Farhang Radjai
email: franck.radjai@univ-montp2.fr

Jean-Yves Delenne

Emilien Azéma

Stéphane Roux

J.-Y Delenne

J.-Y Delenne

Fabric evolution and accessible geometrical states in granular materials

Keywords: Plastic behavior, Granular fabric, Coordination number, Anisotropy, Mohr circle F. Radjai (B)

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Introduction

The plasticity of granular materials is governed by frictional contact interactions among a large number of rigid-body degrees of freedom. The particle velocities are constrained by steric exclusions and, in the absence of thermal randomness, they scale with the driving strain. In a similar vein, the material is devoid of intrinsic stress scale in the limit of rigid particles (compared to the applied confining stress), and thus all internal stresses scale with the confining stress while sat-isfying at the same time the condition of static equilibrium of each particle [START_REF] Roux | Statistical approach to the mechanical behavior of granular media[END_REF][START_REF] Radjai | Contact dynamics study of 2D granular media: critical states and relevant internal variables[END_REF]. These features lead to a perfectly plastic behavior in which the instantaneous flow rule is defined by a strain ratio expressed as a dilatancy angle and the yield surface by a stress ratio expressed as an effective friction angle [START_REF] Wood | Soil Behaviour and Critical State Soil Mechanics[END_REF]. These angles are unequal (non-associated property) due to steric exclusions and, more importantly, they are strongly dependent on the current state of the material, which is fully encoded in the geometry of the granular assembly and evolves with strain (hardening property).

The local geometrical arrangement is nontrivial regarding its generic disorder both in particle positions and in the forcebearing contact network, and its evolution involves local nonaffine velocities induced by steric exclusions [START_REF] Radjai | Turbulentlike fluctuations in quasistatic flow of granular media[END_REF]. However, a coarse-grained representation of this microstructure is a key information for any continuum model of granular plasticity based on physical internal variables. This issue is still far from being settled even when approaching the problem under simplified assumptions such as spherical particles subjected to homogeneous shearing. The heart of the problem is whether one may work with the lowest-order descriptors of the microstructure-possibly at the price of accounting in a less strict sense for the local constraints. In particular, such state variables are expected to provide a minimal description of the reference states such as the critical state and the spectrum of isotropic states with its range of packing fractions, as well as the transients between those reference states.

In this paper, the contact network is portrayed by its lowest-order representation as a tensor combining the coordination number and fabric anisotropy. Its evolution in relation with the velocity field is studied by means of numerical simulations for a few loading paths. The idea is to characterize the range of accessible fabric states constrained by steric effects and force balance condition. A simple model is presented that provides a geometrical interpretation of the critical state.

Tensorial representation of the contact network

The granular microstructure may be described by the contact network, which is directly linked with the plastic behavior as it governs both stress transmission and volume change by shear flow. This granular fabric is disordered in (1) contact positions, (2) contact orientations, as described by the normals to the contact planes or the orientations n of the branch vectors = n joining the centers of contact neighbors, and (3) branch-vector lengths ; see Fig. 2a.

Assuming a statistically homogeneous network, as shown in Fig. 1, the fabric is thus represented by the set { α } of branch vectors and characterized by the probability density P n ( ) ≡ P n ( , n). For non-spherical particles, this function may be replaced by the richer probability density function of contact vectors, i.e. the vectors joining the particle centers to the contact points; Fig. 2a. Note that since the branch vectors α have no intrinsic polarity, the function P n ( ) is π -periodic.

The function P n ( ) is obviously insufficient for the description of individual particle environments (one particle with its contact neighbors), which involve multicontact distributions constrained by angular hinderances δθ as illustrated in Fig. 2b [START_REF] Troadec | Model for granular texture with steric exclusion[END_REF]. Although such a rich representation would be required in order to account for the local geometrical constraints together with higher order statistical information on three (and more) grain locations and orientations, the idea of the proposed approach is to include a minimal geometrical description and to connect it to the mechanical behavior. A scalar variable that provides the lowest-order description of the particle environments is the average number of contact neighbors z (the coordination number), which is a complementary information to P n . Additional local constraints will have to be reintroduced at a later stage from coarse-grained or average properties.

The reduced distribution P ( ) = Ω P n ( )dn of branchvector lengths, where Ω is the space of orientations, reflects the particle size distribution. The correlation between the orientations n and lengths controls the mean branch-vector length

(n) = ∞ 0 P n ( , n) d / ∞ 0 P n ( , n) d as a function of n.
Our simulations show that this function in a moderately polydisperse packing is nearly isotropic even after extended shear [START_REF] Azéma | Quasistatic rheology, force transmission and fabric properties of a packing of irregular polyhedral particles[END_REF]. We focus here on weakly polydis- Yet, P n is too rich to be accessed easily from experiments or handled theoretically. It is proposed to perform a moment expansion of this quantity and retain only the lowest-order terms. The π -periodicity implies n = 0 and therefore the lowest-order moment is the well-known fabric tensor defined by [START_REF] Satake | Fabric tensor in granular materials[END_REF]

F = n ⊗ n ≡ Ω n ⊗ n P n (n)dn
, where ⊗ denotes dyadic product. Since n • n = 1, we have tr(F ) = 1. Hence, the coordination number z and the fabric tensor can be combined in a single enriched tensor defined by

G i j = z 2 n i n j ≡ z 2 Ω n i n j P n (n)dn, (1)
where the roman subscripts denote vector components. Note that G is nothing but the fabric tensor per particle while F is per contact. The tensor G may easily be evaluated from discrete simulation data by G i j = ( α n α i n α j )/N p , where n α is the unit branch vector of contact α and N p is the total number of particles. In 2D, n is parameterized by its orientation θ with respect to a fixed frame and we have n = (cos θ, sin θ). In 3D, using spherical coordinates with azimuth and zenith angles θ and φ, we have n = (sin θ cos φ, sin θ sin φ, cos θ). For convenience, we assume here axial symmetry around the zenith axis so that G and P n are independent of φ. Let {G i } be the principal values and θ = θ c = 0 the major principal direction. We define the following invariants of G in 2D:

p G ≡ G 1 + G 2 2 = z 4 ( 2
)
q G ≡ G 1 -G 2 2 = z 4 cos 2θ (3)
and in 3D:

p G ≡ G 1 + G 2 + G 3 3 = z 6 ( 4
)
q G ≡ G 1 -G 3 3 = z 12 3 cos 2 θ -1 (5)
By symmetry, we have G 2 = G 3 . We note that 3 cos 2 θ -1 is the spherical harmonics function usually denoted Y 0 2 , which together with Y 0 0 = 1 are the only second-order basis func-tions compatible with axial symmetry. The spherical harmonics expansion is the natural 3D counterpart of the Fourier expansion in 2D which is easily read in the above definitions.

The average cos 2θ or 3 cos 2 θ -1 represents the anisotropy and it may be evaluated either from P n (see below) or directly from the numerical data. Given its invariants, the enriched fabric tensor G may be represented by a Mohr circle in the space of its normal and tangential projections G n (m) = Gm • m and G t (m) = Gm • t, respectively, along arbitrary space direction m parameterized by the angle θ . Simple algebra yields

G n (θ ) = p G + q G cos 2θ, ( 6
)
G t (θ ) = q G sin 2θ, (7)
in 2D and

G n (θ ) = p G + q G {3 cos 2 θ -1}, ( 8
)
G t (θ ) = 3 2 q G sin 2θ, (9)
in 3D.

The representation of the fabric by a second-order tensor through Eq. ( 1) keeps only the lowest mode of the Fourier (2D) or spherical harmonics (3D) spectrum of P n [START_REF] Rothenburg | Analytical study of induced anisotropy in idealized granular materials[END_REF]:

in 2D: P n (θ ) = 1 π {1 + a cos 2θ + • • • } , ( 10
) in 3D: P n (θ ) = 1 2π 1 + a(3 cos 2 θ -1) + • • • , ( 11
)
where a is the leading fabric anisotropy. Equation (1) with the expansions (10) and (11) lead to the following relations:

in 2D: a = 2 cos 2θ , ( 12
) in 3D: a = 5 4 3 cos 2 θ -1 . ( 13
)
With this expression of the anisotropy, we see that the description of the fabric by G is equivalent to a description by the truncated expansions:

in 2D: E(θ ) = z 2 P n (θ ) ≈ z 2π {1 + a cos 2θ } , ( 14
) in 3D: E(θ ) = z 2 P n (θ ) ≈ z 4π 1 + a(3 cos 2 θ -1) . ( 15
)
The function E(θ ) may assume various shapes with several modes of unequal amplitudes depending on the history or preparation method. Most of time, however, such configurations are erased as a result of shearing and the subsequent shear histories do not induce modes of order higher than two, so that Eqs. ( 14) and (15) offer a fair approximation of E(θ ). Figure 3 shows two examples of E in the two samples of Fig. 1 before and after shearing. The distribution is fitted by a second-order harmonic approximation, and higher-order terms may be neglected. In this case, the angular dependence of G n (θ ) coincides with that of E. In this way, the fabric state at leading order is represented by G or E(θ ), both characterized by the three variables z, a and θ c . The coordination number z is a simple scalar, but the fabric anisotropy a is associated with the major principal direction θ c of G. When the evolution of the fabric is studied, it is convenient to consider the "signed" anisotropy a ≡ a cos 2(θ 0 -θ c ) in 2D or a ≡ a{3 cos 2 (θ 0 -θ c ) -1} in 3D, where θ 0 is a fixed space direction, which may be the initial value of θ c .

Evolution with shear strain

The fabric evolution is governed by three elementary "events": (1) gain of contact , (2) loss of contact and (3) rotation of branch vectors. The gain and loss events are discontinuous whereas the rotation of branch vectors is a continuous advection of enduring contacts in the angular interval [0, π]. Let ε be the strain rate tensor. These events occur at different rates depending on the space direction θ : a gain rate I + (ε, θ), a loss rate I -(ε, θ) and a current J (ε, θ) of advected contacts, respectively. The evolution of E is thus governed by a detailed balance equation:

∂ E ∂t + ∂ J ∂θ = I = I + -I - (16)
We expect that I + has its largest value along the direction of contraction whereas I -is maximal along the direction of extension. The current J is basically given by the number density of contacts times the angular velocity in each direction. The three functions I + , I -and J depend on the velocity field in the vicinity of the particles. An example of the averaged velocity field is shown in Fig. 4. Without entering the Fig. 4 The field of relative velocities from numerical simulations averaged in the vicinity of all particles during simple shear deformation of a packing of disks along the horizontal direction. The distances are normalized by the average branch-vector length , which defines the radius of the mutual exclusion distance between particles represented by the gray circle in the center details here, we observe that the far field (uniform velocity gradient) is modified in the vicinity of the particles within a distance of the order of the average branch-vector length. However, this velocity field varies in proportion to the macroscopic velocity gradient although with dependencies on the fabric tensor itself. Note that in the absence of J , the relative number E of contacts in each direction would simply be an accumulation of the contacts induced by loss and gain and a steady state would never be reached. However, in the steady state we have ∂ J/∂θ = I , which means that, the contacts are gained on the average in the direction of contraction and advected towards the direction of extension where they are lost. In this way, the memory of the initial configuration is erased by shear deformation and the critical state may be described as a steady shear state in which, in spite of steric exclusions, the fabric reflects the average strain-rate tensor and may thus be described to a good approximation by a second-order fabric tensor or unimodal distributions of contact orientations.

Figure 5 shows the evolution of the fabric parameters z and a during shear as a function of the cumulative shear strain ε q = ε 1 -ε 2 , where ε 1 and ε 2 are the principal values of the strain tensor, for an initially dense system z 0 = z(ε q = 0) = 4 and an initially loose system z 0 = 3.3, as well as in the case where the shear deformation is reversed in the steady state. Notice that the terms "dense" and "loose" will refer throughout this paper to the connectivity of the contact network rather than its packing fraction. The numerical simulations were performed by means of the contact dynamics method [START_REF] Radjai | Contact dynamics as a nonsmooth discrete element method[END_REF]. The samples are composed of 5,000 disks of weak size polydispersity prepared by isotropic compaction so that a 0 = a (ε q = 0) = 0. During shear, the average stress p is kept constant and a constant shear rate εq is applied. We observe an initial rapid decrease of z in the dense case as a consequence of the quasi-instantaneous loss of contacts in the direction of extension. This leads at the same time to an increasingly larger structural anisotropy a with an excess of contacts oriented along the major principal strain-rate direction θ ε . A similar mechanism is at work in the loose system with a gain of contacts along θ ε and an increase of a . After a transient ε q 0.1, a steady state with z * 3.65 and a * 0.2 is reached in both samples. When a reverse strain is applied in the steady state, a falls off to zero but then increases in absolute value to reach a * in the new shear direction. As for z, its steady-state value z * is reached in the new shear-rate direction after a transient where z first declines as soon as the shear strain is reversed and then increases again towards z * . A similar unmonotonic evolution is observed when starting from the dense isotropic state: z first declines to a low value, then increases towards z * .

Accessible fabric states

As a result of local constraints, the fabric parameters do not take arbitrary values. The accessible range of state parameters contains part of the underlying constraints and it may thus partially replace the missing information when a low-order description is used. To illustrate the effect of constraints, let us consider the truncated Fourier function of E(θ ) in 2D and apply it to the contact neighbors of a typical particle. The angular exclusions imply that there can be at most one particle in the angular interval [-δθ/2, δθ/2] (see Fig. 2b) [START_REF] Troadec | Model for granular texture with steric exclusion[END_REF]:

δθ 2 -δθ 2 E(θ ) dθ ≤ 1. ( 17
)
This leads to an upper bound a max on the anisotropy:

a ≤ a max = 2π sin(δθ ) 1 z - 1 z max , ( 18
)
where z max = 2π/δθ is the upper bound on the coordination number due to angular exclusions. However, the function E(θ ) can not be used to describe the local environments because of steric exclusions. We introduce here a simple model based on the fabric tensor G in order to account, though in a less strict way, for the local constraints. In a co-rotating reference frame with axes oriented along θ c (ε q ) and θ c (ε q )+π/2, the fabric state is defined by its position in the space of coordinates z and a. We assume that z is bounded between two limits z min and z max . The upper bound z max is dictated by steric exclusions as discussed above. The lower bound z min reflects the condition of mechanical equilibrium. For example, stable particles often involve more than three contacts in 2D and more than four contacts in 3D. We first consider exclusively the 2D case. The results for the 3D system will be given at the end of this section.

We define two limit states: (1) the loosest isotropic state, characterized by G t = 0 and G min n = z min /4, and (2) the densest isotropic state, characterized by G t = 0 and G max n = z max /4. All accessible fabric states, represented by a Mohr circle on the (G n , G t ) plane as shown in Fig. 6, are enclosed between these two limit states, so that

G 1 ≤ G max n and G 2 ≥ G min n .
Within this range a and z, which describe the current geometrical state, are independent parameters unless one of the two equalities

G 2 = G min n or G 1 = G max n
occurs, in which case p G and q G may no more evolve independently. The condition G 2 = G min n implies p G -q G = z min /4, which together with Eqs. (3) and (12) yields a = 2(1z min /z). This limit corresponds to a loss saturation limit where the mechanical equilibrium implies that no contacts may be lost along the direction of extension. In the same way, the condition G 1 = G max n implies p G + q G = z min /4, which together with Eqs. (3) and (12) yields a = 2(z max /z -1). This is the gain saturation limit where no contacts may be gained along the direction of contraction. Hence, all accessible fabric states in the space (z, a) belong to a domain defined by

a max (z) = 2 min 1 - z min z , z max z -1 . ( 19
)
The largest anisotropy a occurs when both conditions (20)

G 1 = G max
This state corresponds to a Mohr circle touching the line G t = tan α G n where sin α = a /2, as shown in Fig. 6b. Figure 7 shows the joint evolution of a and z in a simulated biaxial compression of two initially isotropic samples with coordination numbers z 0 = 3.1 and z 0 = 3.7. In both simulations, the fabric tends to the critical state with z 3.35 and a 0.24 and coinciding with the state of maximum anisotropy in the model. Remarkably, the anisotropy of the dense packing reaches and then follows closely a limit line given by Eq. ( 19). The values of z max and z min were obtained from the those of z and a using Eq. (20). In the case of the loose sample, the path remains entirely inside the accessible domain and the boundary is reached only at the critical state. This description provides a geometrical interpretation of the critical state. The gain and loss saturation lines meet at the critical state. In this sense, the critical state corresponds to a fully saturated state where both contact gain along the direction of contraction and contact loss along the direction of extension are saturated although the shearing continues to induce gain and loss of contacts in different directions. From this viewpoint, the critical state simply reflects the limit isotropic states. The critical-state anisotropy a increases with z maxz min . The shape, size and frictional characteristics of the particles may therefore influence a via z min and z max . Figure 8 shows az as a function of z for five different simulations with the same sample of 13,000 disks with z 0 = 3.72 but different values ofμ. 1 According to (19), the limit states in this representation are straight lines. The data follow different gain-saturation limit lines with z max varying between 3.7 and 3.8, and reach a peak state followed by a critical state in which z decreases as μ increases whereas a increases nearly as 1/z . Since z max varies slightly with μ, the dom-Fig. 8 Fabric evolution in the space (z, az) for different values of the friction coefficient μ and the same initial isotropic state. The dashed lines are approximate fits to the reference lines predicted by the model inant effect of increasing μ is to allow for lower values of z min and thus larger values of a . It is to be noted that a small systematic deviation from the linear law is observed in the final approach to the critical point. Such a small effect may well be due to the truncation of P n to second order, as, at maximum anisotropy, our (angularly) local interpretation of E from its first two moments becomes fragile.

The accessible fabric states may be modeled also in 3D by means of the Mohr circle on the fabric plane. It can be shown that, under axial symmetry, the critical state is characterized by

z * = 1 3 (z max + 2z min ), (21)
a * = 5 2 z max -z min z max + 2z min . ( 22
)
For example, for the typical values z min = 4 and z max = 6 of a sphere packing, we get z 4.6 and a 0.35, which are plausible critical state values in 3D. It is obvious that z can take much larger values, but such states are not reachable by shearing. When a packing with a high value of z is sheared, the accessible range of fabric parameters, as defined by the model, may be reached from the outside, but the subsequent evolution is likely to take place only inside this range. Various protocols might be designed for the estimation of z max and z min as limit values of the range of accessible fabric states. In this respect, our model may also be regarded as an executive framework in which those values are simply determined from the (z, a) data and interpreted as state-independent parameters characterizing a granular material.

Conclusion

In this paper, a general framework for fabric evolution was presented with the guiding idea that a physical plastic model of granular materials should be based in the first place on low-order parameters pertaining to the granular microstructure but accounting more or less strictly for steric exclusions and mechanical equilibrium of the particles. We introduced a fabric tensor that, by combining the coordination number and fabric anisotropy, allows for a simple tensorial representation of the fabric states by means of Mohr circles. We also briefly discussed how low-order fabrics are induced by homogeneous shearing, and numerical data of fabric evolution were presented for loose and dense isotropic states and reversed strains from the critical state. Finally, a model was introduced for the range of accessible fabric states. Assuming limit values of the coordination number, reflecting the local constraints, this model predicts in a simple way limit lines and an upper bound on the fabric anisotropy. It provides also a geometrical interpretation of the critical state as a "saturated" state reflecting particle-scale constraints.

Fig. 1

 1 Fig. 1 Contact networks in 2D and 3D granular packings

Fig. 2 a

 2 Fig. 2 a Contact geometry; b A particle environment with angular exclusions perse systems where P n ( , n) ≈ P ( )P n (n) and the distribution of branch length P is narrow. Thus the orientation part P n (n) contains most of the information.Yet, P n is too rich to be accessed easily from experiments or handled theoretically. It is proposed to perform a moment expansion of this quantity and retain only the lowest-order terms. The π -periodicity implies n = 0 and therefore the lowest-order moment is the well-known fabric tensor defined by[START_REF] Satake | Fabric tensor in granular materials[END_REF] F = n ⊗ n ≡ Ω n ⊗ n P n (n)dn,where ⊗ denotes dyadic product. Since n • n = 1, we have tr(F ) = 1. Hence, the coordination number z and the fabric tensor can be combined in a single enriched tensor defined by

Fig. 3

 3 Fig.[START_REF] Radjai | Turbulentlike fluctuations in quasistatic flow of granular media[END_REF] Distribution of branch-vector orientations in 2D a and 3D b before (circles) and after (triangles) shearing with major principal strain-rate direction along θ = 0. The solid lines are harmonic fits

Fig. 5

 5 Fig. 5 The evolution of fabric parameters, coordination number (a) and "signed" anisotropy (b). Different curves represent different initial values of the packing fraction

n and G 2

 2 = G min n are satisfied simultaneously. Hence,a =a max (z )=2 z maxz min z max + z min with z = 1 2 (z min + z max ).

Fig. 6 aFig. 7

 67 Fig. 6 a Fabric state represented by a Mohr circle. L 1 and L 2 are the limit isotropic states. b The critical state on the fabric plane. The cone represents the limit of maximum anisotropy reached when the Mohr circle touches the limit states in the critical state

See www.http://cgp-gateway/ref011 for animation videos.


L Éo Terziman

Maud Marchal

Franck Multon

Bruno Arnaldi

Anatole L Écuyer

The King-Kong Effects: Improving Sensation of Walking in VR with Visual and Tactile Vibrations at each Step

Keywords: I, 3, 7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Virtual Reality

In this paper we present novel sensory feedbacks named "King-Kong Effects" to enhance the sensation of walking in virtual environments. King Kong Effects are inspired by special effects in movies in which the incoming of a gigantic creature is suggested by adding visual vibrations/pulses to the camera at each of its steps. In this paper, we propose to add artificial visual or tactile vibrations (King-Kong Effects or KKE) at each footstep detected (or simulated) during the virtual walk of the user. The user can be seated, and our system proposes to use vibrotactile tiles located under his/her feet for tactile rendering, in addition to the visual display. We have designed different kinds of KKE based on vertical or lateral oscillations, physical or metaphorical patterns, and one or two peaks for heal-toe contacts simulation. We have conducted different experiments to evaluate the preferences of users navigating with or without the various KKE. Taken together, our results identify the best choices for future uses of visual and tactile KKE, and they suggest a preference for multisensory combinations. Our King-Kong effects could be used in a variety of VR applications targeting the immersion of a user walking in a 3D virtual scene.

INTRODUCTION

Immersion is fundamental in Virtual Reality (VR) where the user experiences and interacts with virtual worlds. In particular, navigation is one essential task in VR [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF] for which immersion can be crucial. Many techniques have been developed to produce a strong sense of immersion in VR. However, immersion is not a notion exclusive to the VR. Indeed, breathtaking special effects have become common in Hollywood movies to make the experience of the spectators more intense and more realistic, creating a strong immersion feeling. For example, some movies use special visual effects to suggest the incoming of a gigantic creature. Visual vibrations/pulses are applied on the camera at each of the creature steps.

To improve immersion when navigating in Virtual Environments (VE), researchers have developed numerous kinds of sensory feedback. For example, Lécuyer et al. [START_REF] Lécuyer | Camera motions improve the sensation of walking in virtual environments[END_REF] proposed Camera Motions (CM) to reproduce the walking oscillations of the user's point of view during navigation in VR. The point of view oscillates in the VE to generate the visual flow that would be produced by a real walk. They demonstrated that their approach not only improves the user's sensation of walking in VE, but also his immersion [START_REF] Lécuyer | Camera motions improve the sensation of walking in virtual environments[END_REF].

Another solution to improve immersion in VR consists in cumulating different types of sensory feedback [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]. Visual feedback can be combined with audio or haptic feedback, for example, resulting in a fully multimodal walking simulation [START_REF] Visell | Interaction capture in immersive virtual environments via an intelligent floor surface[END_REF]. A typical example can be found in video games which provide not only visual feedback, but also auditory and sometimes vibrotactile modalities. In this paper, we propose a new technique to enhance the sensation of walking in VE inspired by special effects in Hollywood movies. The King Kong Effects (KKE) (Figure 1) provide a new kind of sensory feedback that simulates the feet touching the ground at each step by producing Visual and Tactile vibrations. The KKE can be used in a seated position, and we designed different vibration patterns based on physical and metaphorical models. As a result, we propose a new concept of sensory feedback effects, which correspond to the four main innovations claimed in our paper:

• A step simulator. We introduce a simple biomechanicallybased model to compute the footstep events, and the different contacts of the feet with the ground.

• New visual effects. We introduce the use of visual vibration patterns which simulate the contact of the feet with the ground at each step. Moreover, we simulate both the contacts of heel and toe. We propose and study vibration patterns along either vertical or horizontal directions.

• New vibrotactile effects. We also introduce new vibrotactile feedbacks generated under the feet of the user to reproduce the step sensation. Again, we simulate both the contacts of the heel and toe. Moreover, we propose two different vibration pattern metaphors: (1) a physically based metaphor and (2) a metaphor where the stimulation is proportional to the force pressure applied by the feet on the ground.

• Evaluation. Finally, we evaluated the different vibration patterns for both modalities. We also investigated the influence on the Visual KKE of another visual technique, the Camera Motions [START_REF] Lécuyer | Camera motions improve the sensation of walking in virtual environments[END_REF]. We also evaluated the KKE in a multimodal context with the modalities taken individually or all together.

The remainder of this paper is organized as follows: Section 2 provides an overview of the existing sensory feedback techniques for enhancing walking in VE. In Section 3 we introduce the King Kong Effects (KKE). We detail the Step Simulator used to generate the footstep events and we introduce our novel visual and vibrotactile patterns. In Section 4 we describe the results of a set of experiments conducted to identify the best parameters and best combinations of KKE.

RELATED WORK: IMPROVING SENSATION OF WALKING IN VE

Navigation is one of the essential tasks necessary to interact in VR [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]. To improve the range of sensory feedback available, many techniques require the users to physically walk in the real world. However, the working space is usually smaller than the VE, thus the users can not walk freely. The first solution uses treadmills [START_REF] Iwata | Walking about virtual environments on an infinite floor[END_REF] to keep the user at the center of the workspace. Other possibilities include mechanically actuated platforms under the user's feet [START_REF] Iwata | Gait master: A versatile locomotion interface for uneven virtual terrain[END_REF] or moving tiles [START_REF] Iwata | Circulafloor: A locomotion interface using circulation of movable tiles[END_REF] to keep the user in place. However, these locomotion interfaces provide only kinaesthetic information to the user.

Other devices can be used to improve the range of simulated modalities (audio, visual, tactile). For example, the ALive Floor [START_REF] Toshiaki | The terrain surface simulator alf[END_REF] is composed of actuated tiles that can simulate uneven grounds. Vibrotactile information can also be provided through shoe-based devices [START_REF] Papetti | Audio-tactile display of ground properties using interactive shoes[END_REF]. Using contact sensors and vibrators, the shoes can simulate different ground types by producing adapted vibrotactile feedback at each step. Another possibility uses tiled floors where each tile possesses force sensors and a vibrotactile actuator [START_REF] Visell | A vibrotactile device for display of virtual ground materials in walking[END_REF]. With this device, the user walks on a tiled floor and the foot pressure is used to track its position.

Audio feedback can also be adapted to simulate the different types of ground in the VE. For example, Serafin et al. [START_REF] Serafin | Extraction of ground reaction forces for real-time synthesis of walking sounds[END_REF] proposed a technique to extract the components of a real footstep sound in real time. This technique allows them to dynamically generate a new footstep sound, in real time, matching the properties of the virtual ground to be simulated. For example, a user walking on concrete in the real world could hear sounds of footsteps on snow. The physical simulation used allows a large variety of simulation of footstep sounds matching many types of virtual grounds such as snow, water, leafs, wood, concrete, gravel and so on [START_REF] Nordahl | Sound synthesis and evaluation of interactive footsteps for virtual reality applications[END_REF].

Different modalities can also be combined to improve immersion when walking in VE. For example, shoe-based devices can be associated with real time audio simulation [START_REF] Papetti | Audio-tactile display of ground properties using interactive shoes[END_REF]. Contact sensors embedded in shoes are used to detect footsteps and both vibrotactile and auditory feedback are provided to match a specific virtual ground surface. Tactile tiles can also be used with spatialized audio in a CAVE to provide a complete simulation using the haptic, auditory and visual modalities [START_REF] Visell | Interaction capture in immersive virtual environments via an intelligent floor surface[END_REF].

However, some applications should be able to run in desktop mode, i.e. when the user is seated and is using a basic computer. This includes training applications that need to be massively deployed, or video games. To give the sensation of walking, video games use auditory feedback intensively and footstep sounds to simulate steps. Visual information can also be used to enhance the sensation of walk in VE. For example, effects on the virtual camera can be created [START_REF] Lécuyer | Camera motions improve the sensation of walking in virtual environments[END_REF]. Camera Motions (CM) simulate the motions of the user's head and its associated visual flow during the walk [START_REF] Lécuyer | Camera motions improve the sensation of walking in virtual environments[END_REF]. The point of view of the user oscillates to follow the head motions that would be produced by a real walk. Compensation of the head orientation to focus the gaze on the objects is known as the oculomotor reflex. This reflex can also be added to CM [START_REF] Hillaire | Using an eyetracking system to improve camera motions and depth-of-field blur effects in virtual environments[END_REF], which are then adapted to follow the user's gaze in real time [START_REF] Hillaire | A real-time visual attention model for predicting gaze point during firstperson exploration of virtual environments[END_REF]. Moreover, CM can also be used to suggest slopes for example [START_REF] Marchal | Walking up and down in immersive virtual worlds: Novel interactive techniques based on visual feedback[END_REF]. The height, advance speed and orientation of the camera can be used separately or together to inform about the slope. When the three parameters are combined the perception of slopes increase [START_REF] Marchal | Walking up and down in immersive virtual worlds: Novel interactive techniques based on visual feedback[END_REF].

Unfortunately, the range of available modalities for sensory feedback in desktop mode is still very limited. In particular, to our best knowledge, vibrotactile feedback has been used scarcely in such context, and no visual feedback has been specifically developed to stress the footsteps during the walk. In the remainder of this paper, we present the King Kong Effects (KKE), a new set of sensory feedbacks designed to improve the sensation of walking by focusing on the perception of the footsteps during the walk through visual and tactile modalities.

CONCEPT OF KING KONG EFFECTS

We propose a new technique to enhance the sensation of walking in VE in desktop mode. The KKE is based on visual and tactile vibration patterns generated at each virtual step to simulate the contacts of the feet with the ground. Our technique can be used in static position, such as when seated or standing, whereas the user controls the virtual walk with any input device (joystick, keyboard, etc.).

The KKE are inspired from famous Hollywood movies such as King Kong or Godzilla where the walk of gigantic creatures is emphasized to make the spectators "feel" the steps of the incoming creature. For instance, Jurassic Park's T-Rex produces earth vibrations which generate waves in water, while Godzilla generates electrical disturbances. More recently, complex motions of the camera were used in movies like King Kong or Transformers to achieve the same goal. Our effects are, in a way, reproducing the special effects demonstrated in these movies for desktop VR technologies for the user himself, and are thus named "King Kong Effects" (KKE).

At each virtual step, the user can feel the sensation of hurting the ground with both visual vibration of the camera and vibrotactile feedback under his feet. For each modality, we propose a set of different vibration patterns. Visual and tactile vibrations can be used individually or together for a multimodal simulation. Moreover, existing auditory simulation of footstep sounds could be easily added to the KKE for even higher immersion.

The software architecture behind KKE is composed of three parts: the Step Simulator (1) is designed to compute footstep events based on a simple biomechanical model. Then, two different sensory feedback components corresponding to the visual (2) and vibrotactile (3) modalities have been developed to enhance the walk in the VE based on the generated footstep events (Figure 2).

Step Simulator

To compute the virtual footsteps events in the VE, we need a simple yet realistic biomechanical model. Indeed, contrarily to existing techniques, the KKE do not rely on force sensors to detect the footsteps. Indeed, the users should have the possibility to use the KKE even when seated at a desk using desktop VR applications, in which case the force sensors would be ineffective. However, they must be generated in a realistic way accurately reflecting the different user's interactions. Furthermore, the system must be compatible with different kinds of interaction devices, like a keyboard or a joystick for example.

To solve this problem, we propose a biomechanically inspired model that can generate footstep events in real-time based on a given advance speed. First, we use the classical decomposition of the walking motion in events given by Vaughan [START_REF] Vaughan | Dynamics of Human Gait[END_REF]: (1) Heel Strike (HS), (2) Heel Off (HO), (3) Toe Strike (TS) and (4) Toe Off (TO). Using this formalism, most of the human walking gaits can Figure 3: Walking cycle. The events of contact and separation of the heels and toes are placed accordingly to the position of the feet during the cycle. The parts of the cycle in orange correspond to the moments while either foot is in contact with the ground. be described as a succession of different events in a precise time and order. For example, walking, running, sprinting or even jumping can be described by such sequences associated with a global advance speed. The positions of these events during the walking cycle are shown in Figure 3.

The position of each event in the gait cycle can be predicted. In our implementation, we used the data presented by Novacheck [START_REF] Novacheck | The biomechanics of running[END_REF] for a regular walking motion, ie if the half-cycle is defined to start with HS, TS and HO will be at 21% and 60% of the half-cycle respectively, while TO and HS of the opposite foot will happen at 24% and 100% respectively (Figure 3).

To generate the events, we need to determine the gait half-cycle length T , to know the accomplished percentage of the cycle. For the walk, the advance speed is given by [START_REF] Alexander | Principles of Animal Locomotion[END_REF]:

v = L * f
where v is the advance speed, L the step length and f the step frequency defined as 1 T . Thus we have:

T = L v
However, while v is obviously dependent on the users interactions, it can be easier to compute it from a factor of the interactions and the speed of comfort v c for a given gait. For the walk, the speed of comfort can be estimated using a simple model interpolated from experimental data [START_REF] Kuo | A simple model of bipedal walking predicts the preferred speed-step length relationship[END_REF]:

L ∼ v β c
with β being a constant which equals to 0.42 for adults. With I the input speed in percentage of the comfort speed, such as v = I * v c , we obtain:

T = L I * ln(L) β
Thus, T depends on I: the gait cycle length can change in the middle of one step, which would be unrealistic and could induce implementation troubles. Thus, the value of T must be updated only at the beginning of each new step. For example, for a step length L = 1 m, the comfort speed, the duration of one step T is 1 second, and thus the contact of the toe will happen 210 milliseconds after the contact of the heel of the same foot. Similarly, the heel will be off the ground 600 milliseconds after its initial contact, and the toe of the opposite foot will leave the ground 240 milliseconds after the initial heel strike.

Finally, our Step Simulator can generate footstep events using only a percentage of the comfort advance speed and the virtual avatar size as input. The events are generated using a simple biomechanical model, and can be used to synchronize our KKE.

Visual King Kong Effects

Based on the generated events for the heels and toes contact, each step can be visually simulated using KKE: the technique is based on the metaphor of a visual vibrations produced by each step.

A first metaphor considers the vibration from the point of view of the "creature" as a result of the feet hurting the ground and thus producing Vertical (V) Moreover, a second metaphor emphasis the point of view from the environment point of view, as each step produce Horizontal (H) seismic vibrations when the feet of a heavy virtual avatar hit the ground.

simulated two types of vibration patterns with 1 and 2 successive contacts of the feet with the ground respectively: contact of the heel alone (1 contact) as used in most movies, and a combination of the heel and toe strike together (2 contacts) to provide a more realistic biomechanical simulation.

All different combinations were tested resulting in 4 different types of vibrations (Figure 4). Thus, the Visual KKE can be described by the function K V (x, y) with x the direction of the vibrations (x ∈ {V, H}) and y the number of contacts (y ∈ {1, 2}).

The vibration model is based on a Rigid Contact Model (RCM) used in haptic simulations for the contact between two rigid objects [START_REF] Okamura | Vibration feedback models for virtual environments[END_REF]. The vibrations of the camera are based on high frequency sinusoidal oscillations with an exponentially decaying envelope:

Q(t) = A(v)e -Bt sin(ωt)
where Q(t) is the produced vibration, A(v) the attack depending of the starting velocity v, ω the frequency of the oscillations and for a given material B is the decay constant of the envelope. Values of ω and B can be found for different materials in the literature [START_REF] Okamura | Vibration feedback models for virtual environments[END_REF]. However, the visual rendering strongly depends on the display device frame rate. If the frame rate is too low or the vibration frequency too fast, the user will not be able to perceive the vibration correctly. After preliminary testing, the decaying value B was set to be twice the interval between HS and TS, ie 0.3 s and we used a constant amplitude for the attack (A HS = 7 cm and A T S = 3 cm). Finally, our preliminary tests showed that the value of ω should be inferior to twice the display frequency. In our implementation, we chose ω = 67 Hz corresponding to a ground made of wood.

Tactile King Kong Effects

We also designed a set of vibrotactile techniques to stimulate the feet of the users. The vibrations were transmitted using low frequency loud speakers fixed on tiles under the users' feet [START_REF] Visell | A vibrotactile device for display of virtual ground materials in walking[END_REF] (Figure 5). We also developed 2 different metaphors for the vibrations. The first metaphor is a physically-based simulation of the vibrations that would be produced in real life using Rigid Contact Model (RCM) [START_REF] Okamura | Vibration feedback models for virtual environments[END_REF] presented section 3.2. With this model, the user's feet are considered to be a rigid object colliding with a rigid surface. The resulting collision produces high frequency vibrations depending of the nature of the virtual ground. In the second metaphor, the vibrations are proportional to the forces applied on the ground by the feet: the Ground Reaction Forces Model (GRFM) simulates the force that is applied to the ground by each step [START_REF] Novacheck | The biomechanics of running[END_REF]. We used the same vibration frequency as with the RCM model, but different envelopes are used for the signal. The envelopes reproduce the shape of the forces applied to the ground during each step [START_REF] Novacheck | The biomechanics of running[END_REF]. For the heel contact only, only the beginning of the curve is interpolated from the data, while all the data is used otherwise. Thus, the Tactile KKE can be described by the function K T (x, y) with x the model used (x ∈ {RCM, GRFM}) and y the number of contacts (y ∈ {1, 2}). The four resulting different vibration patterns are presented in Figure 6.

Conclusion

To sum up, our approach is composed of (1) a Step Simulator which computes the footstep events during the walk, (2) Visual vibration patterns based on vibrations along two different directions (vertical and horizontal) to produce different effects and (3) Tactile vibration patterns based on two different metaphors. Moreover, the different patterns can simulate both the heel and toe contacts with the ground. The KKE can be used in static position, such as when seated or standing, whereas the user controls the virtual walk with any input device (joystick, keyboard, etc.). Finally, the KKE can be implemented on any kind of computer, requiring only the tiles which are a low cost device.

4

We conducted a set of experiments to evaluate the different components of the KKE and to determine the best combination among We conducted four different experiments. The first three experiments were based on the 2 Alternative Forced Choice (2AFC) paradigm. For each experiment, the participants were exposed to pairs of passive navigations in straight line using different conditions. At the end of each pair, the participants were asked which one of the two navigations gave them the best sensation of walking. In every case, the participants had to choose between the two navigations. The first experiment was designed to evaluate the optimal parameters for the Visual KKE. The second investigated the influence of Camera Motions on the appreciation of Visual KKE. The third experiment focused on the optimal parameters for the tactile KKE. Finally, the fourth experiment tested the preference for the KKE in a multimodal context.

Method

Experimental Apparatus

The participants were seated at 1 m in front of a 24 inch widescreen monitor with a resolution of 1920 × 1200 pixels (physical field of view of 29 • horizontally and 18 • vertically). The rendering was made at a refresh rate of 50 Hz. Their feet were placed on top of the vibrating tiles, with their shoes removed (Figure 7a). Users were wearing headphones filled with white noise to mask any sound produced by the vibrating tiles. At the end of each pair of navigations, the users had to select their preferred navigation in terms of "sensation of walking" using the keyboard keys "1" and "2". The participants had the possibility to take breaks by pressing the "Space" key at any time.

Virtual Environment

The Virtual Environment was composed of an empty room with textured walls (Figure 7b). The room depth was set to 15 m. At the end of the room, in the center of the screen, a cardboard box regularly textured was placed to provide the participants with a point to focus their gaze (Figure 7b). The participants were exposed to pairs of passive navigations of 5.4 m.

Collected Data

For each pair of conditions, we recorded the choices of the participants. At the end of the experiment the participants had to complete a questionnaire in which they had to grade from 1 (very bad) to 7 (very good) the different techniques according to different criteria: (1) Presence, (2) Sensation of walking, (3) Realism of the walk, (4) Visual tiredness (5) Cybersickness and (6) Global appreciation. For the Visual tiredness and Cybersickness conditions, the grade 1 corresponded to "very tiresome" and "make sick" respectively, while the grade 7 was the opposite.

Experiment 1: Selecting the Best Visual Vibration Pattern for KKE

The first experiment was focused on Visual KKE. The goal was to find which set of parameters provides the best sensation of walking in the VE when the Visual KKE are used alone. The parameters were the number of contacts (1 contact versus 2 contacts) and the direction of oscillations (Vertical versus Horizontal).

Population

Twelve participants (10 males and 2 females) aged from 20 to 34 (mean=23.8, standard deviation=3.5) performed the experiment. None of the participants had any known perception disorder. All participants were used to VEs but were naïve with respect to the proposed techniques, as well as to the experimental setup.

Experimental Conditions

We used a within subject design to evaluate five different Visual KKE conditions. The control condition Ctrl was composed of a linear camera motion without any Visual KKE. The four other conditions corresponded to: (1) K V (V, 1), (2) K V (V, 2), (3) K V (H, 1) and (4) K V (H, 2). All the possible combinations of the different conditions were tested 10 times in both orders. For each group of possible combinations, the order between the different pairs was randomized. The experiment lasted approximatively 25 minutes.

Results

We analyzed answers and preferences of participants for the different patterns in order to determine which condition provides the best sensation of walking in the VE. In particular, we analyzed the impact of the direction of the oscillations, as well as the effect of one contact versus two contacts. For a given pair of conditions, each individual performed 20 comparisons. Under the null hypothesis of equal preference between the two conditions, the number of times an individual preferred the first condition follows a binomial distribution with parameters 10 and 1/2. After standardization, such variable can be approximated by a standard normal random variable. Thus, for each pair of conditions, we tested the presence of a preferred condition using a Student's t-test. The p-values were adjusted with a Bonferroni correction. The analysis showed that K V (V, 1) was more often significantly chosen than Ctrl (t(11) = 6.14, p < 0.001), K V (H, 1) (t(11) = 11.49, p < 0.001) and K V (H, 2) (t(11) = 7.58, p < 0.001). Moreover, K V (V, 2) was more often significantly chosen than Ctrl (t(11) = 5.50, p = 0.002), K V (H, 1) (t(11) = 9.57, p < 0.001) and K V (H, 2) (t(11) = 5.45, p = 0.002). Ours results suggest that for the Visual KKE, the Vertical vibrations are always preferred over the Horizontal ones. Moreover, the number of contacts with the ground does not change this result. However, the experiment failed to found any significant effect on the number of contacts of the feet (1 or 2 contacts) for the Visual vibrations.

Concerning the subjective questionnaires, we performed a Friedman test. The reported p-values were adjusted for multiple comparisons. We found a significant effect for 5 criteria: Global appreciation (χ 2 = 3.6, p = 0.003), Presence (χ 2 = 3.03, p = 0.02), Realism (χ 2 = 4.27, p < 0.001), Walking sensation (χ 2 = 3.96, p < 0.001) and Fatigue (χ 2 = 4.91, p < 0.001). Post-hoc analysis showed that K V (V, 1) was preferred to Ctrl for global appreciation (p = 0.01), presence (p = 0.02), realism (p < 0.001) and walking sensation (p < 0.001). K V (V, 1) was also significantly better rated than K V (H, 1) for global appreciation (p = 0.003) and realism (p = 0.04), and significantly better rated than K V (H, 2) for global appreciation (p = 0.006), walking sensation (p = 0.02) and fatigue (p = 0.03). Moreover, K V (V, 2) was preferred to Ctrl for realism (p = 0.002), walking sensation (p = 0.046) and fatigue (p = 0.006). Finally, K V (H, 1) and K V (H, 2) were significantly better rated than Ctrl for fatigue only (p < 0.001 and p < 0.001). The results of the questionnaires are displayed in Figure 8. results of the subjective questionnaires confirm the results of the 2AFC. Indeed, the Vertical vibrations with 1 contact scored better than the other conditions with Horizontal vibrations in terms of global appreciation and either realism or walking sensation.

Conclusion

To sum up, for the Visual KKE the Vertical direction provides a better sensation of walking than the Horizontal direction. In particular, the Vertical Visual KKE with 1 contact was preferred by the participants over the other conditions.

Experiment 2: Testing the Combination of Visual KKE and Oscillating CM

The second experiment tested the influence of standard oscillating Camera Motions on Visual KKE. The goal was to check if the preference for Visual KKE (number of contacts, direction of oscillations) would change when associated with another visual technique composed of standard camera motions.

Population

The same twelve participants as the first experiment performed this experiment. However, to avoid any bias, half of the participants started with this experiment while the other half started with the first one.

Experimental Conditions

We used a within subject design to evaluate five different Visual KKE when added to the regular Camera Motions proposed by Lécuyer et al. [START_REF] Lécuyer | Camera motions improve the sensation of walking in virtual environments[END_REF]. The control condition Ctrl was made of classical camera motions composed of sinusoidal oscillations along the three axes, without any Visual KKE. The conditions were all composed of the camera motions combined respectively with: (1) K V (V, 1), (2) K V (V, 2), (3) K V (H, 1) and ( 4) K V (H, 2). All the possible combinations of the different conditions were tested 10 times in both orders. For each group of possible combinations, the order between the different pairs was randomized. The experiment lasted approximatively 25 minutes.

Results

We performed the same statistical analysis as for Experiment 1. The analysis showed that K V (V, 1) was more often significantly chosen than K V (H, 1) (t(11) = 4.99, p = 0.004). Moreover, K V (V, 2) was more often significantly chosen than K V (H, 2) (t(11) = 4.17, p = 0.016). Ours analysis suggest similar results to the first experiment: the Vertical vibrations are still preferred to the Horizontal ones. The presence and combination with Camera Motions do not change this result. Using the KKE in combination with standard Camera Motions does not modify the way users perceive the KKE, and the most efficient type of KKE remains the same. The analysis also failed to found any significant effect the number of contacts of the feet (1 or 2 contacts) when combined with Camera Motions.

Concerning the subjective questionnaires, we performed a Friedman test. The reported p-values were adjusted for multiple comparisons. We found a significant effect for 5 criteria: Global appreciation (χ 2 = 4.52, p < 0.001), Presence (χ 2 = 3.14, p = 0.01), Realism (χ 2 = 4.38, p < 0.001), Walking sensation (χ 2 = 4.89, p < 0.001) and Fatigue (χ 2 = 3.23, p = 0.01). Post-hoc analysis showed that K V (V, 1) was preferred to Ctrl for presence (p = 0.04). K V (V, 1) was also significantly better rated than K V (H, 2) for presence (p = 0.02) and fatigue (p = 0.01). Moreover, K V (V, 2) was significantly better rated than K V (V, 1) for global appreciation (p < 0.001), realism (p < 0.001) and walking sensation (p < 0.001). K V (V, 2) was also preferred to K V (H, 2) for global appreciation (p < 0.001) and walking sensation (p = 0.01). Finally, K V (H, 1) was significantly better rated than K V (V, 1) for global appreciation (p = 0.008), realism (p = 0.01) and walking sensation (p = 0.02). The results of the questionnaires are displayed in Figure 9. The results of the questionnaires are more contrasted. Indeed, Vertical vibrations scored better than Horizontal vibrations for walking sensation and global appreciation with 2 contacts, while it is the opposite for 1 contact.

Conclusion

To sum up, combined with CM, the Vertical direction of the Visual KKE is still preferred over the Horizontal direction.

Experiment 3: Selecting the Best Tactile Vibration Pattern for KKE

The third experiment focused on tactile KKE. The goal was to find which set of parameters provides the best sensation of walking in the VE when the tactile KKE are used alone. The parameters were the number of contacts (1 contact versus 2 contacts) and the model used (RCM versus GRFM).

Population

Twelve participants different from the participants from the two previous experiments (8 males and 4 females) aged from 21 to 59 (mean=30.1, standard deviation=12.6) performed the experiment. None of the participants had any known perception disorder. All participants were used to VEs but were naïve with respect to the proposed techniques, as well as to the experimental setup.

Experimental Conditions

We used a within subject design to evaluate five different Tactile KKE conditions. The control condition Ctrl was composed of no Tactile KKE. The conditions were respectively: (1) K T (RCM, 1), (2) K T (RCM, 2), (3) K T (GRFM, 1) and (4) K T (GRFM, 2). All the possible combinations of the different conditions were tested 10 times in both orders. For each group of possible combinations, the order between the different pairs was randomized. The experiment lasted approximatively 25 minutes.

Results

We performed the same statistical analysis as in Experiment 1. The analysis showed that K T (RCM, 1) was more often significantly chosen than Ctrl (t(11) = 36.38, p < 0.001), K T (RCM, 2) (t(11) = 4.26, p = 0.0133) and K T (GRFM, 2) (t(11) = 4.64, p = 0.0072). Moreover, K T (GRFM, 1) was more often significantly chosen than K T (RCM, 2) (t(11) = 5.46, p = 0.002) and K T (GRFM, 2) (t(11) = 8.25, p < 0.001). Concerning the vibration patterns, no significant effect on the metaphor was found during the 2AFC analysis. Rigid Contact Model (RCM) with only heel strikes was preferred to the Ground Reaction Forces Model (GRFM) with heel and toe strikes.

On the other hand, the GRFM with only heel strikes was also found significantly preferred to the RCM with heel and toe strikes. Thus, the number of contacts seems to be a more important criterion for the Tactile vibrations. Indeed, the Heel Strike alone simulation is preferred to the simulation of Heel Strike and Toe Strike. The metaphor used to design the vibration pattern does not influence these results. Thus, both metaphor can be used. Concerning the subjective questionnaire, we performed a Friedman test. The reported p-values were adjusted for multiple comparisons. We found a significant effect for 4 criteria: Global appreciation (χ 2 = 3.99, p < 0.001), Presence (χ 2 = 4.2, p < 0.001), Realism (χ 2 = 4.65, p < 0.001) and Walking sensation (χ 2 = 4.83, p < 0.001). Post-hoc analysis showed that K T (RCM, 1) was preferred to Ctrl for global appreciation (p = 0.001), presence (p < 0.001), realism (p < 0.001) and walking sensation (p < 0.001). K T (RCM, 1) was also significantly better rated than K T (GRFM, 2) for realism (p = 0.009). Moreover, K T (RCM, 2) was significantly better rated than Ctrl for walking sensation (p = 0.01). Finally, K T (GRFM, 1) was preferred to Ctrl for global appreciation (p = 0.03), presence (p < 0.001), realism (p < 0.001) and walking sensation (p < 0.001). K T (RCM, 1) was also significantly better rated than K T (RCM, 2) for global appreciation (p = 0.03). The results of the questionnaires are displayed in Figure 10.

The subjective questionnaires did not reveal any clear significant preference on the number of contacts for any of the experiments. The results suggest a small preference for 1 contact over 2 contacts, only for the Tactile vibrations. For each of the criteria, the mean and standard deviation for each condition are represented.

Conclusion

To sum up, the Tactile KKE provide a better sensation of walking when only 1 contact of the foot with the ground is simulated. More complex simulations are perceived as less natural for the users.

Population

Ten new participants (9 males and 1 female) aged from 21 to 27 (mean=24.1, standard deviation=2.2) performed the experiment. None of the had any known perception disorder. All participants were used to VEs but were naïve with respect to the proposed techniques, as well as to the experimental setup.

Experimental Conditions

We used a within subject design where the participants could freely navigate on a museum scene (Figure 11). They had the possibility to switch at will from one condition to the others. The visual (V ) was composed of K V (V, 1), haptic (vibrotactile) (H) modality was composed of K T (RCM, 1), and the audio (A) modality was rendered using recorded playback of a wooden floor. All the possible combinations of these 3 modalities were available, from one modality alone to the 3 combined together, resulting in the following conditions: V , H, A, V H, VA, HA and V HA. The experiment lasted approximatively 15 minutes.

Collected Data

The participants had to grade from 1 (very bad) to 7 (very good) the different conditions based on the following criteria: (1) Presence, (2) Sensation of walking, (3) Realism of the walk, (4) Fun and (5) Global appreciation.

Results

Concerning the subjective questionnaires, we performed a Friedman test. The reported p-values were adjusted for multiple comparisons. We found a significant effect for all criteria: Fun (χ 2 = 4.49, p < 0.001), Global appreciation (χ 2 = 2.97, p = 0.047), Presence (χ 2 = 4.90, p < 0.001), Realism (χ 2 = 5.29, p < 0.001) and Walking sensation (χ 2 = 4.90, p < 0.001). Post-hoc analysis showed that H was preferred to V for fun (p = 0.02). VA was preferred to V for fun (p = 0.03), global appreciation (p = 0.047), and walking sensation (p = 0.04). HA was preferred to V for fun (p < 0.001), presence (p = 0.007), realism (p < 0.001), and walking sensation (p = 0.001). HA was also significantly better rated than V H for fun (p = 0.008). V HA was preferred to V for presence (p < 0.001), realism (p < 0.001), and walking sensation (p < 0.001). V HA was significantly better rated than H for presence (p = 0.01), realism (p = 0.03), and walking sensation (p = 0.04). V HA was significantly better rated than A for presence (p = 0.03), realism (p = 0.01), and walking sensation (p = 0.009). V HA was significantly better rated than V H for realism (p = 0.04) and walking sensation (p = 0.02). V HA was significantly better rated than VA for realism (p = 0.002). V HA was significantly better rated than HA for fun (p = 0.003).

The multimodal evaluation of the KKE showed that the the effects produced by each modality are reinforced when used in conjunction with the other modalities. In particular, conjunctions of two modalities scored higher on the fun criteria. Finally, the combination of the three modalities resulted in higher grades for presence, realism and walking sensation compared to each modalities taken alone or by two. Thus, our results suggest that a multimodal approach for the perception of the walk in the VE is preferred by the participants. The results of the questionnaires are displayed in Figure 12. For each of the criteria, the mean and standard deviation for each condition are represented.

Conclusion

To sum up, the sensation of walking is increased when the different modalities of the KKE are taken together. A multimodal simulation is not only possible but is also recommended.

Our results suggest that the KKE allow more immersive and more enjoyable navigation compared to the classic FPS paradigm. Moreover, it clearly increases the sensation of actually walking inside the VE, while only using low cost devices. Finally, the KKE can be used to simulate different morphologies of virtual avatars, like height or weight, while still using the FPS paradigm.

The results of the experiments show that for the Visual KKE, the Vertical vibrations are always preferred to the Horizontal ones, and the type of contact of the feet with the ground does not change this result. Indeed, with Heel Strike only or with both Heel Strike and Toe Strike simulations, the Vertical is always preferred. The Vertical vibrations were designed to reproduce the point of view of a big creature walking like King Kong, while the Horizontal vibrations are more related to an external point of view. Thus, it seams that the Vertical vibrations are more immersive and thus provide a better sensation of walking in the VE.

Moreover, the standard Camera Motions do not change this result. Using the KKE in combination with standard Camera Motions does not modify the way a user perceives the KKE, and the most efficient type of KKE remains the same. This result gives a good hope that the KKE could be associated with many other techniques while keeping its properties. In particular, the best type of Visual KKE can probably be used safely in most situations.

Concerning the Tactile vibrations, the number of contacts was significant but the model used was not found to be significant. Thus, it seems that the simplicity of the model is a key to provide the best sensation of walking. However, some more complex models could be tested to simulate more complex virtual grounds, such as aggregate grounds made of sand or gravel for example [START_REF] Nordahl | Sound synthesis and evaluation of interactive footsteps for virtual reality applications[END_REF]. The model of vibration could also be adapted dynamically to reproduce more accurately all the changes of the properties of the virtual ground during the navigation.

Interestingly, for some participants, the KKE modified their self perception inside the VE. Some participants quoted that they "felt like if they were heavier", or "like if they were a big creature like a troll". One of the women who participated also quoted that the double contact felt "as if walking with high heels".

Finally, our results clearly suggest the importance of a multimodal approach. In this paper we did not focus on the auditory modality. However, further integration of this modality would be highly interesting. Indeed, this modality fits particularly well into the scope of a desktop VR, and can provide a lot of useful information about the footsteps and the virtual ground properties.

CONCLUSION AND FUTURE WORK

this paper, we proposed a new navigation technique inspired from movies to increase the sensation of walking in a VE. This technique simulates each virtual step made in the VE by producing a visual and/or tactile vibration. The KKE simulates the contact with the ground of the heel and eventually the toe. Moreover, we proposed two different directions for the Visual vibrations corresponding to different points of view. Finally, we proposed two different models for the Tactile vibration patterns.

We conducted an evaluation of each of the components of the KKE, as well as a multimodal evaluation in order to determine which components and parameters provide the best sensation of walking in VE. We found that vertical visual vibration simulating only heel contact were preferred by the participants. We also found that the Rigid Contact Model with only Heel Strike simulation was also preferred for the vibrotactile pattern. Finally, the participants showed that using the best patterns for each modality, multimodal feedback was preferred to navigations using only one modality.

Future work will focus on increasing the range of navigation motions such as running or jumping for example. Moreover, using the Step Simulator, a more developed auditory feedback rendering could be designed, or other parts of the body could be simulated.

Moreover, new models for the Tactile vibration patterns could be tested to check whether the vibrations could be improved to conveyed more information. In particular, different ground properties (like aggregate grounds for example) may be simulated with other different model, extending the perception of the virtual ground properties. Moreover, these models could be associated with real time audio synthesis to simulate the footstep sounds based on the ground properties.

Figure 1 :

 1 Figure 1: Concept of the King Kong Effects: Visual and Tactile vibrations inspired by special effects in movies enhance the sensation of walking in VE. Visual and Tactile feedbacks are generated at each step made in the VE.

Figure 2 :

 2 Figure 2: Architecture of the KKE. The user inputs are processed by the Step Simulator to generate footstep events. The events are used to generate visual and vibrotactile feedback.

Figure 4 :

 4 Figure 4: Vibration patterns for the Visual KKE. Amplitude (in meters) over Time (in seconds) of the vibration patterns used for the Visual KKE: 1 or 2 contacts in Vertical and Horizontal directions.

Figure 5 :

 5 Figure 5: Vibrotactile tiles. The vibrations are produced by low frequency loud speakers fixed under the tiles [18].

Figure 6 :

 6 Figure 6: Vibration patterns for the tactile KKE. Amplitude over Time (in seconds) of the vibration patterns used for the Tactile KKE: Rigid Contact Model (RCM) and Ground Reaction Forces Model (GRFM) simulating 1 or 2 contacts.

Figure 7 :

 7 Figure 7: Experimental apparatus.

Figure 8 :

 8 Figure 8: Results of the questionnaires for the first experiment. For each of the criteria, the mean and standard deviation for each condition are represented.

Figure 9 :

 9 Figure 9: Results of the questionnaires for the second experiment. For each of the criteria, the mean and standard deviation for each condition are represented.

Figure 10 :

 10 Figure 10: Results of the questionnaires for the third experiment.For each of the criteria, the mean and standard deviation for each condition are represented.

4. 5

 5 Experiment 4: Testing Participant Preference for Multimodal KKEThis last experiment focused on multimodal rendering of KKE. For this experiment, we introduced audio feedback of prerecorded footsteps to the KKE. The footstep sounds were synchronized with the other KKE by the Step Simulator. We used audio, visual and vibrotactile modalities individually or all together. For each modality, we selected the best components found in the previous experiments to yield the best sensation of walking in the VE and we tested every possible combinations of modalities.Taken together, our previous results suggest guidelines for the best Visual and Tactile KKE. Heel strikes (1 contact) only should be used for the Tactile vibrations. Moreover, Vertical oscillations should be used for the Visual vibrations. Because the metaphor used has no clear significant effect on the Tactile vibrations, we chose to use the Rigid Contact Model for both modalities to simplify our model. Moreover, to keep the model as simple as possible, we chose to use only heels strikes for the Visual Vibrations also.

Figure 11 :

 11 Figure 11: Museum scene of the 4th experiment.

Figure 12 :

 12 Figure 12: Results of the questionnaires for the fourth experiment. For each of the criteria, the mean and standard deviation for each condition are represented.