑅 𝑠 + 1 𝑅 𝑑 ) (2.1)
𝑅 𝑡 = 𝑅 𝑗 + 𝑅 𝑠 𝑅 𝑑 (𝑅 𝑑 + 𝑅 𝑠 ) (2.2)
This total unit cell resistance can be related to the conductivity and dimensions of the constituent phases and subsequently equated to the conductivity and dimensions of the unit cell itself as

𝑅 𝑡 = 𝑡 𝑗 𝜎 𝑗 𝑑 2 + 1 𝜎 𝑠 𝑡 𝑠 • 1 𝜎 𝑑 𝑑 ( 1 𝜎 𝑑 𝑑 + 1 𝜎 𝑠 𝑡 𝑠 ) = 𝑡 𝑗 𝜎 𝑗 𝑑 2 + 1 𝜎 𝑠 𝑡 𝑠 + 𝜎 𝑑 𝑑 ≡ 1 𝜎𝑑 (2.3) 𝜎 = [𝑑( 𝑡 𝑗 𝜎 𝑗 𝑑 2 + 1 𝜎 𝑠 𝑡 𝑠 + 𝜎 𝑑 𝑑 )] -1 (2.4) 𝜎 = 1 𝑑 ( 𝑡 𝑗 𝜎 𝑗 𝑑 2 + 1 𝜎 𝑠 𝑡 𝑠 + 𝜎 𝑑 𝑑 ) -1 (2.5)
Depending on whether the conduction is dominated by the droplets and surfaces or the junctions, this model will be dominated by the former or latter terms respectively. Where the droplets are much more conductive than the junctions, such as for water droplets stablized by thick conductive nanosheet films in a very insulating oil phase, this leads to a decreasing conductivity with increasing loading, as observed in our previous work [1] In practise, the all-liquid emulsion networks studied in this manuscript exhibit some intermediate behaviour which can be fitted to the original model but is also functionally equivalent to a power law in the range studied, as shown in Fig. 2c.

Non-Newtonian rheology

Proof-of-concept rheological measurements were performed to demonstrate the non-Newtonian behaviour of these nanosheet-stablized emulsions. Shear-rate dependent shear stress and viscosity are shown for a representative graphene-stablized water-in-CHO emulsion in Fig. S2 exhibit the expected shear thinning behaviour with a characteristic power law scaling 𝜂 ≈ 0.1 𝛾̇-0.58 and little hysteresis. In order to be suitable for inkjet deposition, these emulsions also require a viscosity (~0.01 Pa.s) greater than that of common solvents at the shear rates applied during jetting (~10 6 s -1 ). While these water-in-cycloketone emulsions reach the viscosity required for inkjet printing at 100 s -1 , 10 4 times lower shear rate than during jetting, it it possible that viscosity will rapidly saturate at higher shear rates as shown previously for clay-stablized water-in-oil emulsions [2] . Alternatively, it may be possible to use dilute emulsions (with lower ratio of droplet to continuous phase) which are known to exhibit Newtonian behaviour with viscosity independent of shear rate [3] to ensure the desired viscosity during jetting. However, this does mean reducing the concentration of the emulsion ink and potentially using a high viscosity (likely high boiling point) continuous phase, the selection of which must also satisfy other criteria for surface energy, nanosheet dispersability, etc. A more practical alternative might be to manipulate the shear rate-dependent viscosity by controlling emulsion droplet size. It is well known that smaller droplets in a concentrated emulsion give rise to increased viscosity [3,4] which presents a route to ensure sufficient viscosity during inkjet.

Surface energy model

The stability and orientation of solid-stablized emulsions can be related to the spreading coefficients and constituent interfacial energies. The spreading coefficients for the solid-oil and solid-water are given by Based on the geometric and harmonic mean models, it can be intuitively argued that it is most easily satisfied by 𝛾 𝑜 ≪ 𝛾 𝑤 (giving large 𝛾 𝑜𝑤 ) and 𝛾 𝑠 ≈ 𝛾 𝑜 and 𝛾 𝑠 ≈ 𝛾 𝑤 (giving 𝛾 𝑠𝑜 ≈ 𝛾 𝑠𝑤 ) and the difference is small), which requires that 𝛾 𝑜 < 𝛾 𝑠 < 𝛾 𝑤 , as illustrated in Fig. 4a, although this is more challenging to demonstrate rigourously.

𝑆 𝑠𝑜 =
However, in order to explicitly state this condition, interfacial energy models are required. The orientation of an emulsion (o/w or w/o) is also determined by the spreading coefficients, i.e. whichever is more negative forms the droplet phase; o/w for 𝑆 𝑠𝑜 < 𝑆 𝑠𝑤 and w/o for 𝑆 𝑠𝑜 > 𝑆 𝑠𝑤 . As such, the point at which they are equal can be considered the inversion threshold for an emulsion. This can be simplified (by definition and without any empirical models) as

𝛾 𝑠𝑜 = 𝛾 𝑠𝑤 (4.7)
Subsequently, simple models for interfacial energies can be substituted such as [5] 𝛾 𝑎𝑏 = 𝛾 𝑎 + 𝛾 𝑏 -2√𝛾 𝑎 𝛾 𝑏 (4.8)

𝛾

Figure 1 .

 1 Figure 1. (a) Schematic diagram of emulsification process where nanosheets in water-immiscible solvent are homogenised with water to give water-in-oil emulsion and illustration of nanosheets on surface of a droplet. (b) Interfacial energies at three-phase boundary, which dictate emulsion stability and orientation. (c) Venn diagram illustrating solvent selection considerations for nanosheet-stablized emulsions. (d) and (e) optical micrographs of water-in-cycloketone droplets stablized by graphene and MoS2 respectively, scale bar 100 μm.

Figure 2 .

 2 Figure 2. (a) Droplet diameter as a function of nanosheet volume fraction for graphene and MoS2 emulsions showing 𝜙 -0.5 dependence, attributed to 〈𝑁〉 increasing with 𝜙 (b) Layer number, inferred from geometric model, as a function of nanosheet volume fraction with corresponding 𝜙 -0.5 scaling. (c) Conductivity of liquid emulsions as a function of nanosheet volume fraction. (d) Conductivity-volume fraction comparison to pristine graphene composites from the literature 5 , highlight the appreciable conductivity at ultra-low loading level in the nanosheet-stablized emulsions.

Figure 3 .

 3 Figure 3. (a) Schematic illustration of emulsion droplet deposition, drying and collapse. (b) Low magnification optical micrographs of deposited droplets on PET showing eventual areal percolation and formation of densely-packed films. (c) Conductivity of graphene film deposited from emulsion as a function of film thickness, showing scaling attributed to deposition uniformity, which reaches expected bulk-like value. Inset: Scanning electron micrograph of film cross section (false coloured), showing dense-packed nanosheet network, scale bar 1 µm. (d) Atomic force micrograph of nanosheet film confirming dense and uniform areal packing of nanosheets deposited from a single emulsion droplet, scale bar 500 nm. (e) Raman spectrum of deposited droplet, characteristic of few-layer graphene. Inset: 2D/G ratio mapped over droplet, 30 x 30 µm image. (f) Raman map of G peak intensity illustrating uniformity of deposited film, 30 x 30 µm image.

Figure 4 . 3 ) 4 )

 434 Figure 4. (a) Spreading coefficients for emulsions of graphene and water as a function of oil phase surface tension. (b) Surface tension phase diagram showing different compositions giving rise to o/w (blue) and w/o (green) emulsions which is well described by equation (6) with a surface energy of ~71 mJ/m 2 for all pristine nanosheets studied here. (c) Surface tension of oil and surface energy of graphene as function of pentane volume fraction as inversion experiment to determine surface energy, giving a value in good agreement with above measurement. (d) Volume fraction of ethylene glycol required for inversion as a function of oil phase surface tension for washed surfactant-exfoliated graphene, indicating that stabilisation is still dictated by the nanosheets. (e) Nanosheet surface energy as a function of pH of water phase, determined by pentane/CHO inversion. Inset: photograph of buoyant cycloketone droplets in water continuous phase, inverted at elevated pH, shown for graphene (left) and MoS2 and BN (right).

Figure S1 :

 S1 Figure S1: Unit cell of droplet network in simple conductivity model.

Figure S2 :

 S2 Figure S2: (a) shear stress as a function of shear rate during parallel plate rheology of representative graphene-stablized water-in-CHO emulsion. Inset: photograph of experimental setup (b) viscosity as a function of shear rate.

  𝛾 𝑠𝑜 -𝛾 𝑠𝑤 -𝛾 𝑜𝑤 (4.1) 𝑆 𝑠𝑤 = 𝛾 𝑠𝑤 -𝛾 𝑠𝑜 -𝛾 𝑜𝑤 (4.2) Where the 𝛾 𝑠𝑜 , 𝛾 𝑠𝑤 and 𝛾 𝑜𝑤 are the interfacial energies at the solid-oil, solid-water and oil-water interfaces. The above definitions can be combined to give 𝑆 𝑠𝑜 + 𝑆 𝑠𝑤 = -2𝛾 𝑤𝑜 (4.3) Since interfacial tensions/energies are positive, spreading coefficients can only have the same sign (and thereby form a stable emulsion) if that sign is negative. If both spreading coefficients are negative, the stability criteria can be expressed as 𝛾 𝑠𝑜 -𝛾 𝑠𝑤 < 𝛾 𝑤𝑜 (4.4) 𝛾 𝑠𝑤 -𝛾 𝑠𝑜 < 𝛾 𝑤𝑜 (4.5) Since 𝑆 𝑠𝑜 -𝑆 𝑠𝑤 = -(𝑆 𝑠𝑤 -𝑆 𝑠𝑜 ), one of the above equations will always be satisfied and the criterion reduces to |𝛾 𝑠𝑜 -𝛾 𝑠𝑤 | < 𝛾 𝑜𝑤 (4.6)

  𝑎𝑏 = 𝛾 𝑎 + 𝛾 𝑏 -4 𝛾 𝑎 𝛾 𝑏 𝛾 𝑎 + 𝛾 𝑏 (4.9) Incorporating the geometric mean model (Equation 4.8) into Equation 4.7 gives an expression which describes the inversion threshold of emulsions as a function of the constituent surface energies 𝛾 𝑠 + 𝛾 𝑜 -2√𝛾 𝑠 𝛾 𝑜 = 𝛾 𝑠 + 𝛾 𝑤 -2√𝛾 𝑠 𝛾 𝑤 (4.10) 𝛾 𝑜 -2√𝛾 𝑠 𝛾 𝑤 = 𝛾 𝑤 -2√𝛾 𝑠 𝛾 𝑤 (4.11) Substituting 𝛾 𝑜 = 𝑥 2 and 𝛾 𝑤 = 𝑦 2 𝑥 2 -2√𝛾 𝑠 𝑥 = 𝑦 2 -2√𝛾 𝑠 𝑦 (4.12)

	𝑥 2 -𝑦 2 = 2√𝛾 𝑠 𝑥 -2√𝛾 𝑠 𝑦	(4.13)
	(𝑥 -𝑦)(𝑥 + 𝑦) = 2√𝛾 𝑠 (𝑥 -𝑦)	(4.14)
	Cancelling (𝑥 -𝑦) gives
	𝑥 + 𝑦 = 2√𝛾 𝑠	(4.15)
	Finally, re-expressing in terms of surface energies yields
	√𝛾 𝑜 + √𝛾 𝑤 = 2√𝛾 𝑠	(4.16)


Wisama Khalil

Maxime Gautier

Philippe Lemoine

Identification of the payload inertial parameters of industrial manipulators

Keywords: Identification, inertial parameters, dynamic modeling, least squares, payload parameters

I. INTRODUCTION

Several schemes have been proposed in the literature to identify the dynamic parameters [START_REF] Atkeson | Estimation of inertial parameters of manipulator loads and links[END_REF]- [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. Most of the methods use an identification model linear in the parameters, and solve the system using least squares techniques (LS).

The experimental works have been carried out either on prototypes of laboratories or on industrial robots after replacing the industrial controller by an open loop controller specially developed for the application. In this paper, we show that new standard controllers allow now the identification of the dynamic parameters of the robot by providing the joint positions and torques for any trajectory. We show also that the classical trajectories of the manufacturer are sufficiently exciting for the identification. Very small number of papers were devoted for the identification of the payload [START_REF] Atkeson | Estimation of inertial parameters of manipulator loads and links[END_REF], [START_REF] Kawasaki | Terminal-link parameter estimation and trajectory control of robotic manipulators[END_REF]. In this paper we put the accent on the identification of the inertial parameters of the payload, but the identification of dynamic parameters of the robot will be also presented because they are needed also in some methods.

II. MODELING OF THE ROBOT RX 90

The Stäubli RX-90 robot has a serial structure with six rotational joints. Its nominal payload is equal to 6 Kg.

A. Description of the kinematics

The robot kinematics is defined using Khalil and Kleinfinger notation [START_REF] Khalil | A New Geometric Notation for Open and Closed Loop Robots[END_REF]. In this notation the link j fixed frame is defined such that the z j axis is taken along joint j axis and the x j axis is along the common normal with z j and z j+1 . The link frames are shown in Figure 1. The main advantage of using this notation is that the identifiable inertial parameters can be determined symbolically using simple closed-form rules [START_REF] Khalil | Modeling, identification and control of robots[END_REF], [START_REF] Gautier | Direct calculation of minimum set of inertial parameters of serial robots[END_REF], [START_REF] Khalil | Symbolic calculation of the base inertial parameters of closed-loop robots[END_REF]. (D3 = 0,45m and RL4 = 0,45m)

B. Dynamic identification model Different identification models have been used in robotics [START_REF] De | Parameters identification of robots manipulators via sequential hybrid estimation algorithms[END_REF], [START_REF] Gautier | Chapter 4 : Identification des paramètres des modèles[END_REF], [START_REF] Gautier | Identification of the dynamic parameters of a closed loop robot[END_REF], [START_REF] Gautier | Dynamic Identification of Robots with Power Model[END_REF], [START_REF] Gautier | Contribution à la Modélisation et à l'Identification des Robots[END_REF], [START_REF] Khosla | Parameter identification of robot dynamics[END_REF], [START_REF] Prüfer | Identification of Robot Dynamics with Differential and Integral Models: A Comparison[END_REF], [START_REF] Raucent | Identification of barycentric parameters of robotic manipulators from external measurements[END_REF], [START_REF] Liu | A Base Force/Torque Sensor Approach to Robot Manipulator Inertial Parameter Estimation[END_REF]. In the following, we use the inverse dynamic model, which was found to be the best one. It is represented for a robot with n joints by: Γ = ID(q, . q , .. q ,χ s ) (1) q, . q and .. q are the (nx1) vectors of joint positions, velocities and accelerations. Γ is the (nx1) vector of motor torques. χ s is the (Nsx1) vector of standard dynamic parameters (inertial parameters and friction parameters) of the robot.

A common friction model at non zero velocity is given by:

j f τ = j Fs sign( j q ) + j Fv j q (2)
where j q is the velocity of joint j, sign(.) denotes the sign function, Wisama KHALIL 1 , Maxime GAUTIER 2 , Philippe LEMOINE 1 1 École Centrale de Nantes 2 Université de Nantes IRCCyN UMR CNRS 6597, 1 rue de la Noë, BP 92101, 44321 Nantes Cedex 3, France Mailto: wisama.khalil@irccyn.ec-nantes.fr, maxime.gautier@irccyn.ec-nantes.fr, philippe.lemoine@irccyn.ec-nantes.fr (3)

Identification of the payload inertial parameters of industrial manipulators

We take into account j IA , j Fv and j Fs in χ s .

A particular choice of the link inertial parameters allows obtaining the dynamic model linear in relation to the dynamic parameters, such that:

Γ = Φ S (q, . q , .. q ) χ s = s i=1,N i s ∑ Φ χ s i (4)
where:

Φ S is a (nxNs) matrix, and i s Φ is the i th column of Φ S , χ s i is the i th element of χ s , χ s is the vector of standard dynamic parameters given by:

χ s = [χ s 1 T χ s 2 T … χ s n T ] T
(5) with χ s j is the dynamic parameters of joint and link j:

χ s j = [XXj XYj XZj YYj YZj ZZj MXj MYj MZj M j IAj F sj F vj ] T (6) where:
-XX j , XY j , XZ j , YY j , YZ j , ZZ j are the six components of the inertia matrix of link j at the origin of frame j; -M j is the mass of link j; -MX j , MY j , MZ j are the components of the first moments; j IA is the inertia moment for rotor and gears of actuator j.

The columns of the matrix Φ s (q, . q , .. q ) can be obtained using the recursive algorithm of Newton-Euler, which calculates Γ in terms of the same set of standard dynamic parameters, such that the i th column of Φ s is equal to: Φ s i = ID (q, . q, .. q with χ s i = 1, χ sj = 0 for j ≠ i) [START_REF] Ha | An efficient estimation algorithm for the model parameters of robotic manipulators[END_REF] To increase the efficiency of this algorithm, we use the customized symbolic technique. Moreover, this technique is convenient for the computation of the observation matrix, which is composed of the concatenation of Φ s on all the points of a given trajectory, using an array multiply operator (.* of Matlab) without using loop calculations.

C. Particularities of the robot RX 90

The RX 90 robot is characterized by: 1. There is a spring on the joint 2, which compensates the gravity torque due to the robot links. The corresponding torque is added to the torque Γ 2 of joint 2, it is calculated using the manufacturer data in terms of θ 2 .

2. There is a coupling among the joints 5 and 6 such that:

qr q K qr q K K ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (8)
Thus, the duality relation of torques gives:

c r K K c r K Γ Γ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Γ Γ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (9)
where: j qr : is the velocity of the rotor of motor j, j q : is the velocity of joint j, Γc j : the motor's torque of joint j, with coupling effect, Γr j : electro-magnetic torque of the rotor of motor j, K5 : the transmission gain ratio of axis 5, K6 : the transmission gain ratio of axis 6.

The coupling between joints 5 and 6, leads also to add the effect of the inertia of rotor 6 and new viscous and Coulomb friction parameters Fv m6 and Fs m6 to both Γ c5 and Γ c6. Thus, we can write:

Γ c5 = Γ 5 + IA 6
.. q 6 + Fv m6 . q 6 + Fs m6 sign( . q 6 ) (10) Γ c6 = Γ 6 + IA 6

.. q 5 + Fv m6 . q 5 + Fs m6 sign( . q 5 ) (11) where: Γ 5 , Γ 6 are obtained from (4), they have already the terms (IA j .. q j + Fvj . q j + Fs j sign( . q j )) for j =5 and 6

IA 5 = K5^2 JA 5 + K6^2 JA 6, IA 6 = K6^2 JA 6, JA j is the moment of inertia of the rotor j, Fv m6 and Fs m6 are the friction parameters due to coupling.

D. Calculation of the base dynamic parameters

It has been seen that some inertial parameters cannot be identified because they have no effect on the dynamic model, and some parameters are grouped with some others. The identifiable parameters, called also base inertial parameters [START_REF] Mayeda | Base parameters of manipulator dynamic models[END_REF], can be determined for the serial robots using simple closed-form rules if the Khalil and Kleinfinger notations are used to describe the robot kinematics [START_REF] Khalil | Modeling, identification and control of robots[END_REF], [START_REF] Gautier | Direct calculation of minimum set of inertial parameters of serial robots[END_REF]. For the robot RX 90 the following parameters cannot be identified, they do not belong to the base inertial parameters: -YYj, MZj and Mj for j = 1, … , 6. -XX1, XY1, XZ1, YZ1, MX1, MY1, IA1 and IA2.

The grouped parameters are:

ZZ1R=IA1 + D3^2*(M3 + M4 + M5 + M6) + YY2 + YY3 + ZZ1 XX2R= -D3^2*(M3 + M4 + M5 + M6) + XX2 -YY2 XZ2R= -D3*MZ3 + XZ2 ZZ2R=IA2 + d3^2*(M3 + M4 + M5 + M6) + ZZ2 MX2R=D3*(M3 + M4 + M5 + M6) + MX2 XX3R=2*MZ4*RL4+(M4 + M5 + M6)*RL4^2 + XX3 - YY3+YY4 ZZ3R=2*MZ4*RL4 + (M4 + M5 + M6)*RL4^2 + YY4 + ZZ3 MY3R=MY3 -MZ4 -(M4 + M5 + M6)*RL4 XX4R=XX4 -YY4 + YY5 (12) ZZ4R=YY5 + ZZ4 MY4R=MY4 + MZ5 XX5R=XX5 -YY5 + YY6 ZZ5R=YY6 + ZZ5 MY5R=MY5 -MZ6 XX6R=XX6 -YY6
A numerical method based on the QR decomposition can also be used to determine the base inertial parameters [START_REF] Gautier | Numerical calculation of the base inertial parameters[END_REF].

The dynamic model of the RX90 model is a function of 40 inertial parameters. It is also a function of 14 friction parameters: (Fvj and Fs j for j=1,…,6) and (Fv m6 and Fs m6 ). The total number of the dynamic base parameters, denoted by N b ,is equal to 54.

The dynamic model can be written using the base inertial parameters as follows:

Γ = Φ(q, . q , .. q ) χ = b i=1,N i ∑ Φ χ i ( 13
)
where Φ is obtained from Φ s by eliminating the columns corresponding to the non identifiable parameters.

III. RECALL OF THE IDENTIFICATION METHOD

We consider off-line identification of the base dynamic parameters χ , given measured or estimated off-line data for Γ , , , q q q , collected while the robot is tracking some planned trajectories.

The principle is to sample the identification model ( 13) at a sufficient number of samples i t , for i=1, ..., e n , with e (nxn ) >> N b , in order to get an over-determined linear system of e (nxn ) equations:

+ = χ ρ Y W (14)
ρ is the e ((nxn ) 1)

x vector of errors between the data in Y (measurement or estimation of the torques), and the data χ W , predicted by the model. ( , , ) W q q q is the ( e (nxn ) x b N ) observation matrix.

Y and W are obtained by grouping together the equations of each joint on all the trajectory such that:

Y = ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ Y 1 … Y n , W = ⎣ ⎢ ⎢ ⎡ ⎦ ⎥ ⎥ ⎤ W 1 … W n (15)
where Y i and W i represent all the equations of joint i.

Most of the schemes in robotics solve the system (14) using ordinary (OLS) or weighted (WLS) least squares methods [START_REF] De | Parameters identification of robots manipulators via sequential hybrid estimation algorithms[END_REF], [START_REF] Kozlowski | Modelling and identification in robotics[END_REF], [START_REF] Khalil | Modeling, identification and control of robots[END_REF]. A maximum likelihood approach proposed by [START_REF] Swevers | Optimal robot excitation and identification[END_REF], reduces in practice to a WLS method. An approach based on LMI (linear matrix inequality) tools takes into account perturbations in the observation matrix but without improving the estimation given by the WLS. To our knowledge the best experimental results have been obtained with the WLS method, which is presented in the following.

The estimation of χ is obtained as the OLS solution of the over-determined linear system (14

): χ = 2 Arg.min || || χ ρ = + W Y ( 16
)
where

+ W = T 1 T ( ) - W W
W is the pseudo inverse of W . If W is a full rank matrix the LS solution χ is unique. W numerical rank deficiency can come from two origins: -structural rank deficiency, which is solved by calculating the base or minimal parameters; -data rank deficiency due to a bad choice of noisy experimental , , q q q samples in W , which can be solved by a good planning of exciting trajectories.

In practice, the experimental measurements or estimations of ( )

i t Γ
, ( ), ( ), ( )

i i i t t t q q q
give noisy data. The matrices Y and W are perturbed and the LS solution may lead to a bias estimation if the two random matrices are not independent. Because the coefficients in the observation matrix ( , , )

Φ q q q are non linear functions of , , q q q , it is not possible to get a theoretical expression of the bias and variance. To overcome this difficulty we adopt a practical strategy which reduces bias and variance of the LS solution with twofold: -data filtering to decrease noise effect in ( 14); -closed loop identification for the robot to track exciting trajectories which ensure persistency excitation. In order to cancel high frequency torque ripple in Γ, the vector Y and the columns of the observation matrix W are both low pass filtered and decimated. This parallel filtering procedure [START_REF] Richalet | Pratique de l'identification[END_REF] can be carried out with the Matlab decimate function [START_REF] Gautier | Chapter 4 : Identification des paramètres des modèles[END_REF], [START_REF] Gautier | Dynamic Identification of Robots with Power Model[END_REF], [START_REF] Khalil | Modeling, identification and control of robots[END_REF].

Standard deviations are estimated considering W to be deterministic, and ρ to be a zero mean additive independent Gaussian noise, with standard deviation σ ρ such that [START_REF] Gautier | Chapter 4 : Identification des paramètres des modèles[END_REF]:

C ρ = E(ρ ρ T ) = σ ρ 2 I r ( 17
)
where E is the expectation operator and I r is the (rxr) identity matrix. An unbiased estimation of σ ρ can be:

σ ρ 2 = ||Y -W χ|| 2 (r -c) (18)
The covariance matrix of the estimation error is given by:

Cχ = E[(χ -χ ) (χ -χ ) T ] = σ ρ 2 (W T W) -1 (19)
The standard deviation σχ j and its relative value σχ jr % by:

σχ j = Cχ (j,j) (20)
σχ jr % = 100

σχ j | χ j | (21)
The equations of joint i will be weighted with the inverse of the standard deviation of the error calculated using the equations of joint i [START_REF] Gautier | Chapter 4 : Identification des paramètres des modèles[END_REF], [START_REF] Gautier | Dynamic Identification of Robots with Power Model[END_REF]. This weighting operation normalises the errors to give the WLS estimation of the parameters.

The WLS solution is obtained by a recursive formulation of the QR decomposition of the weighted matrix W [START_REF] Gautier | Chapter 4 : Identification des paramètres des modèles[END_REF]. Some small parameters remain poorly identifiable because they have no significant contribution in the torque. These parameters can be cancelled in order to simplify the dynamic model. Parameters such that σχ jr % is greater than a bound between 5 and 10 are cancelled to keep a set of essential parameters of a simplified dynamic model without loss of accuracy [START_REF] Gautier | Chapter 4 : Identification des paramètres des modèles[END_REF]. The essential parameters are calculated using an iterative procedure starting from the base parameters estimation. At each step the base parameter which has the largest relative standard deviation is cancelled. A new parameter estimation of the simplified model is carried out with a new error standard deviation σ ρe . The procedure ends when σ ρe ≥ 1.02 σ ρb , where σ ρb is the initial error standard deviation obtained with all the parameters.

IV. IDENTIFICATION OF THE DYNAMIC PARAMETERS OF THE ROBOT RX 90

A. Data Acquisition and identification

The identification of the dynamic parameters without load has been carried out using 18 trajectories. The path of each trajectory consists of about 14 intermediate points. The trajectory between the points is carried out using the interpolation function of the controller CS8 of the Stäubli robots. The joint positions and torques are read with a sampling frequency equal to 250 Hz. The condition number of the observation matrix of all the trajectories is equal to 58, which is considered as being sufficiently exciting. The total number of equations is about r = 63000. A supplementary trajectory is used in the validation of the estimated values of the dynamic parameters.

The estimation of . q and .. q are carried out from q by a low pass Butterworth filter and a central derivative algorithm. The Matlab function filtfilt, which is a zero-phase forward and reverse digital filtering, can be used.

After elimination of the parameters with large relative standard deviation (RSD), the robot dynamics is represented by 28 essential parameters. The estimated values of these parameters are not given here. They are close to the first 28 parameters of Table 2.

B. Validation of the identification results

The validation of the identification result is carried out by comparing the measured joint torques mes Y and the estimated torques est χ W on a trajectory which has not been used in the identification (cross validation). Owing to the limited number of pages, the validation curves are not presented in this paper. However, the identification of the payload inertial parameters presented in the following section constitutes another validation procedure.

V. IDENTIFICATION OF THE INERTIAL PARAMETERS OF THE PAYLOAD

A. Presentation of the different methods

Having estimated the dynamic parameters of the robot without load, we present in this section four methods for the identification of the inertial parameters of a payload fixed on the terminal link of the robot.

1) Making use of the values estimated without payload

When the robot is carrying a payload, the dynamic equations can be written as follows:

Y T = W(q, . q , .. q ) χ + W L (q, . q , .. q ) χ L + ρ ( 22
)
where Y T : the measured torques when the robot is loaded, W : the observation matrix without load, χ : the vector of the dynamic parameters of the robot without load (this is already estimated), χ L : the(10x1) vector of the inertial parameters of the load, W L : the observation matrix corresponding to the load inertial parameters.

The load inertial parameters are estimated from ( 22) by: χL

= (W L ) + (Y T -W χ) (23)
We note that physically the 10 inertial parameters are identifiable.

In this method we suppose that the friction parameters are invariant with respect to the payload. If this hypothesis is not satisfied these parameters should be taken into account in both W L and χ L .

2) New identification of the robot parameters with the payload

We identify once more all the dynamic parameters while the robot is carrying the payload. The inertial parameters of the load can be computed using the variation on some estimated inertial parameters with and without payload. In this case we make use of the grouping relations [START_REF] Prüfer | Identification of Robot Dynamics with Differential and Integral Models: A Comparison[END_REF]. Let ∆χ i be the variation of an inertial parameter: ∆χ i = χj (robot with payload)-χj (robot without payload) [START_REF]Theory of Robot Control[END_REF] The load inertial parameters XY L , XZ L , YZ L , ZZ L , MX L , MY L can be obtained directly as ∆XY6, ∆XZ6, ∆YZ6, ∆ZZ6, ∆MX6, ∆MY6. Using ( 12) and ( 24), the parameters XX L , YY L , MZ L and M L can be obtained by solving the following relations:

∆ZZ1R= D3^2* M L ∆XX2R= -D3^2* M L ∆ZZ2R= D3^2* M L ∆MX2R= D3*M L (25) ∆XX3R= RL4^2 * M L ∆ZZ3R= RL4^2* M L ∆MY3R= -RL4* M L ∆XX5R= YY L ∆ZZ5R= YY L (26) ∆MY5R= -MZ L (27) ∆XX6R= XX L -YY L (28)
The parameters M L and YY L are estimated by the least squares solution of equations ( 25) and (26) respectively, whereas the parameters MZ L and XX L are directly obtained from equations ( 27) and (28) respectively.

3) Using the difference between the joint torques before and after loading the robot on the same trajectory

In this method we make use that the term W χ of ( 22) is equal to the joint torques without load. Consequently the payload parameters could be identified by [START_REF] Gautier | A comparison of filtered models for dynamic identification of robots[END_REF] after replacing W χ by Y wL representing the joint torques on the same trajectory without load, which gives:

χL = (W L ) + (Y T -Y wL ) (29)
The unknowns of this method are only the 10 inertial parameters of the payload, which lead to the possibility of reducing considerably the number of points needed for the construction of the observation matrix. In our experiment just one trajectory (out of the planned 8) was sufficient.

This method supposes that the control system is efficient such that the difference between the joint positions with and without payload is negligible and that the friction parameters are the same under the two conditions. For the RX 90 robot the maximum error on joint positions due to the load is 10 -3 rad while carrying a load of 7.025 Kg, which allowed obtaining good results with this simple method.

4) Global identification of the robot parameters and the load parameters

In this method the identification model is constructed by grouping two sets of equations, the first representing a trajectory without load, and the second representing a trajectory with load. These two trajectories could be different:

a b L W W W L ⎡ ⎤ ⎡ ⎤⎡ ⎤ = ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ 0 a b Y Y χ χ (30)
where the first row represents the equations of the trajectory without payload, whereas the second row corresponds to the trajectory with payload. The link parameters χ and the payload inertial parameters L χ are estimated by solving (30) using weighted least squares solution. This method has the advantage of using a global identification procedure that can avoid the accumulation of errors, which may occur for sequential identification methods.

B. Experimental results

To test the proposed methods, a calibrated payload is used. The inertial parameters are provided by the software package CATIA. These values, denoted in the following as a priori, are sufficiently accurate due to the simplicity of the shape of the load. The mass of the reference payload is estimated as 7.025 Kg using a weighing machine whose accuracy is ± 0.050 Kg. The four methods have been applied and the results are nearly the same. We give the results of methods 3 and 4. Method 3 is considered as the simplest one, whereas method 4 is considered as the best scientific one.

The results of methods 3 and 4 are given in Tables 1 and2 respectively. We note that the parameters XX L , YY L , MZ L and M L are very well identified in both methods in terms of relative standard deviation. The reference values of the other parameters are almost zero. In method 3, they have big relative standard deviations, but the estimated values are almost zero. In method 4, the application of the elimination strategy led to eliminate automatically these parameters because they have negligible effect on the dynamics of the robot.

The robot parameters identified in method 4 are very close to those estimated in section 4.

VI. CONCLUSION

In this paper we dealt with the problem of the identification of the inertial parameters of the payload of industrial robots. In the experimentation we used the Stäubli RX 90 robot with its CS8 controller for the data acquisition and for the trajectory generation. Four methods have been proposed and validated by comparing the inertial parameters of a reference-calibrated payload. We put the accent on two methods, the method 3 makes use of data of a given trajectory executed twice, once without the payload and then with the payload. This method relies on a good control tuning in order that the position error between the two executions will be small enough. The method 4 identifies both robot parameters and payload parameters in one step by a global procedure. This method avoids the accumulation of errors, which may occur in methods 1 and 2, and does not impose any hypothesis as in method 3. The experimental results, for the RX 90 robot, are very close for the four methods, thus any of them can be used for the RX90 robot. This may be not the case for other robots using method 3.

4 Figure 1 .

 41 Figure 1. Link frames of Stäubli RX-90 robot.(D3 = 0,45m and RL4 = 0,45m)

  the viscous and Coulomb friction parameters.The inertia of the rotors is generally taken in this model as:

Table 1 .

 1 Identification of the payload parameters using method 3

	Parameter	A priori	Identified	2 σ χ		σ χr %
	XX L	0.161	0.157		0.019		5.931
	XY L	0	0.007		0.009		66.152
	XZ L	0	0.008		0.009		62.705
	YY L	0.161	0.153		0.018		5.738
	YZ L	0	-0.005		0.009		91.418
	ZZ L	0.035	0.025		0.012		24.612
	MX L	0	0.034		0.015		21.869
	MY L	0	0.006		0.015		119.190
	MZ L	1.003	0.977		0.020		1.015
	M L	7.025	7.216		0.045		0.313
	Parameter A priori Identified	2 σ χ	σ χr %
		ZZ1R		12.600		0.040	0.158
		FV1		29.300		0.139	0.237
		FS1		21.700		0.228	0.526
		XX2R		-7.770		0.053	0.339
		XZ2R		-0.866		0.039	2.227
		ZZ2R		11.400		0.046	0.204
		MX2R		15.900		0.026	0.081
		FV2		27.300		0.218	0.399
		FS2		21.300		0.286	0.670
		XX3R		1.230		0.042	1.723
		ZZ3R		1.300		0.017	0.655
		MY3R		-3.170		0.012	0.189
		IA3		1.480		0.019	0.636
	Robot	FV3		7.090		0.067	0.472
	parameters	FS3		12.800		0.102	0.399
		MX4		-0.052		0.004	4.252
		IA4		0.866		0.004	0.231
		FV4		6.000		0.023	0.191
		FS4		5.800		0.048	0.412
		IA5		0.280		0.010	1.847
		FV5		4.730		0.049	0.519
		FS5		2.310		0.063	1.369
		YZ6		-0.020		0.001	3.246
		IA6		0.104		0.002	0.997
		FV6		1.200		0.015	0.612
		FS6		0.820		0.022	1.371
		FVM6		1.050		0.012	0.552
		FSM6		0.700		0.024	1.679
		XX L	0.161	0.157		0.008	2.572
	Payload parameters	YY L ZZ L	0.161 0.035	0.151 0.027		0.008 0.002	2.567 4.487
		MZ L	1.003	0.979		0.010	0.495
		M L	7.025	6.700		0.039	0.294

Table 2 .

 2 Identification results using method 4

ACKNOWLEDGEMENTS

We would like to thank Dr. Luc JOLY, chief of the automatic control department of Stäubli, for providing us with the CS8 controller, M. Paul MOLINA for the design and manufacturing of the reference calibrated payloads.