Sylvain Fréour
email: freour@lamm.univ-nantes.fr

David Gloaguen

Manuel François

Ronald Guillen

R Guillén

Modelling and simulation of multi-phase effects on X-Ray Elasticity Constants

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Introduction

Experimenters require proportionality constants, called X-ray Elasticity Constants (XEC), in order to determine the residual stress state of a material from X-ray strain measurements [START_REF] Noyan | Residual Stress Measurement by Diffraction and Interpretation[END_REF].

Apart from effects due to crystallites morphology or orientation (crystalline texture), XEC depend on several parameters such as the diffracting planes chosen, the nature of each phase through their elastic mechanical properties, and the volume fraction of the different phases constituting the multi-phase polycrystal [START_REF] Tanaka | [END_REF]. Methods leading to XEC values determination are well known in the case of single-phase materials. Experimental and scale transition simulation schemes (based for example on works [START_REF] Voigt | Lehrbuch der Kristallphysik[END_REF][START_REF] Voigt | [END_REF][5][6][7][8][START_REF] Eshelby | Proc. R. Soc. Lond[END_REF]) yield concordant numerical results in this particular case. For two phases materials, a more elaborate approach is proposed in the present paper. Knowing the volume fraction and the mechanical behaviour of the single crystal of each phase, one needs to calculate the XEC necessary for his experiments. In fact, it is usually assumed that XEC remain the same in two-phases materials as in single phase ones. This hypothesis yields straightforwardly to introduce the classical, available in the literature, values of singlephase XEC in the "sin 2 ψ" relation, in order to process the experimental pseudo-macroscopic strain data obtained through measures in two-phases samples, e.g. [START_REF] Hadmar | Proc. ECRS[END_REF][START_REF] Van Acker | Proc. ECRS[END_REF][START_REF] Ceretti | Colloque de métallurgie[END_REF][START_REF] Hauk | Residual Stresses[END_REF]. This methods implicitly neglects the effects of the second phase on the mechanical elastic behaviour of the diffracting volume. It implicitly considers the phase as a connex domain in which the diffracting volume is fully embedded. The aim of this work consists in the evaluation of the two-phases effect on the XEC values and its consequences on the stresses determined by XRD analysis in cubic and hexagonal crystallographic structures.

2 One site self-consistent calculation model 2.1 The three scales representation for multi-phase materials: scale transition formalism XRD stress determination amounts to a sampling of the material with a set of crystallites which orientations are well defined. The elastic properties of this set are therefore anisotropic. It means that a multiscale modeling of XEC should start at the crystallite scale with an elastically anisotropic formulation. To account for intergranular heterogeneities occurring during elastic strain, self-consistent approaches are used to express the behavior of polycrystals. As part of the model, ellipsoidal inclusions, representing a crystallite, are one after another embedded in an infinite homogeneous equivalent matrix representing the two-phases polycrystalline aggregate. It should be remembered that XEC concept is based on an isotropic elastic behaviour of the matrix. It implies that the material should be assumed as perfectly disordered. In fact, in the particular case, when the principal axis of ellipsoidal anisotropic inclusions are statistically oriented, macroscopic stiffness remains isotropic, even in two-phases materials. This question has been thoroughly investigated by Qiu and Weng [START_REF] Qiu | [END_REF]. In the general case, however, when the principal axis are preferentially oriented along a specific direction, the elastic properties of the homogneous equivalent medium become anisotropic. Macroscopic anisotropy can appear in the presence of crystalline texture too [START_REF] Tomé | Texture and Anisotropy[END_REF]. In the considered context, the main consequence of macroscopic anisotropy is that the classical XEC are no more suitable for the interpretation of XRD strains measurements. It will often be necessary to introduce the matrix of the stress factors, calculated from the Orientation Distribution Function (ODF). An example is given by Sprauel et al. in [START_REF] Sprauel | Proc. ICRS[END_REF].

First, let us introduce the description of the effective macroscopic mechanical behavior (σ Ι , ε Ι ) of the polycrystal. The statistical self-consistent framework assumes the following regular equations deduced from [6,[START_REF] Eshelby | Proc. R. Soc. Lond[END_REF][START_REF] Hill | [END_REF] :

II : = C σε , with () () { } 1 1,
::

ii iN ΩΩ - =  =+- Cc I P c C (1)
σ and ε are the stresses and strains. Superscripts I or II on strain and stress tensors denote respectively macroscopic (first order) and mesoscopic (second order) quantities.

C is the effective elastic modulus of the homogeneous equivalent medium (polycrystal).

I is defined by ()

2 ik jl il jk ijkl I δδ δδ + =
, where δ is the Kronecker symbol.

() i Ω c
denotes the single crystal elastic stiffness (of the phase i) referred to a sample-fixed coordinate system whose orientation, in which the orientation of the crystallite is symbolically denoted by Ω. Hill's tensor P expresses the local interactions depending on the morphology assumed for the crystallites [START_REF] Hill | [END_REF]. It is linked to Eshelby tensor S esh by P = C : S esh . The integration of the Green's tensor of the homogeneous equivalent medium over the surface of an inclusion provides the expression of P. The method is detailed, for example, in [18].

A : B denotes the double scalar product A ijkl B klmn .

N is the number of phases in a Representative Volume Element (RVE) of the polycrystal. It is assumed that the average of equation ( 1) is equivalent to a spatial average over the RVE (ergodic hypothesis). The average operation is successively done over every orientation Ω for each phase.

At mesoscopic scale, the stresses σ ΙΙ (Ω) and strains ε ΙΙ (Ω) of any inclusion can be expressed by a similar equation () () ()

II II :. i ΩΩΩ = c σε (2)
The scale transition formalism relating the macroscopic first order and mesoscopic second order mechanical states verifies the strain localization law usually described through () ()

II I : ΩΩ = A εε , where the localization tensor A(Ω) is expressed by () () { } 1 : i ΩΩ -  =-+  AP cC I (3)
A(Ω) depends on the macroscopic stiffness C which can be calculated iteratively from the implicit equation (1). Following Hill [8], the macroscopic strain is obtained by ()

II I . Ω = εε (4)
In order to study a multi-phase polycrystal, it is necessary to introduce a pseudo-macroscopic, intermediate, scale between the mesoscopic and the macroscopic ones. This new level of representation has been defined by Sprauel so as to simulate the behavior of the different phases constituting the whole material [START_REF] Sprauel | Analyse des contraintes résiduelles par diffraction des rayons X et des neutrons[END_REF]. Remind that the term "phase" is here employed in its crystallographic sense: each phase is constituted by every crystallite exhibiting the same properties (e.g. single crystal elastic constants, …), whatever their orientation Ω. The pseudo-macroscopic strain of a given α phase can be defined as the average of the strains over all the crystallites composing this phase:

() II . Ω α α = εε (5)
The localization law can be obtained from equation (3):

() ()

II :: . ΩΩ α α α == AA εε ε (6)
In consequence, the pseudo-macroscopic stress concentration law would be expressed through

() () () ()() II II I :: : . ΩΩ ΩΩ Ω αα α αα α == = cc A σσ ε ε (7)
Using equation (6), it comes:

()() () 1 :: : . α ΩΩ Ω - αα α α = cA A σε (8)
This relation enables us to define the pseudo-macroscopic stiffness corresponding to a phase with

: αα α = C σε (9)
Comparison between (8) and (9) leads to

()() () () () { } () { }
This equation is quite different from the corresponding expression of the isolated polycrystalline single α phase macroscopic stiffness (1) because the properties of the second phase might have an influence on C α through the value of C (and thus of P). In fact, equation ( 1) is established by considering each phase as a single inclusion whereas in equation [START_REF] Hadmar | Proc. ECRS[END_REF] each phase is composed of many crystallites with different orientations. The term factor expressing the average interaction between α crystallites and the infinite equivalent medium. In consequence, in a multi-phase material, the nature and the proportion of the other phases forming the polycrystal affect the average elastic properties of a given α phase. As a result, the pseudo-macroscopic stiffness of a phase arbitrarily isolated might not be identified to its macroscopic stiffness, when it is integrated in a multi-phase material.

The reason why C α is not equal to its single phase value could either be a closure problem of the selfconsistent model or have a physical origin. As a matter of fact, the definition of the pseudo-macroscopic stress and strain in equations ( 6) and (7), although intuitive and consistent with usual definitions, is arbitrary.

Basic tests can be performed to study the expressions obtained. Let us consider a single-phase material. The definition of the average strain II I =〈 〉 εε implies that

1 () () () ΩΩ Ω - αα 〈〉 = = 〈〉 = 〈〉 AI A A.
In consequence, [START_REF] Hadmar | Proc. ECRS[END_REF] leads to the expected regular single-phase implicit relation (1):

()() () ()() () () { } 1 1 :: : : : . ΩΩ Ω ΩΩ Ω Ω - - αα α α α α α αα α ⎡⎤ == = - + ⎣ Cc A A c A c P c CI
Let us focus on the general case of a n-phases polycrystalline aggregate, containing a volume fraction f i of any given phase i. Imposing the homogenization conditions ( 4) and ( 5) over the strain localization law (6) gives :

1,

() .

iN

Ω = 〈〉 = AI
This expression can also be developed :

i () () . i i ff ΩΩ α α ≠α 〈〉 +〈〉 = ∑ AA I
In consequence, α phase average pseudo-macroscopic localization tensor deviates from the identity as follows: 11 () () .

i i i f ff ΩΩ α αα ≠α 〈〉 =- 〈〉 ∑ AIA
Where f i is the volume fractions of phase i and

1 1
.

N i i f = = ∑
In conclusion, self-consistent model constitutive relations imply that the average pseudo-macroscopic localization tensor of a given α phase is necessarily different from the identity. This results in a deviation between pseudo-macroscopic stiffness and the corresponding single-phase macroscopic stiffness. Also, this deviation should be considered as the expression of continuum mechanics fundamental laws rather than as an hypothetical closure problem of the model. As a matter of fact, it can be considered that the α phase is not a connex volume. One could intuitively think that its overall properties should be influenced by the medium that makes it connex. Therefore, as a response to a given average stress, the phase is not free to deform in its own "macroscopic" way, when it is embedded in other phases. In consequence, the deviation between single-phase macroscopic stiffness and pseudo-macroscopic stiffnesses will be strongly related to the volume fraction and elastic mechanical properties of these other phases.

The introduction of the pseudo-macroscopic stiffness C α raises the question of the scale transition relations existing between the phases and the polycrystalline multi-phase matrix. According to the strain localization (3), the macroscopic strain can be developed as follows:

() () () ()

II I I I I I :: : : . i ΩΩ Ω Ω =- + Pc PC εεε ε (11)
An averaging of this expression only on α phase crystallites yields:

() () () ()

II I I I I I :: : : . ΩΩ Ω Ω α αα α α =- + Pc PC εεε ε (12)
I , ε P and C being constant over α phase, these tensors may be factored out of the averages:

() () () () II I I I I I :: : : . α ΩΩ Ω Ω α αα =- + Pc P C εε ε ε (13)
According to the mesoscopic to pseudomacroscopic averaging equations ( 5) and (7), one should obtain:

{ } I :: : : :: : : : .

αα α α αα αα α ⎡⎤ =- + = - + = -+ ⎣ PP C P C P C P C C I εσ ε ε ε ε ε ε (14)
Finally, the pseudomacroscopic stresses and strains can be expressed through the macroscopic strain state, as a function of the pseudomacroscopic and macroscopic stiffness, respectively given by ( 14) and ( 9):

( )

1 I :: , - αα  =-+  PC C I εε (15) ( ) 1 I
:: : .

- αα α  =-+  CP CC I σε ( 16
)
These equations mean that the average stress or strain state of the polycrystalline α-phase can be obtained through a classical multi-phase self-consistent scheme by considering it as a single inclusion of stiffness C α embedded in a homogeneous matrix having the stiffness of the macroscopic material. However, it should be remembered that the stiffness tensor C α is not that of the isolated single α-phase but is given by equation [START_REF] Hadmar | Proc. ECRS[END_REF]. Up to now, the developed equations are very general. They are therefore valid for a macroscopically anisotropic (textured) material. Finally, in order to take into account multi-phase effects in the simulation, it is necessary to consider three levels in the schematization of the material: the classical grain and polycrystal levels (mesoscopic and macroscopic scales) and an intermediate pseudo-macroscopic scale defining the behaviour of each phase. This representation and the associated scale transition formalism will now be applied to XEC determination.

Application to XEC simulation

According to [START_REF] Sprauel | Proc. ICRS[END_REF][START_REF] Sprauel | [END_REF], the lattice strain , ϕψ ε α measured in α phase by XRD corresponds to the integration of the mesoscopic strains (3) over all orientations Ω of the Diffracting Volume (DV), and their projection on the measurement direction ϕ, ψ:

Neglecting effects due to crystallographic or morphologic texture, the usual linear relation between the measured strain and the triaxial pseudo-macroscopic stress undergone by the α-phase could be developed as follows :

( ) ( ) ( ) 22
Substituting the numerical values obtained through [START_REF] Sprauel | Proc. ICRS[END_REF][START_REF] Hill | [END_REF] in (18) provides a relation leading to the determination of the XEC 1/2 2 S α and 1 S α characteristic of the α-phase embedded in the multi-phase polycrystal studied.

According to the theory described by Bollenrath, Hauk and Müller [21], the XEC of a given cubic symmetry phase can be expressed for any {hkl} diffracting plane through: ( ) ( )( )

11 22 22 00 3 . Sh k l Sh h k l S =-Γ
This expression depends on two main factors: the slope S, and the cubic orientation parameter symbolically denoted by Γ. These terms are respectively given by: ( ) ( ) hk hl kl hkl hkl

Γ ++ = ++
, and austenitic phase [23] ferritic phase [23] aluminium [START_REF] Noyan | Residual Stress Measurement by Diffraction and Interpretation[END_REF] β -SiC [23] diamond-C [23] copper [START_REF] Noyan | Residual Stress Measurement by Diffraction and Interpretation[END_REF] c 11 [GPa] 197 A XR denotes the X-Ray anisotropy coefficient of the diffracting phase, corresponding to the following ratio:

( ) [ ] 2X R 1 1 2 SS h h h A =-.
1 2 2 XR 1 2 2 (0 0 ) () Sh A S hhh = .
In the same way, XEC of hexagonal symmetry phases verify the following relation due to Evenschor and Hauk [START_REF] Sprauel | Analyse des contraintes résiduelles par diffraction des rayons X et des neutrons[END_REF] :

( ) ( ) ( ) 2 1 21 2 3 2 ..
. .

Sh k l x x H h k l x H h k l

=+ + Introducing the lattice parameters a and c of the considered structure, the hexagonal orientation parameter symbolically denoted by H is given by: ( ) ( ) In consequence, for phases exhibiting hexagonal crystallographic structure, 1/2 S 2 values varies between 1/2 S 2 (10.0) = x 1 and 1/2 S 2 (00.1) =

x 1 + x 2 + x 3 .
This model has been used to simulate the cases of several two-phases materials: duplex steels, aluminium-β silicon carbide Metal Matrix Composites (MMC), aluminium-diamond carbon composites, copper-Fe α alloys, aluminium-beryllium alloys, aluminium-zinc alloys and cadmium-zinc alloys. The different systems have been especially chosen in order to separate the effects of the following parameters on XEC values: single-crystal anisotropy, crystallographic structure and the ratio of the elastic moduli of the two considered phases.

The micromechanical constants used for each cubic or hexagonal single-crystal inclusion are respectively given in Tables 1 and2. The two-phases elastic behaviour has been simulated through a calculus involving 4000 spherical crystallites randomly oriented. The diffracting volume elastic behaviour has then been determined for each phase and different compositions of the α + β material: 0% β, 25% β, 50% β, 75% β and 100% β, (this last composition corresponds to the limiting case of vanishing α phase). Table 3 summarizes the cases of two usually encountered materials: duplex steels and aluminium-β silicon carbide MMC. The localization of a macroscopic stress in the crystallites of the diffracting volume provides values for the XEC. Results obtained are given in Tables 4 and5 which summarizes the main elastic heterogeneities and sources of anisotropy of the considered polycrystals. Nevertheless, XEC do not depend on σ Ι . In consequences, the macroscopic loading is arbitrary. 3 Interphase interactions in two-phases polycrystals 3.1 Discussion about multi-phase effects on pseudomacroscopic stiffness values (Table 3)

The one site self consistent formalism [START_REF] Hadmar | Proc. ECRS[END_REF] shows that the pseudo-macroscopic stiffness tensor of a given phase integrated in a multi-phase material should be different from the corresponding single-phase material macroscopic stiffness. Pseudo-macroscopic stiffness obviously varies as a function of the nature and volume fraction of the other phase constituting the two-phases polycrystal. Nevertheless, it is shown in table 3 that, in most cases, multi phase effect on pseudo-macroscopic stiffness is usually negligible. Although pseudo-macro-stiffness varies with the nature and the volume fraction of the other phase, the bulk modulus K of each phase remains perfectly constant at pseudomacroscopic scale. It is well known that K values are, since it is an invariant of the stiffness tensor, identical in single crystal (mesoscopic scale) and in cubic structure single phase polycrystals (macroscopic scale). In consequence a variation of the bulk modulus would not be physically acceptable. Results obtained satisfy this criterion. This confirms the numerical validity of the model developed which implies the deviation of pseudo macro stiffness values in order to ensure the consistence between the average moduli determined at macroscopic and pseudomacroscopic scales, and the elastic behaviour of the inclusions.

3.2 Analyse of multi-phase effects on XEC Tables 4,5 show that the XEC of a given α-phase generally vary with the volume fraction of the phase in a given system, and with the nature of the second (β) phase. The model takes into account multi-phase effects on XEC numerical values. In physical terms, this discrepancy between XEC of a given phase, calculated in a single-phase material and in a two phases sample can be explained by the elastic interactions existing between the polycrystalline matrix and any crystallite. Through these interactions, the mechanical behaviour of α-phase diffracting volume depends directly on the polycrystalline macroscopic stiffness denoted by C. Thus, according to (1), the second phase is closely involved in the iterative calculation of C tensor. In consequence, the presence of the β-phase affects α-phase diffracting volume mechanical behaviour through its volume fraction and pseudo-macroscopic stiffness C β tensor. This phenomenon can be schematised on Fig. 1a and 1b: α-phase XEC could be equal in the single-phase sample and in the two-phases polycrystal, if the crystallites of α diffracting volume were only connected to α grains. This corresponds to the ever, in reality, the grains of α diffracting volume often have α and β grains as first neighbours simultaneously, as shown on Fig. 1b. The presence of these first neighbours β grains generates interactions which affect α diffracting volume elastic behaviour. In fact, our model only describes the case of Fig. 1b in an average meaning: the interactions of β grains on the behaviour of α phase are only taken into account through the average stiffness of the matrix, i.e. the homogeneous equivalent medium.

In the case of the MMC aluminium-β SiC, calculated XEC strongly depend on each phase volume fraction. Similar effects have been observed for several two-phases polycrystals: in pure cubic structure systems in the case of ferritic steel-copper sample, in pure duplex hexagonal structure such as cadmiumzinc polycrystal, and in dual structure systems, in the case of aluminium-zinc polycrystal, for example. This property, characteristic to such two-phases polycrystals, results from the mechanical behaviour heterogeneities existing between the phases.

On the opposite, when the macroscopic behaviour of the two phases is similar, such as in the case of duplex steels, the XEC do not vary significantly with the volume fraction of each phase.

In conclusion, in a (α + β) two-phases polycrystal, the XEC of a given α-phase deviates more and more from the values obtained in the single-phase case, when α-phase volume fraction tends towards 0, and when α and β have very different elastic behaviour.

In the case of aluminium-diamond like carbon systems, aluminium XEC verify this rule, while diamond XEC seem not to follow it. In fact, diamond XEC remains almost constant, in spite of the large mechanical discrepancies existing between the two phases. This result could be expected in particular cases: a phase with elastically isotropic single crystal keeps its XEC constant, whatever the two-phases state of the polycrystal in which it is embedded. In fact, this property is only valid for a macroscopically isotropic material. If the second phase exhibit a crystallographic or morphologic texture, the overall stiffness tensor will be anisotropic. This would influence the localisation tensor and thus the elastic behaviour of the diffracting volume. Diamond like carbon has a single-crystal anisotropy constant A c close to 1. In consequence, its XEC are not modified by the polycrystal two-phases state. Similar results have been obtained on tungsten (A c = 0.98) and to some extent in the case of molybdenum (A c = 0.91) and TiC (A c = 0.90), according to the single-crystal elastic constants given in [23][START_REF] Simmons | Single Crystal Elastic Constants and Calculated Aggregate Properties[END_REF][START_REF] Hosford | The Mechanics of Crystals and Textured Polycrystals[END_REF]. On the opposite, results obtained on Zinc and Copper emphasize that multi-phases effects on XEC arise in the case of strongly anisotropic diffracting phases.

Consequences on the stresses determined by XRD analysis on multi-phase samples

As mentioned before, the usual assumption consists in introducing single-phase XEC values in the "sin 2 ψ" equation, in order to process numerically the strains measured in two-phases polycrystals. The results given by our simulations demonstrate that this approach constitutes an approximation of the polycrystalline sample mechanical effective behaviour. In consequence, this assumption generates a systematic error δ on the determined stresses. For a given set of {hkl} diffracting planes, in the case of a material including a volume fraction f of the analysed phase, one should express this systematic deviation as follow :

() ( ) ( ) 22 2 11 1 22 . 1 1 2 Sf Sf Sf δ -= = = (19)
This relation applied to several two-phases systems yields the curves given on Figs. 2 and3. As part of a residual stress analysis through XRD, the relative error on stresses values due to experimental and data processes generally remains between 5% and 15% for the considered phases. The maximal influence of non diffracting phase on the X-ray behaviour of diffracting phase is given in Table 6, where the main properties of the two-phases material simulated are summarized. Figs. 2 and3 show that the systematic relative error δ resulting from the usual assumption is often lower than 5%. δ curves confirm the existence of particular Γ 0 orientation values for which XEC are equal in the single phase and in the two-phases material. It should be noted that our purpose is not to obtain the macroscopic strain in a two phases material. This objective would be limited to a purely elastic loading and could not give any valuable information in the case of residual stresses due to stress-free strains such as thermal or plastic loading or phase transformation (see for example [START_REF] Sprauel | Analyse des contraintes résiduelles par diffraction des rayons X et des neutrons[END_REF]). Our purpose is to provide a rigorous scheme to express the average stress of one phase of a multiphase material from the elastic strains measured by XRD in the same phase, whatever the physical origin of the stress.

To express this mathematically, the measured strain in α-phase can be expressed by ( )

,1 :

ϕψ φψ ε αα = B ɍ , ( ) ( )
IL I L a D v ,1 2 3
. :: :

ϕψ φψ ε α  =+ -  BB B σε ε
Our model provides a mean to calculate operator B 1 in order to measure σ α but, in no way, the expression of operators B 2 and B 3 nor any access to the macroscopic stress σ Ι and the stress-free strains ε LΙ and ε LαD . ε LΙ and ε LαDv are respectively the macroscopic stress-free strain and the stress-free strain of α diffracting volume. In our model, the basic assumption is that one crystallite of the diffracting phase interacts with both phases in proportion to their volume fraction, however, in reality, this might not always be true. For instance, SiC particles in Al/SiC composite are composed of several crystallites; one crystallite is thus surrounded by more SiC crystallites than Al crystallites. To go a little further, the elementary domain in XRD is not the grain but the coherently diffracting domain which size is much smaller than the grain size. Whether such a domain interacts with both phases can be discussed.

Conclusion

The choice of the XEC values is crucial, especially so as to determine residual stress through XRD analysis. A one-site elastic self-consistent calculation scheme based on Kröner-Eshelby's formalism has been developed in order to simulate the diffracting volume elastic behaviour of cubic or hexagonal phases embedded in two-phases polycrystals. It was shown that XEC values of a given phase generally depend on the volume fraction and pseudo-macroscopic stiffness of the other phases constituting the whole material. It has been demonstrated that this dependence could be regarded as an error source affecting the calculation of stresses from measured strains data. It is also necessary to take into account effects due to two-phases interactions in order to minimize the deviation between the residual stresses determined through XRD experiments and true stresses in the sample. To go deeper into this study, an analytical model will be soon proposed in order to express XEC for cubic phases embedded in multiphase polycrystals. The property of elastically isotropic phases at mesoscopic scale, whose XEC remains identical to the corresponding single-phases values, in any multi-phase material, will be demonstrated owing to this analytical solution.

The deviation between pseudomacroscopic and single-phase stiffnesses was also studied. Numerical computations justifies, in most cases, the identification of pseudomacroscopic elastic moduli to the single phases corresponding values. In a further study, it will be taken advantage of this property in order to propose a method suited for the characterization of the macroscopic elastic moduli of a phase embedded in a multiphase polycrystal.

Fig. 1 aFig. 2

 12 Fig. 1 a) Diffracting volume constituted by α grains only connected to other α grains. (Case of equality of the XEC in the single-phase and twophases material). b) Diffracting volume constituted of α grains interacting with α and β first neighbours grains.

Fig. 3

 3 Fig.3Calculated relative systematic error δ, due to multiphase effects, on the stresses determined in several hexagonal structure phases of two-phases polycrystals. f denotes the volume fraction of the analysed phase.

Table 1

 1 Cubic symmetry single-crystal elasticity constants and anisotropy parameter A c .

Table 2

 2 Hexagonal symmetry single-crystal elasticity constants[START_REF] Hosford | The Mechanics of Crystals and Textured Polycrystals[END_REF].c 11 [GPa] c 12 [GPa] c 13 [GPa] c 33 [GPa] c 44 [GPa] c 66 [GPa] A h1

	A h2	c 11 /c 33

NB: The departure of the parameters A h1 , A h2 and c 11 /c 33 from unity can be used as a measure of the elastic anisotropy of hexagonal crystals.

Table 3

 3 Multi phase effects on pseudomacroscopic stiffness.

	Stiffness tensor components and bulk moduli calculated in Al-SiC MMC at pseudomacroscopic scale
	f(Al)	0.00	0.25	0.50	0.75	1.00
	C 11 (Al) [GPa]	112.2	112.2	112.1	112.1	112.1
	C 12 (Al) [GPa]	59.3	59.3	59.3	59.3	59.4
	K(Al) [GPa]	76.9	76.9	76.9	76.9	77.0
	C 11 (SiC) [GPa]	437.7	434.2	430.8	428.1	426.7
	C 12 (SiC) [GPa]	97.6	99.4	101.1	102.5	103.2
	K (SiC) [GPa]	211.0	211.0	211.0	211.0	211.0
	Stiffness tensor components and bulk moduli calculated in duplex steels at pseudomacroscopic scale
	f(Fe α )	0.00	0.25	0.50	0.75	1.00
	C 11 (Fe α ) [GPa]	281.0	281.1	281.2	281.2	281.2
	C 12 (Fe α ) [GPa]	112.9	112.8	112.8	112.8	112.7
	K(Fe α ) [GPa]	168.9	168.9	168.9	168.9	168.9
	C 11 (Fe γ ) [GPa]	250.6	250.8	251.1	251.3	251.5
	C 12 (Fe γ ) [GPa]	97.9	97.8	97.7	97.6	97.5
	K(Fe γ ) [GPa]	148.8	148.8	148.8	148.8	148.8

Table 4

 4 XEC [10 -6 MPa -1 ] determined in cubic symmetry phases of several two-phases polycrystals.

	XEC calculated in the case of a MMC aluminium-β silicon carbide
	diffracting phase volume fraction	100%	75%	50%	25%	∼0%
	aluminium	1/2 S 2 (h00)	19.99	19.75	19.52	19.31	∼19.19
		S	1.71	1.35	0.98	0.69	∼ 0.49
	β-SiC	1/2 S 2 (h00)	3.63	3.82	4.04	4.23	∼ 4.36
		S	1.15	1.40	1.68	1.93	∼2.10
	XEC calculated in the case of duplex steel
	diffracting phase volume fraction	100%	75%	50%	25%	∼0%
	austenitic phase	1/2 S 2 (h00)	8.77	8.68	8.65	8.59	∼8.56
		S	3.68	3.59	3.54	3.48	∼3.43
	ferritic phase	1/2 S 2 (h00)	7.31	7.33	7.35	7.37	∼7.41
		S	2.29	2.32	2.34	2.39	∼2.43
	XEC calculated in the case of an iron-copper two-phases polycrystal
	diffracting phase volume fraction	100%	75%	50%	25%	∼0%
	ferritic phase	1/2 S 2 (h00)	7.31	7.39	7.56	7.71	∼ 7.82
		S	2.29	2.43	2.61	2.80	∼ 2.96
	copper	1/2 S 2 (h00)	13.55	13.32	12.94	12.55	∼12.34
		S	5.36	5.00	4.57	4.15	∼ 3.84
	XEC calculated in the case of an aluminium-diamond carbon two-phases polycrystal
	diffracting phase volume fraction	100%	75%	50%	25%	∼0%
	aluminium	1/2 S 2 (h00)	19.99	19.69	19.34	19.11	∼19.06
		S	1.71	1.23	0.70	0.37	∼0.23
	diamond	1/2 S 2 (h00)	1.29	1.29	1.30	1.30	∼1.30
	carbon	S	0.01	0.01	0.01	0.01	∼0.01

Table 5

 5 XEC [10 -6 MPa -1 ] determined in hexagonal symmetry phases of several two-phases polycrystals

	XEC calculated in the case of an aluminium-beryllium two-phases polycrystal
	diffracting phase volume fraction	-100%	-75%	-50%	-25%	∼0%
	beryllium	x 1 = 1/2 S 2 (10.0)	-3.52	-3.55	-3.58	-3.60	-3.62
		x 2	-0.68	-0.79	-0.90	-0.98	-1.03
		x 3	-0.39	-0.44	-0.47	-0.48	-0.47
		1/2 S 2 (00.1)	-3.23	-3.20	-3.15	-3.10	-3.06
	XEC calculated in the case of a cadmium-zinc two-phases polycrystal
	diffracting phase volume fraction	-100%	-75%	-50%	-25%	∼0%
	cadmium	x 1 = 1/2 S 2 (10.0)	-17.72	-17.80	-17.90	-17.97	-18.05
		x 2	-5.74	-5.80	-5.76	-5.62	-5.43
		x 3	-6.37	-5.16	-4.14	-3.28	-2.61
		1/2 S 2 (00.1)	-29.83	-28.76	-27.80	-26.87	-26.09
	zinc	x 1 = 1/2 S 2 (10.0)	10.21	-10.21	-10.23	-10.24	-10.27
		x 2	-0.84	-1.26	-1.80	-2.43	-3.17
		x 3	-8.61	-9.83	-11.28	-12.90	-14.71
		1/2 S 2 (00.1)	-17.98	-18.78	-19.71	-20.71	-21.81
	XEC calculated in the case of an aluminium-zinc two-phases polycrystal
	diffracting phase volume fraction	-100%	-75%	-50%	-25%	∼0%
	zinc	x 1 = 1/2 S 2 (10.0)	-10.21	-10.21	-10.21	-10.22	-10.24
		x 2	-0.84	-1.24	-1.75	-2.38	-3.19
		x 3	-8.61	-9.73	-11.05	-12.60	-14.51
		1/2 S 2 (00.1)	-17.98	-18.70	-19.51	-20.44	-21.56

Table 6

 6 Maximum multi-phase influence on XEC.

	material diffracting	crystallographic	diffracting	non	ratio of	maximum influence
		phase	structure	phase	diffracting	moduli of	of not diffracting
				anisotropy	phase	the two	phase on the X-ray
					anisotropy	phases	behaviour of
							diffracting phase (δ max )
	Fe(α + β)	Fe(α) Fe(γ)	cubic cubic	medium strong	strong medium	weak	1.4% 2.4%
	Fe(α)/Cu	Fe(α) Cu	cubic cubic	medium strong	strong medium	medium	7.0% 8.9%
	Al/SiC	Al SiC	cubic cubic	weak medium	medium weak	strong	4.0% 20.1%
	Al/C	Al C	cubic cubic	weak none	none weak	strong	4.7% 0.1%
	Al/Be	Al Be	cubic hexagonal	weak weak	weak weak	strong	3.3% 5.3%
	Al/Zn	Al Zn	cubic hexagonal	weak strong	strong weak	1.1% medium 19.9%
	Cd/Zn	Cd Zn	hexagonal hexagonal	medium strong	strong medium	12.5% medium 21.3%


Member, IEEE Francesco Chinello
email: chinello@btech.au.dk

Member, IEEE Monica Malvezzi
email: malvezzi@diism.unisi.it

Fellow, IEEE Domenico Prattichizzo
email: prattichizzo@diism.unisi.it

Member, IEEE Claudio Pacchierotti
email: claudio.pacchierotti@irisa.fr

A modular wearable finger interface for cutaneous and kinesthetic interaction: control and evaluation

Keywords:

We present a novel modular wearable interface for haptic interaction and robotic teleoperation. It is composed of a 3-DoF fingertip cutaneous device and a 1-DoF finger kinesthetic exoskeleton, which can be either used together as a single device or separately as two different devices. The 3-DoF fingertip device is composed of a static body and a mobile platform. The mobile platform is capable of making and breaking contact with the finger pulp and re-angle to replicate contacts with arbitrarily oriented surfaces. The 1-DoF finger exoskeleton provides kinesthetic force to the proximal and distal interphalangeal finger articulations using one servo motor grounded on the proximal phalanx. This paper presents the wearable device as well as three different position, force, and compliance control schemes, together with their evaluations. We also present three human subjects experiments, enrolling a total of 40 different participants: the first experiment considered a curvature discrimination task, the second one a robotassisted palpation task, and the third one an immersive experience in Virtual Reality. Results showed that providing cutaneous and kinesthetic feedback through our device significantly improved the performance of all the considered tasks. Moreover, although cutaneous-only feedback showed promising performance, adding kinesthetic feedback improved most metrics. Finally, subjects ranked our device as highly wearable, comfortable, and effective.

I. INTRODUCTION

H APTIC sensations provided by commercially-available wearable devices are often limited to vibrations, reducing the possibility of rendering complex contact interactions. Toward more realistic touch sensations, researchers started to study how to provide other types of cutaneous stimuli in a wearable and unobtrusive way [START_REF] Pacchierotti | Wearable haptic systems for the fingertip and the hand: Taxonomy, review, and perspectives[END_REF]. One of the first wearables able to provide rich cutaneous sensations has been presented by Minamizawa et al. [START_REF] Minamizawa | Gravity grabber: wearable haptic display to present virtual mass sensation[END_REF]. Two motors, placed on the nail side Fig. 1. The proposed wearable finger device. It is composed of a 3-DoF fingertip cutaneous module and a 1-DoF kinesthetic finger module. The 3-DoF fingertip device comprises two platforms: one on the back of the finger (black), supporting three small servo motors, and the other one in contact with the volar surface of the fingertip (white). The 1-DoF finger module is grounded on the proximal phalanx and it provides kinesthetic force to the PIP and DIP joints using one servo motor.

of the fingertip, move a belt in contact with the user's finger pulp. The belt applies a normal force to the user's fingertip when the motors rotate in opposite directions, while it applies a shear force when the motors rotate in the same direction. The device's motor inputs are calculated by implementing a simple proportional law between belt displacement and target stimuli. More recently, Prattichizzo et al. [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF] presented a similar cabledriven 3-degrees-of-freedom (3-DoF) cutaneous device for the fingertip. A static platform, housing three DC motors, is located on the nail, while a mobile platform, acting as the end-effector, is placed in contact with the finger pulp. The mobile platform can press into the user's fingertip and re-angle to simulate contacts with slanted surfaces. This device has been used for applications in robotic teleoperation and virtual reality, mostly employing position-based control approaches.

Although this type of cutaneous devices have been successfully used in various scenarios [START_REF] Tinguy | Enhancing the stiffness perception of tangible objects in mixed reality using wearable haptics[END_REF], [START_REF] Pacchierotti | Cutaneous feedback of fingertip deformation and vibration for palpation in robotic surgery[END_REF], their end-effectors always contact the finger skin. As a result, these devices are not capable of conveying the sensation of making and breaking contact with virtual and remote surfaces, which is known to be important for haptic interaction [START_REF] Edin | Bio-inspired sensorization of a biomechatronic robot hand for the grasp-and-lift task[END_REF]. To overcome this limitation, Kuchenbecker et al. [START_REF] Kuchenbecker | The touch thimble: Providing fingertip contact feedback during pointforce haptic interaction[END_REF] presented a passive contact location display to be attached to a grounded haptic interface. The kinesthetic feedback provided by the grounded interface bends the internal springs of the display and brings a shell in contact with the user's finger, providing the sensation of making and breaking contact with the rendered surface. Frisoli et al. [START_REF] Frisoli | A fingertip haptic display for improving curvature discrimination[END_REF] achieved a similar effect by creating a fingermounted thimble that moves a 5-DoF flat contact plate around the fingertip. More recently, Girard et at. [START_REF] Girard | Haptip: Displaying haptic shear forces at the fingertips for multi-finger interaction in virtual environments[END_REF] developed a 2-DoF wearable haptic device able to render shear forces at the fingertip. It is composed of a parallelogram structure actuated by two DC motors which move a peg responsible for the shear feedback. A recent review on wearable haptic interfaces can be found in [START_REF] Pacchierotti | Wearable haptic systems for the fingertip and the hand: Taxonomy, review, and perspectives[END_REF].

To provide well-rounded sensations, researchers have also worked on designing wearable interfaces able to provide both kinesthetic and cutaneous stimuli. For example, Cempini et al. [START_REF] Cempini | A powered finger-thumb wearable hand exoskeleton with self-aligning joint axes[END_REF] developed an underactuated fingertip exoskeleton with a custom self-alignment mechanism to absorb human/robot joint axes misplacement. It has been designed for close human-robot interaction applications and it is driven using a hierarchical two-layer position controller. Sarac et al. [START_REF] Sarac | Design of an underactuated hand exoskeleton with joint estimation[END_REF] recently presented an underactuated hand exoskeleton able to adapt its shape and size to objects during grasping. It has been designed for applications in virtual and augmented reality. A review of the literature on lightweight exoskeletons for the hand can be found in [START_REF] Pacchierotti | Wearable haptic systems for the fingertip and the hand: Taxonomy, review, and perspectives[END_REF], [START_REF] Wang | Toward whole-hand kinesthetic feedback: A survey of force feedback gloves[END_REF]. Indeed, providing both cutaneous and kinesthetic feedback has often showed better performance than providing either cutaneous or kinesthetic feedback alone. For example, when kinesthetic feedback was enriched with cutaneous cues, Frisoli et al. [START_REF] Frisoli | A fingertip haptic display for improving curvature discrimination[END_REF] found a significantly lower threshold for curvature discrimination (1.51±0.2 m -1 vs. 2.62±0.61 m -1 ) for stimuli constituted of spheres with curvatures ranging in the interval 4-6 m -1 . Pacchierotti et al. [START_REF] Pacchierotti | Cutaneous haptic feedback to ensure the stability of robotic teleoperation systems[END_REF] showed that providing both cutaneous and kinesthetic feedback improved the performance of a teleoperated pick and place task with respect to conveying, separately, either kinesthetic or cutaneous feedback. Similarly, Meli et al. [START_REF] Meli | Sensory subtraction in robot-assisted surgery: fingertip skin deformation feedback to ensure safety and improve transparency in bimanual haptic interaction[END_REF] found that providing cutaneous stimuli alone performed worse than providing cutaneous and kinesthetic stimuli in a hole-in-peg task. More recently, Quek et al. [START_REF] Quek | Evaluation of skin deformation tactile feedback for teleoperated surgical tasks[END_REF] used a teleoperation system to perform two manipulation tasks (peg transfer and tube connection) using kinesthetic feedback, skin deformation feedback, and the combination of both. The combined kinesthetic and skin deformation feedback achieved better performance and higher participant ratings compared to kinesthetic or skin deformation feedback alone.

This paper presents the control and evaluation of a novel modular interface for haptic interaction, shown in Fig. 1. The device is composed of a 3-DoF fingertip cutaneous device and a 1-DoF finger kinesthetic exoskeleton, which can be either used together as a single device or separately as two different devices. The 3-DoF fingertip module is composed of a static upper body and a mobile platform: the static body is located on the nail, supporting three small servo motors, and the mobile platform is placed in front of the finger pulp. The two parts are connected by three articulated legs, according to a Revolute-Revolute-Spherical (RRS) kinematic chain. The legs are actuated by the servo motors and can move the platform away and toward the user's finger skin as well as rotate it to mimic contacts with arbitrarily oriented surfaces. With respect to the cable-driven fingertip devices presented in [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF], [START_REF] Pacchierotti | Cutaneous feedback of fingertip deformation and vibration for palpation in robotic surgery[END_REF], our 3-DoF fingertip module solves the indeterminacy due to the underactuation of the platform. Moreover, it is one of the most compact and lightweight device ever presented [START_REF] Pacchierotti | Wearable haptic systems for the fingertip and the hand: Taxonomy, review, and perspectives[END_REF]. The 3-DoF fingertip module alone weighs 24 g for 35×50×43 mm dimensions, and its design is inspired by [START_REF] Chinello | A three revolute-revolute-spherical wearable fingertip cutaneous device for stiffness rendering[END_REF].

The finger exoskeleton is a 3-DoF planar mechanism: two DoF allow the adaptability to different finger sizes, while the third one provides the actuation. Once the exoskeleton is worn, a static part is fixed on the proximal phalanx, while a mobile part is fixed close to the Distal-Interphalangeal (DIP) joint axis. In this way, the finger kinematic structure, composed of the proximal and intermediate phalanges connected through the Proximal-Interphalangeal (PIP) joint, constraints the exoskeleton kinematics, reducing the overall system mobility to 1 DoF. Consequently, only one motor is needed to actuate the finger/exoskeleton system and provide kinesthetic stimuli to the PIP/DIP articulation. With respect to similar interfaces [START_REF] Cempini | A powered finger-thumb wearable hand exoskeleton with self-aligning joint axes[END_REF], [START_REF] Sarac | Design of an underactuated hand exoskeleton with joint estimation[END_REF], [START_REF] Lemerle | Design and evaluation of a remote actuated finger exoskeleton using motion-copying system for tendon rehabilitation[END_REF], the proposed 1-DoF finger exoskeleton is extremely compact and lightweight: it weighs only 18 g for 117×30×42 mm dimensions. The complete kinesthetic and cutaneous device weighs 42 g for 117×50×43 mm dimensions. Moreover, it can be easily adjusted to fingers of different sizes. Finally, to the best of our knowledge, it is the first time two wearable devices are designed in such a modular way. A video showing the device can be found at https://youtu.be/N9 jFOafkKk

II. CONTROL

According to the type of interaction to be rendered, our device can be controlled in different ways. We present here four control schemes for our two modules, along with their evaluation. These control schemes will be then used in the experiments of Secs. III and IV.

A. Cutaneous device position control

The commanded tactile sensations can be provided to the user's fingertip by controlling the position and orientation of the 3-DoF platform. In the underactuated wearable devices of [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF], [START_REF] Chinello | A three dofs wearable tactile display for exploration and manipulation of virtual objects[END_REF], where the platform is controlled by three cables, position and force controls are coupled. In both schemes, the estimation of the platform displacement is approximated assuming a linear contact stiffness model. However, this solution presents several issues. The most evident is that the controller heavily relies on several parameters that cannot be measured or properly estimated, such as the size of the fingertip, or the age and sex of the user. The 3 DoF device presented in this paper, which is not underactuated, overcomes this indeterminacy. Its position and force control approaches can be in fact decoupled, and they do not require a model of the fingertip compliance.

In this section, we present a position control scheme for our fingertip device. Since the device has 3 DoF, only three of the six parameters describing the platform configuration can be controlled. As common for these devices [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF], [START_REF] Chinello | A three dofs wearable tactile display for exploration and manipulation of virtual objects[END_REF], we choose to control the translation in the direction normal to the platform plane, p z , together with its roll ψ and pitch θ rotations (see Fig. 1). We collect these variables in the vector ξ . Therefore, when we control the device in position, the objective is to make the mobile platform follow a desired trajectory

ξ d = [p zd , ψ d , θ d ] T .
Two position control schemes are possible. In the first one, shown in Fig. 2a, the desired configuration ξ d is compared with the actual one ξ . Using an inverse kinematics procedure [START_REF] Chinello | A three revolute-revolute-spherical wearable fingertip cutaneous device for stiffness rendering[END_REF], [START_REF] Tsai | Robot analysis: the mechanics of serial and parallel manipulators[END_REF], the error in the configuration space ∆ξ is transformed into an error in the input rotation space ∆q, where q = [q 1 , q 2 , q 3 ] T and the generic q i represent the rotation of the i-th proximal link, actuated by one of the three actuators (see Fig. 1). A PID controller is then used to define the torques τ = [τ 1 , τ 2 , τ 3 ] to be applied by the device actuators. In this case, the device dynamics depends on the applied torques τ and on the resulting force and moment generated by the interaction with the fingertip, collected in the sixdimensional wrench w. Finally, rotational encoders on the actuators measure actual rotations q of proximal links, and then, through a direct kinematic procedure, we evaluate the actual platform configuration. The main drawback of this approach is that it needs to solve the direct kinematic problem to estimate the platform configuration ξ from the measured actuator rotations q. This estimation is not straightforward and it requires iterative procedures that may be computationally heavy.

In the second control scheme, shown in Fig. 2b, the desired configuration ξ d is directly transformed into reference values for the actuators q d through a standard inverse kinematics procedure [START_REF] Chinello | A three revolute-revolute-spherical wearable fingertip cutaneous device for stiffness rendering[END_REF]. The actual values q are then directly measured by the encoders on the actuators. Finally, the torque applied by the motors is evaluated as a function of the error ∆q, using a standard PID controller.

Evaluation. Fig. 3a shows the performance of the second control scheme, shown in Fig. 2b. We commanded a step signal as the desired position

ξ d = [p zd , ψ d , θ d ] T = [3 mm, 15 • , 10 • ] T .
Results show that the device reaches the desired position with a rise time of about 0.35 s and an error in the stationary phase lower than 2%. The rise time is defined as the time required for the signal to rise from 10% to 90% of its steady value. Fig. 3b shows the behavior of the device when the desired trajectory is sinusoidal: p zd (t) = 1.5 sin(4t) + 1.5 mm, ψ d (t) = 7.5 sin(4t) + 7.5 deg, θ d (t) = 6 sin(4t) + 6 deg.

After this, to quantitatively evaluate the performance error, we asked the device to simulate the contact with n s = 100 arbitrarily oriented surfaces. At each repetition, the system selected a random target platform configuration ξ d,n , n = 1, . . . , 100, with

0 ≤ p zd ≤ 4 mm, -20 • ≤ ψ d ≤ 20 • , -20 • ≤ θ d ≤ 20 • ,
For each repetition, we evaluated the error ε n = ξ d,n -ξ a,n , where ξ d,n and ξ a,n indicate the n-th value of the desired and actual configurations, respectively, while ξ a,n is measured through the actuators' encoders and the direct kinematic procedure. For each sample, ε n = e z,n , e ψ,n , e θ ,n T , where ε z,n

is the position error in the z direction, and ε ψ,n and ε θ ,n are the angular errors, according to the definition of the platform configuration vector ξ previously introduced. The mean ± std. deviation along all the repetitions of ε z,n , ε ψ,n , and ε θ ,n , are 0.8 ± 0.5 mm, 1.72 ± 0.71 • , and 1.14 ± 0.29 • for p z , ψ, and θ , respectively. A Pearson's product-moment correlation is run to assess the relationship between the platform's position in the workspace and the above error. A preliminary analysis show that the relationship was linear with normally distributed variable, according to the Shapiro-Wilk test, and there are no outliers. There is no strong correlation between the platform position vs. rendering error (|r| < 0.3).

The same type of evaluation has also been carried out for the coupled control scheme shown in Fig. 2a. In this case, the mean error ± std. deviation is 1.1 ± 0.9 mm, 2.35 ± 0.87 • , and 1.72±0.56 • for p z , ψ, and θ , respectively. The same Pearson's product-moment correlation as above shows again no strong correlation between the platform position vs. rendering error.

B. Cutaneous device force control

Another way to control the fingertip device is to command a target force f d to be applied by the platform to the fingertip. To do this, let us assume that the mobile platform is equipped with a force sensor able to measure the contact force f m .

We consider a force control loop with an internal position control loop, shown in Fig. 2c [START_REF] Sciavicco | Modelling and control of robot manipulators[END_REF]. The desired contact force f d and the measured contact force f m are transformed into platform configurations by the compliance terms C f d and C f m , respectively. The difference between such values is the desired platform configuration ξ d , i.e.

ξ d = C f d f d -C f m f m (1)
The dimensions of f d and f m depend on the target application and sensing system. For example, if the platform is instrumented with a 3-DoF force sensor,

f m ∈ R 3 . In this case, if the target force f d is also in R 3 , C f d and C f m are 3 × 3 matrices, e.g., C f d = C f m =      0 0 c θ 0 c ψ 0 c z 0 0      , (2)
where c z is dimensionally a compliance, since it transforms the force error along z in a reference displacement in the same direction, while c ψ and c θ are rotations over a force. These compliance values depends on several parameters, including the user's finger dimension and mechanical characteristics [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF].

In the literature [START_REF] Pacchierotti | Wearable haptic systems for the fingertip and the hand: Taxonomy, review, and perspectives[END_REF], these values are often fixed and evaluated as an average of the human's fingertip skin compliance. In our prototype of Fig. 1, the platform is only instrumented with a 1-DoF force sensor, which measures the contact force orthogonal to the platform surface. In this case, we consider a simplified form for C f m , e.g., C f m = [c z , 0, 0] T . In other words, the force measured can be only used to regulate the reference value p zd . In this case, C f d can be defined in different ways: we can choose again C f d = C f m , i.e., for the reference rotations we have ψ d = θ d = 0; or we can define C f d as a 3 × 3 matrix, as in eq. ( 2), i.e., we can set a three-dimensional force reference that is transformed by C f d in a reference configuration vector. This latter approach is the one adopted for the teleoperation experiment of Sec. IV. The internal position control is similar to the one previously described and shown in Fig. 2b. In Fig. 2c, we also highlight the possibility of regulating the proportional term of the PID position controller, so as to simulate a platform desired stiffness K d [START_REF] Chinello | A three revolute-revolute-spherical wearable fingertip cutaneous device for stiffness rendering[END_REF]. In this case, it is necessary to evaluate the corresponding actuator stiffness matrix K q , which depends both on the desired stiffness and the platform current configuration, according to the following relationship

K q = JT K d Jδ q, ( 3
)
where J represents the device Jacobian matrix, which depends on device configuration and can be evaluated by analyzing the differential kinematics problem [START_REF] Tsai | Robot analysis: the mechanics of serial and parallel manipulators[END_REF]. The torque τ is then evaluated as

τ = K q (q d -q) . (4)
Since J depends on the platform's position and orientation, to simulate the interaction with a surface with a desired stiffness, K q needs to be evaluated at runtime, according to the current configuration. Moreover, a derivative term D q is present in the internal control loop so as to achieve a suitable damping in the system's response.

Evaluation.

To quantitatively evaluate the performance of our device in rendering a certain target force f d = [ f xd , f yd , f zd ] T , we placed an ATI Mini 25 sensor (ATI Industrial Automation, Inc.) between the mobile platform and the upper body (i.e., where the fingertip should be). Then, we asked the mobile platform to apply n s = 100 random force vectors on the sensor. At each repetition, the system selected a random force vector f d,n , n = 1, . . . , 100, with

0 ≤ f xd ≤ 2 N, 0 ≤ f yd ≤ 2 N, 0 ≤ f zd ≤ 4 N.
Then, we compared each desired force f d,n with the actual force f m,n , measured through the ATI sensor. The platform was held in each configuration for 1 s, and the values gathered by the ATI were arithmetically averaged. The mean error ± std. deviation is 0.32 ± 0.14 N, 0.18 ± 0.12 N, and 0.43 ± 0.19 N for f x , f y , and f z respectively. Similarly to Sec. II-A, a Pearson's product-moment correlation is run to assess the relationship between the platform's position in the workspace and the above error. There is no strong correlation between the platform position vs. rendering error (|r| < 0.3). Fig. 4 shows the performance of this control scheme.

C. Exoskeleton position control with variable stiffness

For the kinesthetic exoskeleton, we implemented a position control in which the stiffness can be varied. This control scheme is depicted in Fig. 5. A desired rotation of the finger phalanx θ d is set according to the considered interaction, i.e., to make the finger follow a certain trajectory (see Fig. 1). A straightforward kinematic analysis of the system composed by the exoskeleton and the user's finger allows to evaluate the corresponding value for the motor angle α d , that is then compared with the measure of the resolver. Finally, the error signal is used to evaluate the torque τ d applied by the motor by means of a simple proportional control scheme

τ d = k e (α d -α) , (5)
where the proportional constant (stiffness) k e is varied according to the target interaction. A video showing the control of the exoskeleton with different k e can be found at https: //youtu.be/dw-o8nslgT0. Evaluation. To quantitatively evaluate the performance of our exoskeleton in rendering a desired stiffness k ed , we fixed the proximal clamp to a rigid support and the distal clamp to a dynamometer. Then, we applied a constant force to the distal clamp and we registered the actuator's displacement with different desired stiffness values 0 ≤ k ed ≤ 2 Nmm/rad. We ran n s = 100 iterations of this performance evaluation experiment. The exoskeleton was held in each configuration for 1 s, and the values gathered by the potentiometer were arithmetically averaged. We registered a mean error ± std. deviation of 0.67 ± 0.10 Nmm/rad.

III. EXPERIMENT #1: CURVATURE DISCRIMINATION

To test the effectiveness of the proposed haptic device in rendering shapes, we carried out a curvature discrimination experiment. This experiment is inspired by the work of Prattichizzo et al. [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF] and Frisoli et al. [START_REF] Frisoli | A fingertip haptic display for improving curvature discrimination[END_REF], who evaluated their fingertip systems using a similar approach. A video showing this curvature discrimination experiment can be found at https://youtu.be/ dKfbKOf14c.

A. Experimental setup and participants

Similarly to [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF], [START_REF] Frisoli | A fingertip haptic display for improving curvature discrimination[END_REF], we used the same-different procedure to measure the just noticeable difference (JND) for curvature [START_REF] Gescheider | Psychophysics: the fundamentals[END_REF]. Six participants (4 males, 2 females, age range 24 -33) took part to the experiment. One of them had previous experience with haptic interfaces. The experimental setup was composed of our complete wearable haptic device, shown in Fig. 1, and a Leap Motion tracking system.

B. Experimental task and conditions

Participants were blindfolded and asked to wear the proposed device on their right index finger. According to the aforementioned same-different procedure, participants were required to explore, in succession, two virtual spheres. The surface of each sphere was rendered at 20 cm from the table. The task consisted in judging if the curvature of the two virtual spheres was different or the same, relying on the feedback provided by the proposed wearable device. We registered the rate of correct responses given when the two spheres had different curvatures (hit rate, p h ) and the rate of incorrect responses when the two spheres had the same curvature (false alarm rate, p f ) [START_REF] Gescheider | Psychophysics: the fundamentals[END_REF].

We considered two different feedback conditions:

• Condition CK: kinesthetic and cutaneous feedback, provided by the complete device (3-DoF fingertip module + 1-DoF finger exoskeleton), and • Condition K: kinesthetic feedback, provided by the 1-DoF kinesthetic module only. In condition CK, both the cutaneous fingertip module and the kinesthetic finger module provide haptic cues to the subject. The fingertip display provides cutaneous feedback about the local geometry of the surface, according to the decoupled position control scheme described in Sec. II-A; while the finger exoskeleton provides kinesthetic force normal to the surface, according to the variable stiffness position control scheme described in Sec. II-C. In condition K, only the kinesthetic finger module is active, while the 3-DoF mobile platform does not move. On each repetition of each series, two spheres with random curvature (κ a, * or κ b ) were rendered. In this way, the probability of exploring a pair of spheres with the same (different) curvature was 0.5. To minimize learning effects, the order of presentation of the series and feedback conditions was randomized.

C. Results

False alarm and hit rates were recorded for each subject and converted to z scores of the normal distribution [START_REF] Gescheider | Psychophysics: the fundamentals[END_REF], [START_REF] Stanislaw | Calculation of signal detection theory measures[END_REF]. According to the criterion already adopted in [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF], [START_REF] Frisoli | A fingertip haptic display for improving curvature discrimination[END_REF], [START_REF] Gescheider | Psychophysics: the fundamentals[END_REF], we evaluated the discrimination threshold as the difference between the curvatures for which d ′ = z hz f = 1, where d ′ is the so-called "sensitivity index" [START_REF] Gescheider | Psychophysics: the fundamentals[END_REF]. The JND was then computed for each subject and for each condition CK and K, assuming d ′ to increase linearly. Finally, the overall JND was computed as the average of the JND values of all the participants. Fig. 6 shows the mean sensitivity measure d ′ vs. ∆κ and the average interpolating lines for condition K (dashed lines), CK (solid lines), κ b,1 (blue lines), and κ b,2 (red lines). A Shapiro-Wilk normality test confirmed the normality of the registered data. A parametric two-tailed paired t-test showed that the average JND values were significantly lower (t = 7.633, p = 0.001) for condition CK than for K, with an average ± std. of 2.41 ± 0.11 m -1 and 3.00 ± 0.16 m -1 for conditions CK and K, respectively. Time needed to complete the given task was recorded as well. No statistical difference was found between the two conditions for this metric.

Results show that the combination of cutaneous and kinesthetic feedback leads to better performance than employing solely kinesthetic feedback. These results confirm the effectiveness of the proposed cutaneous/kinesthetic solution and that the cutaneous display of surface orientation helps the haptic perception of shape. The discrimination threshold for curvature observed in this work is in fact in agreement with previous results in the literature [START_REF] Prattichizzo | Towards wearability in fingertip haptics: a 3-DoF wearable device for cutaneous force feedback[END_REF], [START_REF] Frisoli | A fingertip haptic display for improving curvature discrimination[END_REF]. An interesting work analyzing the role of kinesthetic and cutaneous cues in curvature discrimination can be found in [START_REF] Wijntjes | Local surface orientation dominates haptic curvature discrimination[END_REF].

At the end of this experiment, we asked the participants to rate the perceived effectiveness of the two feedback conditions, along with the comfort and level of wearability of the proposed haptic system. The responses were given using a slider that ranged from 0 to 10, where a score of 0 meant "very low" and a score of 10 meant "very high". Wearability of the system was rated 8.2 out of 10 and the comfort of the system was rated 7.7 out of 10. The perceived effectiveness of conditions CK and K were rated 8.0 and 6.7 out of 10, respectively.

IV. EXPERIMENT #2: ROBOT-ASSISTED PALPATION

To evaluate the effectiveness and viability of our haptic system in a robotic scenario, we carried out a robot-assisted palpation task enrolling 20 human subjects. This experiment is inspired by the work of Pacchierotti et al. [START_REF] Pacchierotti | Cutaneous feedback of fingertip deformation and vibration for palpation in robotic surgery[END_REF], who evaluated their cutaneous fingertip device in a palpation experiment using a da Vinci Surgical System. A video showing this palpation experiment can be found at https://youtu.be/P2oABhEcBUI.

A. Experimental setup and participants

Fig. 7 show the experimental setup. The slave system is composed of a six-axis ATI Nano25 force sensor mounted on the end-effector of a 6-DoF Universal Robot UR5 manipulator. It has high speed output, span temperature compensation, and signal-to-noise ratio. The master side is composed of our complete wearable haptic device attached to the end-effector of a Novint Falcon device. The Falcon is a 3-DoF parallel grounded haptic interface, but here it is only used to track the position of the user's finger, and it did not provide any force feedback. The remote environment is composed of a simulated prostate tissue model made from Ecoflex 0010 (Smooth-On Inc., USA) and brown dye. To simulate the presence of a tumor [START_REF] Lee | Prostate cancer: comparison of transrectal us and digital rectal examination for screening[END_REF], a stiff plastic sphere with a diameter of 1 cm is embedded into the tissue model at 3 mm from the surface (see Fig. 8). The plastic sphere is not visible from the outside. To minimize the utility of any incidental visual cues and to reduce any learning effect, the study used four copies of the prostate model, that were interchanged between trials. The location of the stiff sphere changed across the different copies, but its depth was kept constant at 3 mm from the surface. Twenty participants took part in the experiment, including 5 women and 15 men. Seven of them had previous experience with haptic interfaces. The experimenter explained the procedures and spent about five minutes adjusting the setup to be comfortable before the subject began the experiment. Each subject then spent about two minutes practicing controlling the slave robot through the master interface.

B. Implementation of the robotic teleoperation system

The teleoperation system was managed by a GNU/Linux machine (Ubuntu 16.04 with Linux Kernel 4.4).

We use the Novint Falcon to measure the position of the subject's finger and set the one of the slave end-effector. The Falcon does not provide any force feedback. The velocities of the robotic manipulator joints qr ∈ R 6 are then commanded as

qr = J # r ṗh υ m , (6)
where J r ∈ R 3×6 represents the manipulator Jacobian matrix [START_REF] Sciavicco | Modelling and control of robot manipulators[END_REF], apex # indicates its pseudo-inverse, ṗh ∈ R 3 are the velocities of the haptic interface's end-effector, and υ m ∈ R is the scaling factor between the master and slave workspaces. Scaling factor υ m is tunable to enable coarse and fine gestures.

In this experiment, υ m was set to 1.3 in order to enable the subject to explore all the surface of the tissue phantom without clutching. The 3-DoF haptic interface registers the motion of the operator at 500 Hz. The Universal Robot manipulator then follows the motion commanded by the haptic interface at 125 Hz, according to eq. ( 6). Finally, the ATI force/torque sensor registers interaction forces at the slave side at 1 kHz. Since cutaneous feedback does not affect the stability of the teleoperation loop, we did not implement any specific stability controller for conditions N and C [START_REF] Pacchierotti | Cutaneous haptic feedback to ensure the stability of robotic teleoperation systems[END_REF], [START_REF] Ryu | Stable teleoperation with time-domain passivity control[END_REF]. On the other hand, the stability of condition CK was guaranteed by the passivity controller described in [START_REF] Franken | Bilateral telemanipulation with time delays: A two-layer approach combining passivity and transparency[END_REF], [START_REF] Pacchierotti | Steering and control of miniaturized untethered soft magnetic grippers with haptic assistance[END_REF].

C. Experimental task and conditions

The task consisted of exploring the tissue model to try to detect the location of the hidden plastic sphere. The task started when the Nano25 sensor touched the tissue model for the first time and ended when the subject told the experimenter that he found the hidden sphere. Each participant performed twelve trials of the palpation task, with four repetitions for each of the following three feedback conditions:

• Condition N: no haptic feedback, • Condition C: cutaneous feedback only provided by the 3-DoF fingertip module, • Condition CK: cutaneous and kinesthetic feedback provided by the complete device. In all conditions, the user was asked to wear the complete wearable device as shown in Fig. 7b. In condition N, the cutaneous and kinesthetic modules were not active. The mobile platform was always in contact with the subject's fingertip and the kinesthetic module was driven to apply no force to the subject's finger. In condition C, the cutaneous module provided cutaneous stimuli to render the interaction with the tissue phantom, according to the force control scheme described in Sec. II-B. Again, the kinesthetic module was driven to apply no force to the subject's finger. In condition CK, both the cutaneous and kinesthetic modules provided cutaneous and kinesthetic stimuli, respectively, to render the interaction with the tissue phantom, according to the control schemes described in Secs. II-B and II-C. The slave system was placed at 1 m from the master console, so that subjects were always able to see the operative environment.

Each subject performed all four repetitions of a single feedback condition as a block, and the order of the conditions was randomized to test all six possible combinations exactly three times. At the end of each condition, each subject was asked to rate, on two sliders going from 0 to 10, "how easy was it to detect the location of the stiff sphere?" and "how confident were you in detecting the location of the stiff sphere?" A score of 0 meant "very difficult" or "not at all confident," and a score of 10 meant "very easy" or "very confident." At the end of the experiment, each subject was asked to choose which feedback conditions were the most and least effective at enabling detection of the sphere's position.

D. Results

To evaluate the subject's performance under each of the considered feedback conditions, we evaluated (1) the absolute error in detecting the location of the plastic sphere, (2) the task completion time, and (3) the root mean square (RMS) pressure exerted by the Nano25 sensor on the tissue model. A low value of these three metrics denotes the best performance.

Fig. 9a shows the absolute position error results for the three experimental conditions. The position error is calculated as the planar distance between the location indicated by the subject on the phantom surface and the projection of the center of the sphere on the same surface. A repeated-measures ANOVA showed a statistically significant difference between the means of the three feedback conditions (F 2,38 = 9.245, p = 0.001, a = 0.05). Post-hoc analysis (Games-Howell posthoc test) revealed statistically significant differences between conditions N and C (p = 0.007) and between N and CK (p = 0.004). Fig. 9b shows the completion time results. The collected data passed the Shapiro-Wilk normality test and Mauchly's Test of Sphericity. A repeated-measures ANOVA showed a statistically significant difference between the means of the three feedback conditions (F 2,38 = 6.132, p = 0.005, a = 0.05). Post-hoc analysis (Games-Howell post-hoc test) revealed statistically significant differences between conditions N and C (p = 0.037) and between N and CK (p = 0.013). Fig. 9c shows the RMS force exerted by the slave end-effector on the tissue model, registered as the norm of the 3-DoF force sensed by the ATI Nano25. A repeated-measures ANOVA showed a statistically significant difference between the means of the three feedback conditions (F 2,38 = 22.922, p < 0.001, a = 0.05). Post-hoc analysis (Games-Howell post-hoc test) revealed statistically significant differences between conditions N and C (p = 0.016), N and CK (p < 0.001), and C and CK (p = 0.011).

Finally, we analyzed the ratings given by the subjects at the end of each feedback condition. Fig. 9d shows the ratings that the three feedback conditions received for the first question ("how easy"). A repeated-measures ANOVA showed a statistically significant difference between the means of the three feedback conditions (F 2,38 = 44.707, p < 0.001, a = 0.05). Post-hoc analysis (Games-Howell post-hoc test) revealed statistically significant differences between conditions N and C (p < 0.001) and between N and CK (p < 0.001). Fig. 9e shows the ratings that the three feedback conditions received for the second question ("how confident"). A repeated-measures ANOVA showed a statistically significant difference between the means of the three feedback conditions (F 2,38 = 8.034, p = 0.001, a = 0.05). Post-hoc analysis (Games-Howell posthoc test) revealed statistically significant differences between conditions N and C (p = 0.007) and between N and CK (p = 0.029). Fourteen subjects chose condition CK as the most effective feedback condition, four subjects chose condition C, and only two chose condition N. The two outlier subjects who chose condition N said that they considered the haptic feedback useless. Subjects were asked to interact with the virtual environment while wearing two wearable cutaneous devices, one on the thumb and one on the index finger. The virtual environment is composed of seven different objects lying on a table. Interaction forces in the virtual environment are provided to the user by the cutaneous device as indicated in Sec. II.

V. EXPERIMENT #3: INTERACTION IN A VIRTUAL REALITY (VR) ENVIRONMENT

Finally, we carried out a third experiment in a virtual reality environment. This experience has been inspired by the experiments carried out in [START_REF] Chinello | A three revolute-revolute-spherical wearable fingertip cutaneous device for stiffness rendering[END_REF], [START_REF] Scheggi | Touch the virtual reality: using the leap motion controller for hand tracking and wearable tactile devices for immersive haptic rendering[END_REF], [START_REF] Meli | Wearable haptics and hand tracking via an rgb-d camera for immersive tactile experiences[END_REF], [START_REF] Meli | Combining wearable finger haptics and augmented reality: User evaluation using an external camera and the microsoft hololens[END_REF].

A. Experimental setup and participants

Fig. 10 shows the experimental setup. It is composed of our complete wearable haptic device and a virtual environment with 7 different objects. Fourteen participants (10 males, 4 female, age range 25 -30) took part to the experiment, all of whom were right-handed. Eight of them had previous experience with haptic interfaces.

B. Experimental task and conditions

Users are asked to wear two prototypes of our complete haptic devices on the thumb and index fingers of their right hand, and interact with the virtual environment. The users hand pose is tracked using an Optitrack system, and a virtual hand mimicked the subjects hand pose in the virtual environment. Every time one of the virtual fingers comes in contact with a Interaction forces in the virtual environment were computed using a virtual proxy algorithm [START_REF] Ruspini | Haptic interaction in virtual environments[END_REF] and they were provided by the haptic device as indicated in Secs. II-B and II-C. This algorithm uses a proxy point attached to the haptic interaction point by a virtual spring (representing the surface stiffness). When the haptic interaction point moves within a virtual object, the proxy point is constrained to the object surface, stretching the spring and defining a virtual interaction force.

C. Results

We evaluated the immersiveness of the haptic-enabled virtual reality scenario through the Usefulness, Satisfaction, and Ease of use (USE) questionnaire [START_REF] Lund | Measuring usability with the use questionnaire12[END_REF]. At the end of each condition, subjects were asked to rate their experience on a slider going from 0 to 15, where a score of 15 was described as "completely agree" and a score of 1 as "completely disagree" with the assertion. Similarly to [START_REF] Chinello | A three revolute-revolute-spherical wearable fingertip cutaneous device for stiffness rendering[END_REF], our USE questionnaire is composed of 21 questions. Questions and answers for each condition are reported in Table I. To determine whether the answers registered differ between the three feedback conditions, we ran twenty-one repeated-measures ANOVA tests (significance level alpha = 0.05), one for each question. Results of this statistical analysis are reported in Table II.

We also asked which condition the subjects preferred. Six subjects preferred condition CK, three condition C, and one condition N.

VI. DISCUSSION, CONCLUSION AND PERSPECTIVES

In this paper, we presented the control and evaluation of a novel wearable modular 4-DoF exoskeleton for haptic interaction. The device is composed of a 3-DoF fingertip cutaneous device and a 1-DoF finger exoskeleton, which can be either used together as a single device or separately as two different devices. It weighs only 42 g for 117×50×43 mm dimensions. With respect to other wearable solutions presented in the literature, the proposed device is extremely compact and lightweight, and it features a quite unique modular structure. Moreover, the 3-DoF fingertip module solves the indeterminacy due to the underactuation of the platform, and the 1-DoF finger module has been designed to be easily adjusted to different finger sizes.

To test its effectiveness and wearability, we carried out three experimental evaluations, enrolling a total of 40 human participants. The first experiment tested the capability of our device in differentiating objects with different curvatures. Results showed that providing both cutaneous and kinesthetic feedback (CK) led to a JND significantly lower than providing only kinesthetic feedback (K). Therefore, users were able to better differentiate the curvatures when also provided with cutaneous information. All subjects found our system wearable, comfortable, and easy to wear. All of them were able to don the device with minimal assistance. It is interesting to notice that the tracking of the hand worked quite well even using a Leap Motion sensor. This may seem quite surprising, as the exoskeleton significantly occludes the user's hand. However, we believe that the Leap Motion still managed to achieve a satisfactory tracking thanks to the fact that users kept their hand mostly open.

The second experiment considered a robot-assisted palpation task. Providing haptic feedback (either C or CK) led to a significantly better performance with respect to not providing any force feedback (N) in all the considered metrics. Moreover, providing both kinesthetic and cutaneous feedback (CK) did not significantly reduce the error in locating the sphere or the completion time with respect to providing cutaneous feedback only (C). This result means that, for this task, providing additional kinesthestic feedback was not particularly useful, and that the local geometry information provided by the 3-DoF cutaneous module was already sufficient. However, in condition CK, users exerted significantly less force than in condition C, which is quite important, as applying excessive force when manipulating tissue may lead to significant damage [START_REF] De | Assessment of tissue damage due to mechanical stresses[END_REF]. Finally, we also evaluated the user's preference when using the system, registering preference ratings for questions "how easy was it to detect the location of the stiff sphere?" and "how confident were you in detecting the location of the stiff sphere?". Providing haptic feedback (either C or CK) led to significantly better ratings with respect to not providing any force feedback (N). As before, providing both kinesthetic and cutaneous feedback did not produce significantly higher ratings with respect to providing cutaneous feedback only (C). These results are in agreement with the experiment carried out by [START_REF] Pacchierotti | Cutaneous feedback of fingertip deformation and vibration for palpation in robotic surgery[END_REF]. Also there, providing cutaneous feedback during a robot-assisted palpation task led to improved performance with respect to providing no haptic feedback at all. However, the device used in [START_REF] Pacchierotti | Cutaneous feedback of fingertip deformation and vibration for palpation in robotic surgery[END_REF] did not provide any kinesthetic feedback and it was not designed to be wearable/portable. In this second experiment, to precisely track the position of the fingers, we used a Falcon interface, which guaranteed a high tracking accuracy (i.e., more than 10 times higher than the Leap Motion used in the first experiment). However, of course, this choice severely reduced the workspace of the system, and it did not enable us to evaluate again the portability and wearability of our system. Moreover, the Falcon interfaces, even when commanded to provide no external force, shows an internal stiffness and friction that is unwittingly displayed to the user. Nonetheless, since this effect was present in all the considered conditions, we expect it to have negligible effect in the final computation of the considered metrics.

The third and last experiment considered a Virtual Reality experiment. Users were asked to report on their experience through a 21-question Usefulness, Satisfaction, and Ease of use (USE) questionnaire, as well as to choose their preferred condition. In most questions, condition CK performed the best, followed by condition C and then by condition N. Similarly, most subjects preferred condition CK.

In the near future, we will work on extending the experimental evaluation, testing a larger set of feedback conditions, scenarios, and applications, enrolling an even higher number of participants. Finally, we will also study how the control of the platform should be adjusted for different finger shapes and sizes.

  (a) Coupled position control scheme. (b) Decoupled position control scheme.(c) Force control scheme.

Fig. 2 . 3 -

 23 Fig. 2. 3-DoF cutaneous device. Position and force control schemes.

Fig. 3 . 3 -

 33 Fig. 3. 3-DoF cutaneous device. Performance of the decoupled position control scheme shown in Fig. 2b.

  Sine wave response.

Fig. 4 . 3 -

 43 Fig. 4. 3-DoF cutaneous device. Performance of the force control scheme shown in Fig. 2c.

Fig. 5 .

 5 Fig. 5. Position control of the kinesthetic module.

Fig. 6 .

 6 Fig.6. Experiment #1. Mean sensitivity measure d ′ vs. ∆κ, together with the average interpolating lines for condition K (dashed lines), CK (solid lines), κ b,1 (blue lines), and κ b,2 (red lines).

Fig. 7 .

 7 Fig.7. Experiment #2: robot-assisted palpation. Setup. The slave system is composed of an ATI Nano25 sensor mounted on the endeffector of a 6-DoF Universal Robot UR5 manipulator. The master system is composed of our complete wearable haptic device attached to the end-effector of a Novint Falcon device. The Falcon is only used to track the position of the user's finger and does not provide any force feedback. The remote environment is composed of a simulated prostate tissue model, with a 1-cm-diameter stiff sphere embedded at 3 mm from the surface.

Fig. 8 .

 8 Fig. 8. Simulated prostate tissue model. The overall diameter and thickness of the model are 80 mm and 25 mm, respectively. A 1-cmdiameter plastic sphere is embedded at a depth of 3 mm from the surface to mimic a tumor.

  Error in locating the stiff sphere (cm).

Fig. 9 .

 9 Fig.9. Experiment #2: robot-assisted palpation. Mean and 95% confidence interval of the (a) error in locating the position of the sphere, (b) completion time, (c) RMS force applied by the slave robot on the tissue phantom, and preference ratings for questions (d) "how easy was it to detect the location of the stiff sphere?" and (e) "how confident were you in detecting the location of the stiff sphere?" are plotted. P-values of post-hoc group comparisons are reported when statistically different.

Fig. 10 .

 10 Fig.10. Experimental setup. Subjects were asked to interact with the virtual environment while wearing two wearable cutaneous devices, one on the thumb and one on the index finger. The virtual environment is composed of seven different objects lying on a table. Interaction forces in the virtual environment are provided to the user by the cutaneous device as indicated in Sec. II.

This research has received funding from Rennes M étropole (2017 AIS) and from the CNRS (2018 PEPS JCJC).