The yielding dynamics of a colloidal gel Thomas Gibaud, Damien Frelat, Sébastien Manneville † Laboratoire de Physique, Université de Lyon -École Normale Supérieure de Lyon -CNRS UMR 5672, 46 allée d'Italie, 69364 Lyon cedex 07, France (Dated: November 4, 2009) Attractive colloidal gels display a solid-to-fluid transition as shear stresses above the yield stress are applied. This shear-induced transition is involved in virtually any application of colloidal gels. It is also crucial for controlling material properties. Still, in spite of its ubiquity, the yielding transition is far from understood, mainly because rheological measurements are spatially averaged over the whole sample. Here, the instrumentation of creep and oscillatory shear experiments with high-frequency ultrasound opens new routes to observing the local dynamics of opaque attractive colloidal gels. The transition proceeds from the cell walls and heterogeneously fluidizes the whole sample with a characteristic time whose variations with applied stress suggest the existence of an energy barrier linked to the gelation process. The present results provide new test grounds for computer simulations and theoretical calculations in the attempt to better understand the yielding transition. The versatility of the technique should also allow extensive mesoscopic studies of rupture mechanisms in soft solids ranging from crystals to glassy materials.

I. INTRODUCTION

Colloidal particles with strong enough short-range attraction are known to aggregate into fractal clusters that may form a space-spanning network leading to solid-like properties even at very low volume fractions [START_REF] Larson | The Structure and Rheology of Complex Fluids[END_REF][START_REF] Russel | Colloidal Dispersions[END_REF]. Such gelation obviously has tremendous consequences in terms of viscosity increase and rheological properties, which are capital for applications ranging from paints and coatings to food products and cosmetics. In the gel state, the system is kinetically arrested and displays non-ergodic features such as ultraslow relaxations, aging, and dynamical heterogeneities [START_REF] Cipelletti | Universal aging features in the restructuring of fractal colloidal gels[END_REF][START_REF] Duri | Length scale dependence of dynamical heterogeneity in a colloidal fractal gel[END_REF][START_REF] Trappe | Investigation of q-dependent dynamical heterogeneity in a colloidal gel by x-ray photon correlation spectroscopy[END_REF]. These glassy-like features have raised the issue of unifying the behaviour of colloidal gels and that of more concentrated amorphous jammed systems, e.g. colloidal glasses or concentrated emulsions [START_REF] Trappe | Jamming phase diagram for attractive particles[END_REF]. Therefore, most recent works on systems of short-range attractive particles have focused on the physics of gelation and on their structural properties at rest [START_REF] Cardinaux | Interplay between spinodal decomposition and glass formation in proteins exhibiting short-range attractions[END_REF][START_REF] Gao | Direct imaging of dynamical heterogeneities near the colloid-gel transition[END_REF][START_REF] Lu | Gelation of particles with short-range attraction[END_REF][START_REF] Royall | Direct observation of a local structural mechanism for dynamic arrest[END_REF][START_REF] Zaccarelli | Colloidal glasses and gels: The interplay of bonding[END_REF].

Of even greater practical importance is the behaviour of such materials under external stress. It is well known that applying stress to a soft glassy material generally triggers a solid-to-fluid transition, in the same way as increasing the temperature T , lowering the interparticle attractive energy U , or lowering the particle volume fraction φ [START_REF] Trappe | Jamming phase diagram for attractive particles[END_REF][START_REF] O'hern | Jamming at zero temperature and zero applied stress: The epitome of disorder[END_REF]. In particular, the yielding phenomenon, i.e. the fact that the material flows like a liquid above a yield stress σ y and remains solid below σ y , has been extensively documented in colloidal gels and glasses through standard rheological measurements and theoretical or numerical models, with emphasis on the scaling of viscoelastic properties and of σ y with φ and U [START_REF] Russel | Colloidal Dispersions[END_REF][START_REF] Trappe | Jamming phase diagram for attractive particles[END_REF][START_REF] Barnes | The yield stress -a review of 'panta rei' -everything flows?[END_REF][START_REF] Brader | Glass rheology: From mode-coupling theory to a dynamical yield criterion[END_REF][START_REF] Fuchs | Theory of nonlinear rheology and yielding of dense colloidal suspensions[END_REF][START_REF] Kobelev | Strain softening, yielding, and shear thinning in glassy colloidal suspensions[END_REF][START_REF] Osuji | Shear thickening and scaling of the elastic modulus in a fractal col-loidal system with attractive interactions[END_REF][START_REF] Potanin | Microrheological modeling of weakly aggregated dispersions[END_REF][START_REF] Rueb | Viscoelastic properties of colloidal gels[END_REF][START_REF] Whittle | Large deformation rheological behaviour of a model particle gel[END_REF][START_REF] Woutersen | The rheology of adhesive hard sphere dispersions[END_REF]. However, rheology provides information that are averaged over the † e-mail: sebastien.manneville@ens-lyon.fr whole sample size and it cannot capture local phenomena such as apparent wall slip [START_REF] Barnes | A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: Its cause, character, and cure[END_REF], shear localization [START_REF] Coussot | Coexistence of liquid and solid phases in flowing softglassy materials[END_REF], or fracture within the material [START_REF] Pignon | Thixotropic colloidal suspensions and flow curves with minimum: Identification of flow regimes and rheometric consequences[END_REF]. Recently, a few original experimental approaches have been reported using confocal microscopy [START_REF] Ballesta | Slip and flow of hard-sphere colloidal glasses[END_REF][START_REF] Besseling | Three-dimensional imaging of colloidal glasses under steady shear[END_REF][START_REF] Isa | Shear zones and wall slip in the capillary flow of concentrated colloidal suspensions[END_REF] and particle imaging velocimetry in microchannels [START_REF] Goyon | Flow cooperativity and breakdown of local constitutive laws for confined glassy flows[END_REF]. Still, these works were restricted to steady-state measurements, and, as a matter of fact, very little is known on how the stress-induced solid-to-fluid transition exactly proceeds in both space and time.

In the present article, we introduce high-frequency ultrasound as a new tool to follow the local deformation and flow of carbon black suspensions, a totally opaque colloidal gel made of weakly attractive soot particles. Both creep tests and oscillatory shear experiments show for the first time that, above a critical yield stress σ y , (i) yielding proceeds from the cell walls in a spatially and temporally heterogeneous way and (ii) the time needed for total fluidization of the sample decreases exponentially with the applied shear stress. These results point out the existence of an energy barrier that linearly decreases with the external stress. Therefore, they provide evidence for the relevance of activated barrier hopping and for processes reminiscent of heterogeneous nucleation, in strong support of recent models [START_REF] Bocquet | A kinetic theory of plastic flow in soft glassy materials[END_REF][START_REF] Fielding | Shear banding, aging and noise dynamics in soft glassy materials[END_REF]. Such features should be common to all weakly attractive colloids and models are expected to become quantitatively predictive, based on more spatially and temporally resolved measurements as described here.

II. MATERIALS AND METHODS

A. Carbon black gel preparation

Carbon black is widely used in industry, e.g. for coatings, printing, rubbers, tires, paints, or batteries [START_REF] Donnet | Carbon black: Science and technology[END_REF]. These colloidal carbon particles result from partial combustion of fuel and are made of unbreakable aggregates of permanently fused nanometric primary particles. These aggregates have a typical diameter of 500 nm. When dispersed in a light mineral oil (from Sigma, density 0.838, viscosity 20 mPa.s) at a weight concentration of a few percents, carbon black particles (Cabot Vulcan XC72R of density 1.8) form a space-spanning network of fractal dimension d f = 2.2 due to weakly attractive interactions of typical strength U ∼ 30k B T [START_REF] Trappe | Investigation of q-dependent dynamical heterogeneity in a colloidal gel by x-ray photon correlation spectroscopy[END_REF]. Our samples are prepared by vigorous mixing of 6 % w/w carbon black particles into the mineral oil. Since the resulting gel is transparent to ultrasound, 1 % w/w hollow glass microspheres of mean diameter 6 µm (Sphericel, Potters) are added in order to provide enough acoustic contrast (see below). The sample is further sonicated for 1 hour.

B. Standard rheology

Rheological measurements are performed in a Plexiglas Couette cell (rotating inner cylinder radius 24 mm, gap width 1 mm, and height 30 mm) by a stress-imposed rheometer (Bohlin C-VOR 150, Malvern Instruments).

To ensure an initial reproducible gel state, all measurements are carried out by first preshearing the suspension at 1000 s -1 then at -1000 s -1 for 20 s each, in order to break up any large aggregate. The sample is then left at rest for 100 s during which the gel structure reforms, a process that is controlled by monitoring the viscoelastic moduli G ′ and G ′′ at f = 1 Hz and σ = 0.2 Pa. A rest time of 100 s is sufficient for G ′ and G ′′ to reach a steady state with negligible aging (Fig. 2d). All measurements were performed at a temperature of 25 • C.

C. Ultrasonic echography under shear

In our experiments, classical rheology in Couette geometry is combined to ultrasonic echography in order to access local properties of the gel such as its velocity or deformation fields. In creep experiments, we use ultrasonic speckle velocimetry (USV) as described in Ref. [START_REF] Manneville | High-frequency ultrasonic speckle velocimetry in sheared complex fluids[END_REF] to record temporally-resolved velocity profiles accross the gap at about 15 mm from the cell bottom (Fig. 1a). USV is based on the interaction between ultrasound and micronsized scatterers embedded in the gel (here, the glass microspheres that act as contrast agents). A highfrequency (36 MHz) immersion transducer first emits a short ultrasonic pulse and collects the pressure signal backscattered from the gel. The backscattered signal is an ultrasonic speckle that directly reflects the spatial distribution of the scatterers within the gel along the acoustic beam (Fig. 1b). As the scatterers move with the gel, cross-correlating two successive speckle signals over small time windows gives access to the local velocity of the gel with a spatial resolution of about 40 µm. This original combination of rheology and ultrasound turns out to be very efficient to follow the onset of flow both spatially and temporally. For oscillation experiments, we take advantage of ultrasonic echography in a new way by focusing on the spatiotemporal diagrams of the successive speckle signals (Fig. 1c-f). Indeed, the spatial variation of the speckle intensity provides crucial information on the local deformation of the sample. When pulses are sent with a sufficiently high repetition frequency f prf compared to the oscillation frequency f , tracking the local oscillatory motion of the gel becomes possible (Fig. 1c ande). Here, we rather set f prf = f so that the material is probed by ultrasound only once per oscillation period. Data analysis is then straightforward: if two successive speckle signals are fully correlated, then the gel is solid-like since it comes back exactly to the same position every period (Fig. 1d). On the other hand, we interpret any significant degree of decorrelation in the speckle signal from one period to the other as the signature of fluid-like behaviour due to motions of the scatterers induced by local rearrangements inside the gel (Fig. 1f). Note that such decorrelation in the fluidized state does not arise from flow irreversibility (since the Reynolds number is Re ≃ 10 -3 ≪ 1) but rather from sedimentation of carbon black clusters due to the density mismatch between the mineral oil and the carbon black particles. We checked that sedimentation of clusters as small as 10 µm is sufficient to produce significant decorrelation after one oscillation period.

spatiotemporal tracking of the transition from correlated speckle signals (i.e. solid-like state) to decorrelated speckle signals (i.e. fluidized state) is made quantitative by using a threshold for the correlation coefficient of successive signals over small time windows, thus identifying the boundary between solid and fluid regions as a function of time up to a spatial resolution of 40 µm (Fig. 4d).

The fraction of fluidized sample φ at a given time t is readily extracted from the red contour separating fluid and solid regions in Fig. 4d. Jumps in φ(t) are interpreted as fast fluidization events and identified using a threshold on the second derivative of φ(t). Statistical analysis of the time intervals δt between two successive jumps then yields the probability distribution function P and the complementary cumulative distribution function C of δt (inset of Fig. 5). C is defined as C(x) = ∞ x (y)dy and experimentally estimated by sorting the measured values {δt j } j=1...N in ascending order and by plotting 1j/N as a function of δt j [START_REF] Berg | From data to probability densities without histograms[END_REF].

III. RESULTS

A. Standard rheology

Our experiments focus on a colloidal gel made of carbon black particles dispersed at 6 % w/w in a mineral oil [START_REF] Trappe | Scaling of the viscoelasticity of weakly attractive particles[END_REF]. Its rheological properties are typical of a soft solid as shown in Fig. 2: the elastic modulus G ′ is about ten times larger than the loss modulus G ′′ and both moduli are frequency independent (Fig. 2a). Moreover, as the amplitude σ of the oscillatory shear stress is increased, the gel yields for σ = σ y ≃ 8 Pa (Fig. 2b), where σ y is defined by G ′ (σ y ) = G ′′ (σ y ). Above σ y , one has G ′′ ≫ G ′ , i.e. the material behaves like a liquid. As seen in Fig. 2c where the gel was fluidized by strong preshearing for times t < 0 and left at rest for t > 0, the elastic modulus recovers within a few seconds. This variation corresponds to the gelation process. The short range attraction between the carbon black particles leads to cluster aggregation. As the clusters percolate, they form a space-spanning network. The backbone of the network gives rise to the observed solid-like structure. For t 100 s, we observe very little aging (Fig. 2d), which is ideal to run long experiments.

Yield stress measurements and global rheological properties are, however, too simple and elude fundamental issues such as the mechanism of fluidization and the role of self-healing processes due to the attractive interparticle forces. To follow the local dynamics of yielding, rheometry in Couette (concentric cylinders) geometry is coupled to ultrasonic speckle velocimetry (USV, see Methods section), which allows us to access deformation and velocity fields with high temporal resolution [START_REF] Manneville | High-frequency ultrasonic speckle velocimetry in sheared complex fluids[END_REF], under either creep tests or oscillatory shear.

B. Creep experiments

Figure 3a gathers the results of creep experiments for a wide range of imposed shear stresses. The critical shear stress σ y = 8.5 ± 0.5 Pa clearly separates two regimes. For σ < σ y , the gel remains solid-like and does not flow, whereas for σ > σ y , the sample eventually flows with a measurable shear rate γ. The shear rate response γ(t) presents a characteristic shape that has already been observed in creep experiments on surfactant columnar phases [START_REF] Bauer | Collective rearrangement at the onset of flow of a polycrystalline hexagonal columnar phase[END_REF]. In order to get better insight into the fluidization process, we correlate the rheological measurements to the local velocity field v(r) measured with USV for the typical creep experiment at σ = 10 Pa shown in Fig. 3b. In the early stage where γ(t) slowly increases (t 1700 s), solid-body rotation is observed, i.e. the sample is in an unsheared solid-like state except for thin lubrification layers along the cell walls that allow the gel to slide as a solid block (Fig. 3c). At the beginning of the sharp upward bend in γ(t), for 1700 s t 1800 s, the bulk sample starts to experience a non-zero shear rate. Fluidization then proceeds in a spatially heterogeneous way. Indeed, in this regime, velocity profiles present a highly sheared region close to the rotor that coexists with a low-shear band next to the stator and whose size increases in time (Fig. 3d). Such shear localization suggests that the gel progressively sticks to the cell walls and gets dragged along the rotor. The fluidized zone expands from the rotor until, for t 1800 s, the gel flows homogeneously without any apparent wall slip, as seen from the linear velocity profiles of Fig. 3e. In this last stage, the increase of the bulk shear rate indicates that further microstructural changes still occur and/or that sedimentation sets in.

Our local measurements refute the simple scenario in which yielding occurs abruptly and homogeneously at σ y (as suggested by the standard stress sweep experiment of Fig. 2b) and rather support a yielding mechanism that is heterogeneous in both space and time, where fluidization starts at the walls and propagates through the whole sample. Such front dynamics is reminiscent of heterogeneous nucleation as predicted by a phenomenological "fluidity" model [START_REF] Picard | A simple model for heterogeneous flows of yield stress fluids[END_REF]. The present experiment also allows us to define unambiguously a characteristic time τ for fluidization as the time when homogeneous shear is first observed, e.g. τ = 1850 ± 100 s for σ = 10 Pa. The fluidization time roughly corresponds to the inflection point in the γ(t) curve (and typically to γ = 1-10 s -1 ), a criterion that has been used previously [START_REF] Bauer | Collective rearrangement at the onset of flow of a polycrystalline hexagonal columnar phase[END_REF] but without the confirmation from local measurements.

Finally, Fig. 3f sums up the behavior of our colloidal gel in creep experiments. Below σ y (region I), the gel remains solid for all times, while above σ y , the gel first undergoes solid body rotation or shear localization for t < τ (region II) and eventually flows homogeneously for t > τ (region III). Remarkably, the fluidization time follows an exponential law: τ ∼ exp(-σ/σ 0 ) where σ 0 = 4.0±0.2 Pa. In other words, with σ 0 = k B T /v, where v is some characteristic volume, this behaviour is equivalent to an Arrhenius law τ ∼ exp(E(σ)/(k B T )), where the energy barrier E(σ) = -σv decreases linearly with the applied stress.

C. Oscillatory shear experiments

In a second series of experiments, the gel is submitted to oscillatory shear stress of given amplitude σ and frequency f = 1 Hz. The strain amplitude γ 0 (t) is monitored and again, we observe that when σ is large enough γ 0 eventually increases with time, suggesting that the sample becomes fluid over time (Fig. 4a,b). To confirm this scenario, we track the local deformation of the gel with USV (see Methods section). For σ < 5 Pa, the gel indeed remains solid: no decorrelation of the ultrasonic speckle signals, i.e. no rearrangement within the gel, was observed during waiting times as long as 10 5 s. However, for stress amplitudes above 5 Pa, rearrangements and fluid-like behaviour are detected first close to the cell walls (see t ≃ 100-300 s in Fig. 4d). The fluidized zone then progressively expands through the bulk material (t ≃ 300-550 s) until the whole sample gets fluid (t 550 s).

It is important to note that fluidization does not have such a dramatic signature on γ 0 (t) under oscillatory stress as on γ(t) during creep tests. Moreover, as shown in Fig. 4c, the loss modulus is larger than the elastic modulus throughout the whole experiment. In view of the USV results, this can only be explained by the presence of lubrication layers at the walls. A naive interpretation of standard viscoelastic data would thus wrongly conclude that the sample is fluid as soon as stress is applied. Here simultaneous spatially-resolved measurements show that solid-fluid coexistence comes into play and reveal a highly heterogeneous fluidization pattern.

In oscillation experiments, the fluidization time, defined as the time τ for which 90 % of the gel is in the fluidlike state, follows an exponential law τ ∼ exp(-σ/σ 0 ) with σ 0 = 3.9 ± 0.3 Pa for σ 8 Pa (Fig. 4e). This is strikingly similar to the behaviour obtained in creep experiments (Fig. 3f). However, under an oscillatory stress of amplitude 5 Pa σ 8 Pa, a strong departure from exponential behaviour is observed, pointing to a divergence at σ ≃ 5.5 Pa. In this stress range, no fluidization was observed during creep tests. Although this regime certainly deserves more attention, in the following, we shall focus on the exponential behaviour observed for stresses above 8 Pa. Shown in Fig. 5 is the fraction of fluidized sample φ(t) for ten different oscillatory experiments at a fixed stress amplitude of σ = 15 Pa. The large deviations from one run to the other reveal the random nature of the local processes at work during yielding. These noisy features are compatible with models that include activated dynamics [START_REF] Kobelev | Strain softening, yielding, and shear thinning in glassy colloidal suspensions[END_REF][START_REF] Fielding | Shear banding, aging and noise dynamics in soft glassy materials[END_REF][START_REF] Sollich | Rheology of soft glassy materials[END_REF]. Overall, the fluidized fraction φ(t) averaged over the various experiments is well fitted by a sigmoidal relaxation φ (t) = 1exp(-t 2 /τ 2 ), a behaviour typically found in nucleation and one-dimensional growth phenomena [START_REF] Berret | Isotropic-tonematic transition in wormlike micelles under shear[END_REF].

IV. DISCUSSION

The creep tests and the oscillation experiments above have shown the benefit of local measurements in establishing a spatiotemporal scenario for yielding: fluidization is initiated at the walls and propagates into the entire sample within a characteristic time τ that follows an Arrhenius law with an energy barrier E = -σv. It is interesting to notice that a similar Arrhenius scaling has been reported in shear-induced aggregation of non-Brownian suspensions [START_REF] Guery | Irreversible shear-activated aggregation in non-brownian suspensions[END_REF]. In both cases of shearinduced fluidization and aggregation, the energy barrier is reduced proportionally to the shear stress.

To the best of our knowledge, the present results provide the first experimental evidence for the relevance of activated processes and barrier hopping in the context of yielding in colloidal gels. Although such ingredients have recently been incorporated into theoretical approaches based on mode-coupling theory [START_REF] Kobelev | Strain softening, yielding, and shear thinning in glassy colloidal suspensions[END_REF], local dynamical descriptions are still missing from microscopic models of the stress-induced solid-to-fluid transition. Our results also suggest a comparison with fracture in heterogeneous materials. Indeed, the propagation of fracture fronts is known to involve activated processes attibuted to the nucleation of microcracks [START_REF] Bonn | Delayed fracture of an inhomogeneous soft solid[END_REF][START_REF] Guarino | Failure time and critical behaviour of fracture precursors in heterogeneous materials[END_REF][START_REF] Tabuteau | Microscopic mechanisms of the brittleness of viscoelastic fluids[END_REF]. Similarly, yielding proceeds by rapid local fluidization events (Fig. 4d) separated by some "waiting times" δt. Previous measurements on fibrous materials have shown that time intervals between microcrack events are power-law distributed [START_REF] Guarino | Failure time and critical behaviour of fracture precursors in heterogeneous materials[END_REF], a feature typical of self-similar avalanche-like phenomena. Here, δt rather follows an exponential distribution (insets of Fig. 5), which points to random uncorrelated events. Moreover, our observation of a linear energy barrier E = -σv is in stark contrast with the Griffith scaling E ∼ σ -4 for fracture [START_REF] Guarino | Failure time and critical behaviour of fracture precursors in heterogeneous materials[END_REF][START_REF] Tabuteau | Microscopic mechanisms of the brittleness of viscoelastic fluids[END_REF]. Following Ref. [START_REF] Guery | Irreversible shear-activated aggregation in non-brownian suspensions[END_REF], the linear dependence E = -σv may be interpreted as the reduction of the energy barrier through hydrodynamic energy. In that case, the volume v reads v = a 2 δ with a the particle radius and δ a microscopic length characteristic of the interparticle potential. Here, assuming that T can be identified with the thermodynamic temperature, k B T /v ≃ 4 Pa and a ≃ 500 nm lead to δ ≃ 4 nm, which provides a reasonable estimate for the range of attractive forces between carbon black particles.

Very recently, a novel rheological model to describe the local dynamics of ductile to brittle fracture was introduced and applied in the context of metallic glass formers [START_REF] Furukawa | Inhomogeneous flow and fracture of glassy materials[END_REF], but the connection with soft gels remains to be made. In particular, it can be noted that in some cases the fluidized fraction φ(t) decreases over short time periods before increasing again at longer times (Fig. 5). This behaviour, absent from fracture phenomena, is a direct illustration of the competition between shear-activated cluster breaking (which favors fluidization and an increase of φ(t)) and self-healing processes (which tend to lower φ(t)) due to the attractive interactions.

In summary, the fluidization scenario depicted in this article highlights the crucial importance of time-resolved local measurements in characterizing the behaviour of soft materials under stress. Ultrasonic echography constitutes a versatile tool to follow yielding and fluidization in both space and time even in optically opaque materials. Its mesoscopic resolution of a few 10 µm bridges the gap between microscopic imaging techniques and global rheological measurements. Indeed, coarse-grained models for soft glassy materials use local strains and stresses averaged over intermediate sizes [START_REF] Bocquet | A kinetic theory of plastic flow in soft glassy materials[END_REF][START_REF] Fielding | Shear banding, aging and noise dynamics in soft glassy materials[END_REF][START_REF] Sollich | Rheology of soft glassy materials[END_REF], so that gathering information at a mesoscopic scale is a key issue for comparing experiments to such models. From the fundamental point of view, our observations not only emphasize the importance of activated processes and noise in yielding but also reveal a complex interplay between boundaries and bulk dynamics. Such results should provide a new test ground for computer simulations and for theoretical calculations in the current attempt to under-stand yielding in a wide variety of soft glassy materials.

FIG. 1 :

 1 FIG. 1: Methods: ultrasonic echography under shear. (a) Schematic of the rheology experiment combined with USV. Temperature is controlled to 25±0.1 • C by a water circulation around the Couette cell. (b) Typical normalized pressure signal backscattered by glass microspheres embedded in a carbon black gel as a function of time after the incident pulse is emitted at t = 0 s. The times-of-flight corresponding to the positions of the stator and the rotor are indicated by a red and blue dashed line respectively. The corresponding gap is 1 mm. (c and d) Spatiotemporal diagrams of the normalized pressure signal during an oscillation experiment at f = 1 Hz and σ = 15 Pa.The pressure is coded in linear grey levels and plotted against the time at which the incident pulse is sent (x-axis) and the radial distance r to the rotor (y-axis). In (c), the pulse repetition frequency f prf is 400 Hz, which is large enough to resolve the deformation due to the oscillations of the rotor, while in (d), f prf = f = 1 Hz. (e and f). Same as in (c and d) but for a higher shear stress σ = 30 Pa.

FIG. 2 :

 2 FIG. 2: Standard rheology of a 6 % w/w carbon black suspension in a mineral oil. Viscoelastic moduli G ′ ( ) and G ′′ ( ) (a) in a frequency sweep in the linear regime at σ = 0.5 Pa, (b) in a stress sweep experiment at f = 1 Hz and (c, d) at σ = 0.2 Pa and f = 3 Hz as a function of time after preshear at high speed during 60 s. The red dashed line in (b) indicates the yield stress σy, which is time and frequency independent.

FIG. 3 :

 3 FIG. 3: Creep experiments in a 6 % w/w carbon black gel. (a) Temporal evolution of the shear rate γ for (from bottom to top) σ = 5, 7, 8, 9, 10, 11, 13, 15, 18, 20, 23, 25, 28, 30, 33, and 40 Pa. (b-e) Combined rheological and USV measurements for σ = 10 Pa. (b) γ(t) as a function of time. The red dashed line indicates the fluidization time τ discussed in the text. (c, d, e) Instantaneous velocity profiles v(r) recorded with USV at the times shown in (b) using the same symbols. (f) The fluidization time τ as a function of the applied shear stress σ. The black line is an exponential fit to the experimental data. Region I: solid-like behaviour. Region II: plug-like flow or shear localization. Region III: homogeneous flow.

FIG. 4 :

 4 FIG. 4: Large amplitude oscillatory shear experiments at f = 1 Hz in a 6 % w/w carbon black gel. (a) Temporal evolution of the deformation amplitude γ0 for (from bottom to top) σ = 5, 5.8, 6, 7, 8, 10, 13, 15, 18, 20, 23, 26 Pa. (b-d) Combined rheological and USV measurements for σ = 15 Pa. (b) γ0 and (c) G ′ (•) and G ′′ (•) as a function of time. (d) Spatiotemporal diagram of the ultrasonic speckle signal coded in linear grey levels. The USV sampling frequency is equal to the oscillation frequency f = 1 Hz. Red dots indicate the boundary between solid-like and fluid-like regions. The red dashed line shows the time τ at which 90 % of the gel across the gap is fluidized. (e) The fluidization time τ as a function of the stress amplitude σ. The black line is an exponential fit to the experimental data (only data above 8 Pa are considered in the fit). Region I: solid-like behaviour. Region II: solidfluid coexistence. Region III: fluid-like response. Between the green and blue dashed lines, τ strongly deviates from the exponential behaviour.

FIG. 5 :

 5 FIG. 5: Temporal evolution of the fluidized sample fraction φ for ten different oscillatory shear experiments at f = 1 Hz and σ = 15 Pa. Black dots show the average behaviour φ(t)and the thick green line is φ(t) = 1exp(-t 2 /τ 2 ) with τ = 320 s. The data are plotted against t/τ where τ = 480 s is the fluidization time defined as φ(τ ) = 0.9. Insets: probability distribution function P (top) and complementary cumulative distribution function C (bottom) of the waiting times δt between two successive jumps in φ(t). The solid lines are exponential fits P (δt) ∼ C(δt) ∼ exp(-t/τ ⋆ ) with τ ⋆ = 0.137τ = 65.8 s.

Acknowledgments

The authors are grateful to L. Bocquet, T. Divoux, J.-F. Palierne, and V. Trappe for fruitful discussions.


L Rakotondrainibe
email: lahatra.rakotondrainibe@insa-rennes.fr

Y Kokar

G Zaharia

G El Zein

Toward a Gigabit Wireless Communications System

Keywords: Wireless communications, 60 GHz system, high bit rate, DBPSK modulation, synchronization, BER

This paper presents the design and the realization of a hybrid wireless Gigabit Ethernet indoor communications system operating at 60 GHz. As the 60 GHz radio link operates only in a single-room configuration, an additional Radio over Fiber (RoF) link is used to ensure the communications within all the rooms of a residential environment. The system uses low complexity baseband processing modules. A byte synchronization technique is designed to provide a high value of the preamble detection probability and a very small value of the false detection probability. Conventional RS (255, 239) encoder and decoder are used for channel forward error correction (FEC). The FEC parameters are determined by the tradeoff between higher coding gain and hardware complexity. The results of bit error rate measurements at 875 Mbps are presented for various antennas configurations.

Introduction

The massive use of high quality multimedia applications such as high definition video streaming, file transfer and wireless Gigabit Ethernet explains the need of 1 Gbps overall throughput. One of the most promising solutions to achieve a gigabit class wireless link is to use millimeter-waves (MMW) for the carrier frequency. Due to the large propagation and penetration losses, 60 GHz Wireless Personal Area Networks (WPANs) are primarily intended for use in short range and single room environments. In addition to the high data rates that can be achieved in this frequency band, the radio waves propagation at 60 GHz has specific characteristics offering many other benefits such as high security and frequency reuse.

High frequency and even MMW analog communication circuits, which were traditionally built on more expensive technologies such as bipolar or Gallium Arsenide (GaAs), are progressively implemented on CMOS. Table 1 summarizes some experimental previous works about the realized transceiver operating at 60 GHz [START_REF] Scott | A Silicon 60-GHz Receiver and Transmitter Chipset for Broadband communications[END_REF], [START_REF] Ohata | 1.25 Gbps Wireless Gigabit Ethernet Link at 60 GHz Band[END_REF]. Different architectures have been analyzed in order to develop new MMW communication systems for commercial applications [START_REF] Collonge | Wideband and Dynamic Characterization of the 60 GHz Indoor Radio Propagation -Future Home WLAN Architectures[END_REF]- [START_REF] Young | An Overview of Multigigabit Wireless through Millimeter Wave Technology: Potentials and Technical Challenges[END_REF]. The selection of a modulation scheme is a primary consideration for any wireless system design and has a large impact on the system complexity and power consumption as well as issues such as amplifier linearity, and oscillator phase noise. Moreover, to determine the appropriate system design at 60 GHz, wireless channel characteristics must be well understood. This includes path loss, material attenuation, multipath effects, and antennas [START_REF] Collonge | Wideband and Dynamic Characterization of the 60 GHz Indoor Radio Propagation -Future Home WLAN Architectures[END_REF], [START_REF] Daniels | 60 GHz Wireless Communications: Emerging Requirements and design Recommendations[END_REF]. Hence, other considerations such as synchronization, coding/error correction and equalization must be taking into consideration for the overall system design. This paper proposes a hybrid communication system derived from the simplified IEEE 802.15.3c physical (PHY) layer [START_REF] Funada | A design of single carrier based PHY for IEEE 802.15.3c standard[END_REF] to ensure near 1 Gbps data rate on the air interface. The first system application in a point-to-point configuration is the high-speed file transfer. The system must operate in indoor, line-of-sight (LOS) domestic environments.

The rest of this paper is organized as follows. Sections 2 and 3 describe respectively the transmitter and the receiver block diagrams. In these sections, the radiofrequency (RF) architecture is first presented. Then, the baseband blocks are described. The byte/frame synchronization method also is discussed. Measurement results and analysis are respectively presented in sections 4 and 5. Section 6 concludes the work.

Transmitter design

Figure 1 shows the block diagram of the realized transmitter (Tx) system.

Figure 1. 60 GHz Wireless Gigabit Ethernet transmitter

The system uses a single carrier architecture based on a Differential Encoded Binary Phase Shift Keying (DBPSK) modulation. A differential encoder is used before the BPSK modulation to remove the phase ambiguity knowing that a differential demodulation is used at the receiver (Rx). Compared to higher order constellations or OFDM systems, this system is more resistant to phase noise and power amplifier (PA) non-linearities. OFDM requires large back-off for PA, high stability and low phase noise for local oscillators. Furthermore, the implementation of a singlecarrier system is simple.

Radiofrequency architecture

After channel coding and scrambling, the input data are differentially encoded using logic circuits (PECL). The differential encoder performs the delayed modulo-2 addition of the input data bit x n with the output bit y n as in [START_REF] Scott | A Silicon 60-GHz Receiver and Transmitter Chipset for Broadband communications[END_REF].

y n+1 = x n ⊕ y n (1)
The obtained data are used to modulate an intermediate frequency (IF) carrier generated by a 3.5 GHz phase locked oscillator (PLO) with a 70 MHz external reference. The IF modulated signal is fed into a band-pass filter (BPF) with 2 GHz bandwidth and transmitted through a 300 m optical fiber link. This IF signal is used to modulate the current of the Vertical Cavity Surface Emitting Laser (VCSEL) operating at 850 nm. In order to avoid signal distortions, the power of the VCSEL input signal must not exceed -3 dBm. A 300 m multimode fiber with a bandwidth-length product of about 4000 MHz.km is enough to cover the bandwidth and the distance range. After transmission via the fiber, the optical signal is converted to an electrical signal by a PIN diode and amplified. The overall RoF link has 0 dB gain and an 8 GHz bandwidth.

Following the RoF link, the IF signal is sent to the RF block. This block is composed of a mixer, a frequency tripler, a PLO at 18.83 GHz and a band-pass filter (59-61 GHz). The frequency of the local oscillator is obtained with an 18.83 GHz PLO with a 70 MHz reference and a frequency tripler. The phase noise of the 18.83 GHz PLO signal is about -110 dBc/Hz at 10 kHz off-carrier. The upper sideband is selected using a BPF. The BPF prevents the spill-over into adjacent channels and removes out-of-band spurious signals caused by the modulator. The 0 dBm obtained signal is fed into the 22.4 dBi horn antenna with 10° vertical half-power beamwidth and 12° horizontal half-power beamwidth (HPBW).

Baseband architecture

The Gigabit-Ethernet interface is used to connect a home server to the 60 GHz wireless link with around 800 Mbps bit rate, as shown in Figure 2.

Figure 2. Transmitter Gigabit Ethernet interface

The Gigabit Media Independent Interface is an interface between the Media Access Control (MAC) device and the physical layer (PHY). This interface defines data rates up to 1 Gbps, implemented using byte data interface at 125 MHz. However, this frequency is different from the clock data (104.54 MHz) generated by a clock manager, as explained latter. This clock asynchronism provides data packet loss. In order to avoid jitter, a programmable circuit (FPGA) is used as part of buffers memory. The transmitter baseband architecture is shown in Figure 3.

Figure 3. Transmitter baseband architecture

The data bytes from the Gigabit Ethernet interface are directly transfered into the dual port FIFO memory (due to the byte streams of the RS encoder). Otherwise, the input bit stream is paralellized by a serial-to-parallel (S/P) converter.

The frame format consists of 4 preamble bytes, 239 data burst bytes, 16 check bytes and a "dummy byte" as shown in Figure 4. The dummy byte is added besides 4 preamble bytes to obtain a multiple of 4 coded data length useful for the scrambling operation.

A known preamble is sent at the beginning of each frame in order to achieve the frame synchronization at the receiver. The used preamble is a Pseudo-Noise (PN) sequence of 31 bits + 1 bit to provide 4 preamble bytes. Hence, owing to the frame structure and the byte operation, two different clock frequencies f 1 and f 2 are used in the baseband blocks: 8)] is determined in order to obtain the lowest value of the maximum cross-correlation between the preamble P and T, given by ( 2) and ( 3), where: For each value of k, the cross-correlation is represented for 1 ≤ i ≤ 8. All the results are analyzed to identify the best dummy byte which gives the lowest cross-correlation values. The best result is obtained for k = 64 which gives d = [0 0 0 0 0 0 1 0]. For k = 64, the cross-correlation for different possible shifts (1 ≤ i ≤ 8) is shown in Figure 6. The maximum cross-correlation is equal to 16. This value is obtained twice; all the other values are smaller. Therefore, this optimal byte gives the smallest value of the PHY preamble false detection probability.

Long sequences of '0' or '1' must be avoided for reliable operation of Clock and Data Recovery (CDR) circuit at the receiver. Therefore, the use of a scrambler at the transmitter and a descrambler at the receiver is essential. The scrambler is a PN-sequence of 31 bits + 1 bit to obtain 4 bytes. This scrambler is chosen in order to provide the lowest crosscorrelation values between the received data and the 4 bytes PHY preamble. This method reduces the number of false detections of the PHY preamble within the scrambled data. The received byte stream is finally parallel-to-serial (P/S) converted just before the differential encoder.

Receiver design

Figure 7 shows the block diagram of the realized receiver system. The system architecture uses a simple non coherent demodulation at the receiver.

Radio frequency architecture

The input band-pass filter, after the Rx antenna, removes the out-of-band noise and adjacent channels interference. The RF signal obtained at the filter output is down-converted to IF = 3.5 GHz and fed into a band-pass filter with 2 GHz bandwidth. An Automatic Gain Control (AGC) with a dynamic range of 20 dB is used to ensure a quasi-constant power level at the demodulator input. A Low Noise Amplifier (LNA) with a gain of 40 dB is used to achieve sufficient gain. The differential demodulation enables the coded signal to be demodulated and decoded.

Compared to a coherent demodulation, this method is less efficient in additive white Gaussian noise (AWGN) channel. However, in the presence of frequency selective propagation channel, the differential demodulation is more robust to multi-path propagation which induces intersymbol interference (ISI). Indeed, the differential demodulator is realized with a delay line (Ts = 1.14 ns) and a mixer. Then, due to the product of two consecutive symbols, the rate between the main lobe and the second lobes of the impulse response of the channel is increased. This operation improves the Signal to Noise + Interference ratio (SNIR). Following the differential demodulator, a Low-Pass Filter (LPF) with 1 GHz cut-off frequency removes the high-frequency components of the obtained signal.

Baseband architecture

Figure 8 shows the receiver baseband architecture. The serial data (after the CDR) are converted into byte streams.

Figure 9. PHY Preamble detection architecture

The preamble detection is based on the cross-correlation between 32 successive received bits and the internal 32 bits PHY preamble. In fact, this preamble detection is performed within a FPGA by using a byte stream in order to reduce the data clock frequency. Because the serial-to-parallel conversion realized by the multigigabit transceiver of the FPGA is arbitrary, a byte-synchronization must be performed. Therefore, 8 identical correlators are used. In addition, each correlator must analyse a 1-bit shifted sequence of 32 bits. Hence, the preamble detection is performed with 32 + 7 = 39 bits (+ 7 because of different possible shifts of a byte). In all, there are 8 corrrelators of 32 bits within each correlator-bank, as shown in Figure 10. Each value of the correlation between the preamble and the received data (32 bits) is compared to the threshold to be determined. Threshold setting plays an important role because the performance of the proposed system is based on the reliable byte/frame synchronization. The frame synchronization performance is characterized by the miss detection probability P m , the false alarm probability P f and the channel error probability p, as described in [START_REF] Urkowitz | Signal Theory and Random Processes[END_REF]. Figure 11 shows the miss detection probability of the PHY preamble as a function of p. The maximum setting threshold S = 32 is equal to the maximum cross-correlation between the PHY preamble (32 bits) and the received data (32 bits). This maximum crosscorrelation is obtained when the preamble of the received frame has no error. The byte/frame synchronization also depends on the false alarm events. Occasionally, due to the random values of the binary transmitted data, the architecture shown in Figure 10 can falsely declare that the preamble is detected. For this reason, the threshold is chosen in order to obtain the best trade-off between a high value of the detection probability and a very small value of the false alarm probability. A false alarm is declared when the same correlators C k of the two correlator-banks indicate the detection of the preamble within two successive bursts of 256 data bytes. Figure 12 shows the preamble false alarm probability for p = 10 -2 . Thus, by taking into account the periodical repetition of the preamble, we obtain:

• for p = 10 -2 and S = 27, Pm ≈ 10 -6 and Pf ≈ 2*10 -10 ;

• for p = 10 -2 and S = 28, Pm ≈ 3*10 -5 and Pf ≈ 10 -11 .

As can be seen, the best trade-off is obtained for a threshold setting S = 28. The simulation of the preamble detector shows isolated miss detection events. This observation can be used to further improve the reliability of the byte/frame synchronization. Indeed, the decision of a preamble loss can be taken when 3 successive PHY headers are not detected. A new byte/frame synchronization can be declared when the same correlators C k of each correlator-block indicate 3 successive preamble detections. After the byte alignment and the preamble detection, the descrambler performs the modulo-2 addition between 4 successive received data bytes and the 4 bytes scrambler. The RS decoder processes the descrambled bytes and attempts to correct the errors. The RS (255, 239) decoder can correct up to 8 erroneous bytes and operates at a high clock frequency (f 2 = 109.37 MHz). The byte streams obtained at the decoder output are written discontinuously in the FIFO dual port memory with the clock frequency f 2 . The other frequency f 1 = 100.54 MHz reads out continuously the data bytes stored in the FIFO dual port memory. Then, the byte streams are finally transmitted to the Gigabit Ethernet interface, as depicted in Figure 13.

Figure 13. Gigabit Ethernet interface

Due to the low transmission rate for the return link, another system such as Wi-Fi or an Ethernet cable can be used.

Measurement results

A vector network analyzer (HP 8753D) was used to determine the frequency response and the impulse response of the both RF blocks (Tx + Rx) including LOS channel, using horn antennas. In this good propagation conditions, the measured response allows to estimate the system bandwidth and the RF components imperfections. The measurements were realized in a corridor with approximate dimensions of 15x2.4x3.1 m³ where the major part of the transmitted power is focused in the direction of the receiver. The RF-Tx and RF-Rx blocks were placed at a height of 1.5 m and at 10 m distance. After measurement set-up and calibration, a frequency response of 2 GHz bandwidth is obtained, as shown in Figure 14 From this eye pattern, the recovered data can be obtained by sampling at half-period in order to be noise resistant. Figure 16 shows the measured BER results at 875 Mbps as a function of the Tx-Rx distance. In our experiment, four antennas were used: two horn antennas and two patch antennas. Each patch antenna has 8 dBi gain with 30° HPBW.

Discussion

The results show that the use of high-directivity horn antennas gives a remarkable BER performance and greatly reduces the harmful effects of the multipath propagation. This leads to a link budget improvement as high as 15 dB, compared to patch antennas. The system with high gain directive antennas is acceptable for point-to-point wireless communications links, with minimal multipath interference. However, a 60 GHz radio link is sensitive to shadowing due to high attenuation of the non line-of-sight (NLOS) propagation. Hence, when directive antennas are used, once the direct path is blocked by moving obstacles, the communication can be completely lost. Furthermore, the Tx-Rx antennas have to be aligned; otherwise the beampointing errors will cause an important reduction of the communication quality due to an increased BER. For LOS and point-to-point configurations, the use of a patch antenna for the receiver has the advantage to assure an important angular coverage, which significantly reduces the misalignment errors. However, if antennas beamwidth is large, equalization should be adopted to overcome the multipath interference at a high data rate [START_REF] Falconer | Frequency Domain Equalization for Single-Carrier Broadband Wireless Systems[END_REF].

In order to improve the system reliability, the choice of a centralized transmitter, preferably located on the center of the room ceiling must be considered. The height of the transmitter (with less directional antenna) can partly reduce the influence of the shadowing. The different receivers forming the network could be equipped with more directional antennas, pointed toward the transmitter. It has been noted that the use of a directional receiving antenna is interesting for its gain and the reduction on the multipath propagation effects. In this configuration, only the contribution of the direct path is sought, and classical single carrier modulations could be used.

The previous measurements were performed in a corridor. In a large hall, the power of the received signal can be lower and thus significantly affects the quality of the communication link. Future work will provide BER measurement results (with channel coding) in different indoor environments.

Conclusion

This paper presented the design and the implementation of a 60 GHz communication system for WPAN applications in point-to-point or point-to-multipoint configurations. The proposed system provides a good trade-off between performance and complexity. An original method used for the byte and frame synchronization is also described. This method allows a high preamble detection probability and a very small false detection. For 1 Gbips reliable communications within large rooms, the transmitting and receiving antennas must have a relatively high gain.

Increasing the data rate (1,75 Gbps) can be achieved using higher order modulations such as QPSK. Implementation of frequency domain equalization methods is still under study. The system will be further enhanced to prove the feasibility of reliable wireless communications at data rates of several Gbps in different configurations, especially in non line-ofsight (NLOS) environments.

Figure 4 .

 4 Figure 4. Frame structure

Figure 5 .

 5 Figure 5. Maximum correlation between P and T

Figure 6 .

 6 Figure 6. Cross-correlation between P and T for k = 64

Figure 7 .

 7 Figure 7. 60 GHz Wireless Gigabit Ethernet receiver

Figure 8 .

 8 Figure 8. Receiver baseband architecture Figure 9 shows the physical preamble detection architecture.

Figure 10 .

 10 Figure 10. Preamble detection with periodicity and control alignment Further, in order to obtain a very low preamble miss detection probability, the periodical repetition of the PHY preamble is taken into consideration. The decision is made from two successive PHY preambles (preamble1 + 256 data bytes + preamble2) stored in a register. If the preamble detection is indicated in each correlator-bank by the same correlator C k , the operation is validated.

Figure 11 .

 11 Figure 11. PHY preamble miss-detection probability

Figure 12 .

 12 Figure 12. False alarm probability of PHY preamble detection with one/two blocks of correlators

  (a).

Figure 14 .

 14 Figure 14. (a) Frequency response and (b) Impulse response A perfect system must have an impulse response with only one lobe. As shown in Figure 14 (b), the measured impulse response presents some side lobes which are mainly due to RF components imperfections. A back-to-back test was also realized. The Tx and Rx antennas were used with a 45 dB fixed attenuator at 60 GHz. Similar results for the frequency and impulse responses were obtained, confirming the previous assumption.Moreover, in order to evaluate the transmission performance, a HP70841B pattern generator was used to generate pseudo-random sequences at the transmitter and a HP 708842B error detector at the receiver. After demodulation, the eye pattern is shown in Figure15. The eye opening indicates a good transmission quality at 875 Mbps for 10 m Tx-Rx distance.

Figure 15 .

 15 Figure 15. Eye diagram from 875 Mbps demodulated signal (10 m Tx-Rx distance)

  Figure 16. BER performance at 875 Mbps (without coding)

Table 1 .

 1 Summary of experimental 60 GHz radio systems

		Wigwam	IBM	NEC	Motorola
		(2005)	(2006)	(2002)	(2004)
	Technologies 0.25 µm	0.15 µm	0.13 µm	AsGa
		SiGe	AlGaAs/	SiGe
		BiCMOS	InGaAs HJFET	BiCMOS
	Modulation	OFDM	OFDM/QPSK	ASK	OOK
		/QPSK	ASK, MSK
	Keywords	LNA, PA, Mixer, PLL	LNA, PA, Mixer, LO	Mod/demod Filter, LNA, doubler	LNA, PA, Mixer, Multiplier
	Intermediate
	Frequency
	Performance

  The frame structure is obtained as follows: byte streams are written into the dual port FIFO memory at 100.54 MHz. The FIFO memory has been set up to use two different clock frequencies for writing at f 1 and reading at f 2 . Therefore, reading can be started when the FIFO memory is half-full in order to avoid the under/overflow of the FIFO memory. The encoding control generates 4 preamble bytes (not processed by the RS encoder) and reads 239 bytes. The RS encoder reads one byte every clock period. After 239 clock periods, the encoding control interrupts the bytes transfer during 17 clock periods. During this interruption, 4 preamble bytes are added; the encoder takes 239 data bytes and appends 16 control bytes to make a code word of 255 bytes. In addition, the encoding control adds the dummy byte for the balanced scrambling operation. In all, the number of data coded bytes (excepting the 4 preamble bytes) is 256, a multiple of 4.

	1 f	=	1 8 F	=	100.54 MHz,	f	2	=	2 8 F	=	109.37 MHz.
	where:
		2 F	=	4 IF	=	875 MHz and	1 2 F F	=	239 260	.

The additional dummy byte d = [d(1) d(2) … d(

Acknowledgements

This work is part of the research project Techim@ges supported by the French "Media & Network Cluster" and the COMIDOM project supported by the "Région Bretagne". The authors especially thank Guy Grunfelder (CNRS engineer) for his technical contributions during the system realization.


G Sartori

A Chemori

F Pierrot

O Company

Nonlinear Dual Mode Adaptive Control of PAR2 : a 2-dof Planar Parallel manipulator, with Real-time experiments

This paper deals with nonlinear dual mode adaptive control of a redundant manipulator for a pick-andplace scenario with high acceleration (20G). For performance comparisons, a conventional Proportional-Derivative (PD) controller has also been implemented. In this context, the experimental testbed is not equipped with velocity sensors. Therefore, a high-gain observer has been implemented to estimate the articular velocities. Real-time experiments show the performance improvements obtained by the proposed control approach in comparison to the conventional one.

I. INTRODUCTION

Par2 is a new two-degree-of-freedom (do f ) parallel manipulator that is able to produce two translations in the vertical plane, and can reach accelerations above 40G. The proposed architecture of this robot can guarantee that it can be lighter and stiffer than a classical 2 do f planar mechanism [START_REF] Pierrot | Above 40g acceleration for pick-and-place with a new 2-dof pkm[END_REF]. One other advantage of this parallel robot is that it has the possibility to keep the motors fixed on the base, allowing fast movements to be performed [START_REF] Honegger | Adaptive control of the hexaglide, a 6 dof parallel manipulator[END_REF].

In order to achieve such accelerations and perform an accurate movement, a good controller must be used. By using simple linear single-axis controllers (such as a Proportional Derivative (PD) controller), the tracking performance can be limited, especially when the robot has highly nonlinear dynamics and/or when the velocities/accelerations are high [START_REF] Honegger | Adaptive control of the hexaglide, a 6 dof parallel manipulator[END_REF].

During last decades, adaptive control systems have been widely developed. The main principle of this control technique is to adjust automatically the controller parameters in the case of unknown and time-varying process parameters in the aim that a desired degree of the performance index is met. One of the main characteristics of adaptive control systems is their capability to tune the controller parameters in real-time from the measurable information of the closedloop system.

In the literature, there are mainly four techniques of adaptive control, namely Gain Scheduling, Model Reference Adaptive Control, Self-tuning controllers and Dual control. They can be classified into two classes: Direct methods and indirect methods. In the first case, the adjustment rules tell us directly how the controller parameters should be updated. This class of schemes includes: Gain scheduling [START_REF] Shamma | Analysis of gain scheduled control for non linear plants[END_REF] and Model Reference Adaptive Control [START_REF] Monopoli | Adaptive control: The model reference approach[END_REF]. In the former, a set of measurable variables that correlate with eventual changes in the process dynamics is used to adapt the controller parameters. The latter was proposed to solve the problem in which the performance specifications are given in terms of a reference model.

The second class concerns indirect methods, where the estimates of the process parameters are updated and the controller parameters are obtained through the solution of a design problem using estimated parameters. This class includes: Self-tuning controllers [START_REF] Bobal | Digital Self-Tuning Controllers[END_REF] and Dual control [START_REF] Feldbaum | Optimal Control Systems[END_REF].

The three first adaptive controllers do not take into account parameter uncertainties in the design stage. However, if nonlinear stochastic control theory is used, it will significantly improve the performance of the control approach. Already in 1960, A. Feldbaum indicated that adaptive control systems based on the certainty-equivalence (CE) approach are often far away from being optimal. Instead of the CE approach, he introduced the principle of adaptive dual control [START_REF] Feldbaum | Optimal Control Systems[END_REF]. The design of such a controller takes into account uncertainties in the estimated parameters, so that it will be able to take special actions when it has poor knowledge about the system.

Different model-based adaptive controllers have been used until now for the control of parallel robots. In [START_REF] Honegger | Adaptive control of the hexaglide, a 6 dof parallel manipulator[END_REF], a nonlinear adaptive feedforward controller was proposed for the control of the Hexaglide (a 6 do f parallel robot), in addiction to a PD feedback term. The main objective of this work was to show the convergence of the adaptive parameters by simulation. In [START_REF] Honegger | Application of a nonlinear adaptive controller to a 6 dof parallel manipulator[END_REF], a scheme similar to the so called computed torque controller 1 [4] was also proposed to control the Hexaglide robot, but with experimental results that showed a good improvement in the trajectory tracking with relation to a PD controller, although no control signal has been presented.

In this paper, a nonlinear Dual Mode (DM) controller, originally refered as 'binary' in [START_REF] Hsu | B-mrac global exponential stability with a new model reference adaptive controller based on binary control theory[END_REF], is proposed to control the Par2 parallel manipulator for a pick-and-place application with high acceleration. This controller consists in the one proposed by Slotine and Li in [START_REF] Slotine | Adaptive manipulator control: A case study[END_REF], with the addition of a projection to the parametric adaptation law, such that it would be possible to guarantee that the estimated parameters would be bounded.

This paper is organized as follows. In section 2, a description of the Par2 parallel robot is presented. Section 3 details the proposed control algorithm. In section 4, the experimental results are presented and commented. In section 5, a conclusion about the current results and the expectations on the future performance of Par2 robot with the proposed controller is made.

II. PAR2 PROTOTYPE: DESCRIPTION AND DYNAMICS

A. Description of the Par2 robot

In the aim of application of the proposed control scheme, the prototype Par2 will be used. It's a two-dof parallel manipulator, illustrated in figure 1, with the following characteristics:

• the platform 6 is a rigid body, The proposed architecture of the prototype has the following advantages:

• the two coupled passive chains made a constrained system supporting almost all the moments and force besides the driving forces, • it can be designed with existing technologies and parts (can be made as light as for Delta [START_REF] Clavel | Delta, a fast robot with parallel geometry[END_REF] or Par4 [START_REF] Nabat | Par4: very high speed parallel robot for pick-and-place[END_REF] parallel robots) • it is symmetric with respect to motion plane and the yoz plane in its centered position

The proper functioning of this two-dof parallel manipulator is guaranteed by the coupling of the rotation of arms 4 . This constrains the robot's platform to evolve in one plane.

The coupling means that the rotation of the first arm in the clockwise direction involves the rotation of second one in the counterclockwise direction.

Remark 1:

The two actuated articulations of the robot are equipped with TPM50 Alpha motors coupled with gears having a reduction ratio of 21. For more details about the prototype Par2, the reader is referred to [START_REF] Pierrot | Above 40g acceleration for pick-and-place with a new 2-dof pkm[END_REF].

B. Dynamic model of the robot Par2

The Lagrangian nonlinear dynamic model [START_REF] Sciavicco | Modeling and control of robot manipulators[END_REF], [START_REF] Spong | Robot Dynamics and Control[END_REF] of the manipulator Par2 is given by:

I eq q1 q2 = τ -f d sign( q1 ) sign( q2 ) -f v q1 q2 - g 2 (M 1 + M 2 ) L 1 cos(q 1 ) L 2 cos(q 2 ) + J T M p J(q, q) q -g M p M p (1)
where I eq = I drv + I f + I a + J T M p J, I drv is the motor driver inertia, I f is the forearm inertia, I a is the arm inertia, J is the jacobian matrix, M p is the mass of the platform of the robot, g is the gravitational acceleration, f v is the viscous friction coefficient, f d is the dry friction coefficient, M 1 and M 2 are the masses of the arms, L 1 and L 2 are their lengths. This dynamic model can be written in the following matrix form:

D(q) q +C(q, q) q + G(q) + f (q, q) = τ (2)
where D(q) ∈ R 2×2 is the inertia matrix, C(q, q) ∈ R 2×2 is the matrix of the centrifugal and Coriolis terms, G(q) ∈ R 2 is the vector of gravitational forces, f (q, q) ∈ R 2 is the vector of friction forces, τ ∈ R 2 is the vector of control inputs (torques generated by the actuators), q = q 1 q 2 ∈ R 2 is the vector of articular positions, q = q1 q2 ∈ R 2 is the vector of articular velocities, q = q1 q2 ∈ R 2 is the vector of articular accelerations.

III. PROPOSED SCHEME: A NONLINEAR ADAPTIVE DUAL MODE CONTROLLER

The Dual Mode controller consists basically in the utilization of a high adaptation gain together with a projection of the estimated parameters. Then, to large tracking errors in the transitory stage, the controller behaves approximately as a sliding mode controller, generating an exponential convergence to a residual domain arbitrarily small. To smaller errors, it behaves as a parametric adaptation law. Other important advantages of the adaptation law in dual mode with respect to other adaptation laws or known robust control algorithms are listed below:

• Generation of continuous control signals,

• Improvement of the robustness of the system, • Limitation of the values of the estimated parameters through a projection, which has the effect of reducing the effective gain of the controller when the tracking error increases (reducing the sensitivity to measurement noises).

A. High-Gain Observer Based Control Algorithm

The desired articular trajectory denoted by q d (t) is supposed uniformly bounded, twice continuously differentiable with its two first derivatives qd (t) and qd (t) also uniformly bounded. In order to obtain this trajectory, the Cerebellum Path Generator [12] based on adept cycle is used. It generates the desired cartesian trajectories, which will be the input for the direct geometric model of the robot [START_REF] Pierrot | Above 40g acceleration for pick-and-place with a new 2-dof pkm[END_REF]. This scheme is illustrated in the diagram of figure 2:

Fig. 2. Generation of the desired articular trajectories

According to [START_REF] Slotine | Adaptive manipulator control: A case study[END_REF], the following errors are introduced: q = q dq , q = qdq , s = q + λ q , qr = qd -λ q (3)

where q is the position error, q d is the desired joint position, q is the measured joint position, q is the velocity error, s is an auxiliary error, λ is a positive constant, and qr is the denominated 'reference velocity' [START_REF] Slotine | Adaptive manipulator control: A case study[END_REF], [START_REF] Slotine | Applied Nonlinear Control[END_REF].

Assume that only the articular positions are measured (which is the case of the Par2 manipulator). Therefore the easiest way to compute the articular velocities consists in a numerical derivation. However, if the measured positions are noisy or do not have a good enough resolution, this technique will amplify the noise/quantization effect. In order to overcome this problem, an observer-based technique is proposed. It consists in estimating the joint velocities by means of a High-gain observer (HGO) [START_REF] Lee | Adaptive output feedback control of robot manipulators using high-gain observer[END_REF], described by the following:

ẋ1 = x2 + 1 ε η 1 (x 1 -x1 ) ẋ2 = 1 ε 2 η 2 (x 1 -x1 ) -qd + F( x, q d , qd ) + Ĥ( x1 , q d )τ s (4)
where • x 1 represents the error of the system (x 1 = q) and x 2 , its derivative, • x1 and x2 represent the estimated states, • ε, η 1 and η 2 are positive constants,

• F( x, q d , qd ) = -D -1 (x 1 , q d )[C(x, q d , qd )(x 2 + qd ) + G(x 1 , q d )], • Ĥ( x1 , q d ) = D -1 (x 1 , q d ),
• τ s is the saturated torque (to avoid the 'peaking phenomena'). The control architecture is illustrated in figure 3, and will be detailed in the following:

As illustrated in figure 3, the control law consists in the sum of three terms: an 'adaptive term', a 'smooth variable structure term' and a 'stabilizing term', and is given by: being d, α and K positive constants, Sat(.) is a continuous function with respect to its argument (with continuous partial derivatives and components limited to the interval [-1, +1]). The vector â represents an estimate of the unknown parameters of the system given by the vector a, and Y is the regressor vector (based on the dynamic model of the system). The adaptation of the parameters is given by: θ = -σ θ -γsY [START_REF] Honegger | Adaptive control of the hexaglide, a 6 dof parallel manipulator[END_REF] where θ = âa nom [START_REF] Hsu | B-mrac global exponential stability with a new model reference adaptive controller based on binary control theory[END_REF] is the difference between the currently estimated values and the nominal values of the parameters. The variable σ is given by the following:

τ = Y â + dSat(αs) + K.s (5)
σ = 0, if || θ || < M θ or σ eq < 0 σ eq , if || θ || ≥ M θ and σ eq ≥ 0 ( 9
)
σ eq = - γsY θ || θ || 2 ( 10
)
being M θ the maximum possible value (supposed known) of the norm of the estimated deviation of the parameters with relation to their nominal values.

Let us now consider the dynamic model ( 2) with the estimates of the system parameters â: D(q, â) q +C(q, q, â) q + G(q, â) + f (q, q, â) = τ [START_REF] Nabat | Par4: very high speed parallel robot for pick-and-place[END_REF] the regressor vector Y = Y (q, q, qr , qr ) can then be computed: D(q, â) qr +C(q, q, â) qr + G(q, â) + f (q, q, â) = Y â (12)

Note that in (12) we use qr and qr instead of q and q. The key idea comes from eliminating undesirable steady-state position errors by restricting them to evolve on a sliding surface [START_REF] Slotine | Adaptive manipulator control: A case study[END_REF] such as in robust sliding mode control [START_REF] Spurgeon | Sliding mode control: theory and applications[END_REF].

By neglecting the mass of the platform and the dry friction, these terms become D(q, â) = I motor +I arm +I f orearm , f (q, q, â) = f v and:

G(q, â) = - g 2 (M 1 + M 2 ) L 1 cos(q 1 ) L 2 cos(q 2 ) (13)
where

I motor = I m 0 0 I m , ( 14
)
I arm = I a 0 0 I a , ( 15
)
I f orearm = L 2 1 m 2 2 0 0 L 2 2 m 2 2 (16)
By rewriting the dynamics of the system such that it has the form given in (12), one can find that:

Y = q1r 0 q1r -L 1 cos(q 1 ) 0 q2r q2r -L 2 cos(q 2 ) (17)
and

â = ⎡ ⎢ ⎢ ⎣ â1 â2 â3 â4 ⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ I m + I a + L 2 1 m 2 2 I m + I a + L 2 2 m 2 2 f v g 2 (M 1 + M 2 ) ⎤ ⎥ ⎥ ⎦ (18)

IV. REAL-TIME EXPERIMENTAL RESULTS

The objective of this section is to present real-time experimental results of the application of the proposed control scheme described in section III to the Par2 parallel manipulator described in section II. A PD controller was also implemented for a performance comparison with respect to the proposed controller.

The platform evolves in the XOZ plane (being X the horizontal axis and Z the vertical axis). The desired cartesian trajectory to be tracked by the Par2 parallel robot is a usual 'pick-and-place' trajectory, and its parameters are described in table I. The parameters of the proposed control approach are summarized in table II. This experimental scenario deals with the control of the parallel manipulator for one cycle of a pick-and-place trajectory, that is, the robot's platform has to go (as illustrated in figure 4) from the initial cartesian position (x d i , z d i ) to the desired final cartesian position (x d f , z d f ) and then return to The trajectory tracking obtained by the PD controller and by the DM controller for 20G is shown in figure 5, the tracking errors are shown in figure 6, and the control inputs are shown in figure 7. The real-time implementation of both controllers was made with a sampling time of 0.5 msec. In such scenario, the main control objective is to have the smallest tracking error possible, specially on the final positions of the trajectory. These positions occur at t ∈ [0.125, 0.175] (final desired position) and at t ∈ [0.325, 0.375] (back to the initial desired position).

By analyzing figures 5 and 6, it is possible to notice that the DM controller has a considerably smaller tracking error than the PD controller during all the trajectories tracking, specially in the stop points, which is the most important objective of this experiment. The PD controller keeps the 7, it is possible to notice that the PD controller was delayed in comparison to the DM controller, and the amplitudes of both signals were roughly similar.

As it was noticed above, Par 2 is equipped with TPM50 Alpha motors of a maximum torque of 483 N.m, therefore according to figure 7 of the generated torques, we conclude that this bound is largely satisfied, but it should also be checked that the power requirement remains within the admissible limit.

To check the admissibility of the required power of the actuators, the idea is to plot their absolute torques versus their absolute angular velocities, and to check if the obtained curves remain within the admissible region given by the manufacturer. These curves are plotted in figure 8 (for the PD controller) and in figure 9 (for the DM controller). According to the obtained results, it is easy to conclude that the required power of the actuators is admissible for both controllers. The performance details of the two controllers are summarized on table III. The obtained movements of the robot while tracking the pick-and-place reference trajectory is illustrated by the video accompanying the paper.

V. CONCLUSIONS AND FUTURE WORKS

This paper deals with a nonlinear adaptive Dual Mode controller for the parallel manipulator (Par2), for a pick-andplace trajectory tracking with an acceleration of 20G. The obtained experimental results show that the Dual Mode (DM) controller has a considerably better performance than the conventional Proportional-Derivative (PD) controller. In the future, the proposed control approach will be evaluated for higher accelerations and also for different load conditions.

Fig. 1 .

 1 Fig. 1. The two-dof parallel manipulator Par2: view of the robot (left), schematic view of its mechanical structure (right)

Fig. 3 .

 3 Fig. 3. Block diagram of the proposed control scheme

Fig. 4 .

 4 Fig. 4. Illustration of the robot's movements (top) and the desired cartesian trajectory x-z in a larger scale (bottom)

Fig. 5 .Fig. 6 .

 56 Fig. 5. Articular reference trajectory for 20G (solid), PD tracking (dashed), DM tracking (dots)

Fig. 7 .Fig. 8 .

 78 Fig. 7. Torques for the PD controller (dashed) and for the DM controller (continuous)

Fig. 9 .

 9 Fig. 9. Absolute torques of the actuators versus their absolute angular velocities for the DM controller, and admissible region

TABLE I PARAMETERS

 I OF THE CARTESIAN REFERENCE TRAJECTORY

	Parameter Description	Value
	(x d i , z d i )	in the plane XOZ Initial desired cartesian position	(-0.35 m,-0.95 m)
	(x d f , z d f )	the plane XOZ Final desired cartesian position in	(0.35 m,-0.95 m)
	ẋmax d ẍmax d	Maximum cartesian velocity Maximum cartesian acceleration	8m/s 20 G

  • , 3.3 • ] during all the trajectories, the DM controller keeps it inside the limits of [-2.8 • , 1.9 • ] (peak-to-peak improvement of approximately 37%). And concerning the stop points, the PD controller generates errors inside [-2.3 • , 2.3 • ] while the DM controller generates errors inside [-0.7 • , 0.7 • ] (peak-to-peak improvement of approximately 70%). Concerning the control signals shown in figure

TABLE III PERFORMANCE

 III COMPARISON BETWEEN THE PROPOSED CONTROL APPROACH AND A CONVENTIONAL PD CONTROLLER Performance PD DM Error peaks [-4.15 • , 3.3 • ] [-2.8 • , 1.9 • ] Stop point errors [-2.3 • , 2.3 • ] [-0.7 • , 0.7 • ] PD controller delayed in comparison to the DM controller Control signals Roughly similar amplitude values

With the modification that the dynamic model is computed from the desired values, instead of the actual joint coordinates.The

IEEE/RSJ International Conference on Intelligent Robots and Systems October 11-15, 2009 St. Louis, USA 978-1-4244-3804-4/09/$25.00 ©2009 IEEE


E Di Salvo

Z J Ajaltouni

Can We Investigate Direct Time Reversal Violation?

Keywords: numbers: Nos.: 11.30.Er, 11.80.Et, 13.25.-k, 13.30.Eg

We illustrate a theoretical condition under which a two-body decay of a resonance violates time reversal invariance. Moreover, we examine some cases for which this condition looks particularly simple. As a consequence, we deduce tests of time reversal violation for weak decays involving spinning particles.

Introduction

The lively interest shown in the last years by high-energy physicists in CP violations [1][2][3][4][5][6][7][8][9][10][11][12][13][14] is mainly due to recent hints [15][16][17][18][19][20][21][22][23][24][25][26] for New Physics (NP). Indeed, such violations -and especially those involving b → s transitions [27,[START_REF] Bona | UTfit coll[END_REF] -constitute a promising door to physics beyond the Standard Model [START_REF] Eilam | [END_REF][30][31][32][33][34] (SM), unsatisfactory under several aspects (for recent reviews see [35,[START_REF] Ellis | International Conference on Particles and Nuclei (PANIC)[END_REF]), although consistent with a wealth of data [START_REF] Bigi | Production, Properties and Interactions of Mesons[END_REF]. Time Reversal Violation (TRV) is commonly regarded as the counterpart of CP violation, in view of the CPT theorem, valid under very mild assumptions and not contradicted by any experiments, even supported by stringent tests [START_REF][END_REF]. However direct TRV has been observed only in the CPLEAR experiment [39], by comparing

K 0 → K 0 to K 0 → K 0 transition.
In fact, it is generally quite difficult to realize experimentally the inverse process of a given decay; this is why people give up showing directly such a kind of violations. Alternatively, TRV may be revealed by the presence, in a hadronic two-body weak decay amplitude, of a "weak" phase, besides the one produced by strong Final State Interactions (FSI) [START_REF] Marshak | Theory of Weak Interactions in Particle Physics[END_REF][START_REF] Bigi | CP Violation, Cambridge Monographs on Particle Physics, Nuclear Physics and cosmology[END_REF][START_REF] Valencia | [END_REF]43]. However also in this case experimental uncertainties of the "strong" phases create serious problems in singling out the "weak" one [START_REF] Marshak | Theory of Weak Interactions in Particle Physics[END_REF]44]. Incidentally, in the SM the "weak" phase is provided by the Cabibbo-Kobayashi-Maskawa (CKM) scheme.

The aim of the present note is to illustrate a theoretical condition corresponding to TRV and to suggest tests for detecting it. Such tests are feasible in the framework of experiments like those recently suggested or realized [1][2][3][4][5][6][7][8][9][10][11][12][13][14] for CP violations and are suitable for two-body decays of resonances discovered in relatively recent timeslike B, B s and Λ b -, characterized by higher masses and nonzero spins of the decay products, and therefore by a greater number of amplitudes. Section 2 is devoted to deducing and illustrating the theoretical condition for TRV. In section 3 we suggest tests to be applied to some particular decays, while in section 4 we draw a short conclusion.

A condition for TRV

We focus on hadronic two-body decays of the type

R 0 → R 1 R 2 , (1)
where R 0 is the original resonance and R 1 and R 2 the decay products, with spin J, s 1 and s 2 respectively.

Our condition for TRV is derived by extending the standard treatment of Time Reversal Invariance (TRI) for two-body decays [START_REF] Marshak | Theory of Weak Interactions in Particle Physics[END_REF][START_REF] Bigi | CP Violation, Cambridge Monographs on Particle Physics, Nuclear Physics and cosmology[END_REF] to the case where more than one non-leptonic decay mode is involved [45]. If (1) is a weak decay, the relative, rotationally invariant amplitude reads, at first order in the weak coupling constant,

A J λ 1 λ 2 = f out |H w |JM , (2)
where H w is the weak hamiltonian, |f out a shorthand notation for the final two-body angular momentum eigenstate |JMλ 1 λ 2 , M the component of the spin of R 0 along the z-axis of a given frame and λ 1 and λ 2 the helicities of, respectively, R 1 and R 2 in the rest frame of R 0 . Assume H w to be TRI, i. e.,

T H w T † = H w , (3)
where T is the Time Reversal (TR) operator. Then, taking into account the antilinear character of T and the rotational invariance of the amplitude, we get[46]

A J λ 1 λ 2 = f in |H w |JM * . (4)
Inserting a complete set of "out" states yields

A J λ 1 λ 2 = n f in |n out * n out |H w |JM * . (5)
The only terms which survive in this sum correspond to the decay modes of R 0 ; furthermore the non-leptonic decay modes give the main contribution, since they involve a much greater coupling constant than the semi-leptonic decay modes. Relaxing the limitation of the state |f in to a two-body one, and expressing the "out" states in terms of the S-matrix -which is unitary and, owing to TRI ‡ , also symmetric with respect to angular momentum eigenstates [START_REF] Martin | Elementary Particle Theory[END_REF] -, eq. ( 5) can be rewritten as

A m = n S mn A * n . (6)
Here, omitting spin and helicity indices,

A m = m out |H w |R 0 (7)
and

S mn = m in |S|n in . ( 8
)
It is worth noting that eq. ( 6) coincides with eq. ( 12) of ref. [45]. The S-matrix is block-diagonal [43], since not all hadronic states are strongly connected to one another.

In particular, such blocks are characterized by flavor [43].

The most general solution to eq. ( 6) can be obtained by diagonalizing the Smatrix. To this end we recall a result by Suzuki [45], that is,

S mn = k O mk e 2iδ k O T kn , (9)
where O is an orthogonal matrix and the δ k are strong phase-shifts. Then the solution to eq. ( 6) is given by

A m = n O mn a n e iδn , ( 10
)
where a n are real amplitudes representing the effects of weak interactions on the decay process. Obviously complex values of one or more such amplitudes -that is, "weak" phases -would imply TRV, analogously to the case when only elastic scattering is allowed between the decay products [START_REF] Marshak | Theory of Weak Interactions in Particle Physics[END_REF].

Before considering applications of our model independent result, an important remark is in order. If a local field theory is assumed -like the SM -, a nontrivial phase of at least one a n implies also CP violation, owing to CPT symmetry. In particular, in the SM this phase is related to the phase of the CKM matrix. ‡ We neglect weak contributions to scattering.

Possible tests for TRV

Unfortunately the result found in the previous section is generally not useful for detecting TRV. Indeed, first of all, the elements of the matrix O are not known: at best one can elaborate models, as in ref. 45. Secondly, the amplitudes A n may be determined, at best, up to a phase per decay mode. In fact, a decay of the type (1), with spinning and unstable decay products, allows to determine, through angular distribution, polarizations and polarization correlations [10,32,[START_REF] Valencia | [END_REF][START_REF] Dighe | [END_REF]48,49], all products of the type

A λ 1 λ 2 A * λ ′ 1 λ ′ 2 .
In particular we recall a previous paper [49], where we showed a method for extracting such products from the observables of the decay

Λ b → ΛJ/ψ. ( 11
)
These products, in turn, amount to inferring all moduli of the amplitudes and their phases relative to a given amplitude, taken as a reference. This is a too poor piece of information, since it is not sufficient for solving the linear system (10) with respect to the products a n e iδn . Therefore generally we cannot elaborate tests for TRV through the method described.

However, relation ( 10) may be considerably simplified, provided we choose suitable decay modes. Let us consider, for example, the decay (11) or the following decay modes of B + and B 0 , already studied, both theoretically [10,11,12,13,14,15,19,30,31,32,[START_REF] Valencia | [END_REF][START_REF] Dighe | [END_REF]48,50,51,52] and experimentally [1][2][3][4][5][6][7][8][9]:

B + → (J/ψK * + ); (12)
B 0 → (J/ψK * 0 ); (13)
B s → (J/ψφ), (J/ψK * 0 ). ( 14)

We point out that the decay modes chosen do not involve isotopic spin, in order to avoid interference among different isospin amplitudes.

We examine the scattering corresponding to FSI between the decay products in decays of the type just illustrated. Here the two decay products constitute the initial hadrons. The momentum of such hadrons in the center-of-mass system is 1.6 to 1.7

GeV /c, therefore the scattering involves at least 15 partial waves. Since the decays considered imply orbital angular momenta not greater than 2, we are faced essentially with central collisions. These give rise mainly to inelastic scattering, whose shadow constitutes the largest part of elastic scattering. Instead, anelastic reactions -that is, with excitation of one or both decay products to higher mass states -are suppressed, as well as spin-flip elastic scattering, since these processes occur only in peripheral collisions, so as to keep coherence with the initial state.

But deep inelastic collisions imply a complete loss of coherence with respect to the initial particles. Indeed, any such process includes, initially, an exchange of gluons between the two hadrons, then hard collisions among partons, followed by parton fragmentations and/or recombinations to final hadrons, and, lastly, again gluon exchange among final hadrons. Therefore the sum (10) consists of quite a lot of terms.

Moreover it appears logical to assume for the phases in (10) a very rapid dependence on the index n (n = m), that is, a sudden variation of the phase-shift from state to state. On the contrary, there is no reason for the terms a n (weak amplitudes) and O mn (kinematic quantities) to depend so strongly on n. Then, for n = m, the sum can be approximated by an integral, whose integrand consists in the product of a slowly varying function times a rapidly varying phase. Therefore only the term with n = m survives in that sum, i. e.,

A m = O mm a m e iδm . ( 15
)
This result was obtained also by Wolfenstein [44] with a different line of reasoning (see also refs. [53,54]).

Eq. ( 15) can be tested by comparing the decays considered with the CP -conjugate ones and assuming the CP T symmetry. Indeed, under this assumption, a m differs from a m just by a phase, while O mm and δ m are CP -invariant, since they depend only on strong interactions. Then we have

|A m | = |A m |, (16)
the barred amplitude referring to the CP -conjugate process, obviously with opposite helicities, denoted synthetically by m. We stress that this test is not mandatory for the method we are going to suggest for detecting TRV.

In order to state tests for TRV, we define, preliminarily, a particular observable that we can extract from analyses of decays, that is, for a given decay mode,

Φ m = arg(A m ) -arg(A m 0 ). (17)
Here, as explained before, A m 0 is conventionally taken to be real. Then we define the following asymmetries:

A CP = Φ m -Φ m Φ m + Φ m , A C = Φ m -Φ m Φ m + Φ m , A P = Φ m -Φ m Φ m + Φ m . (18)
Here the barred quantities refer to the phases of the C-conjugate amplitudes. Given the wealth of b -b pairs to be produced at LHC per year (10 12 ), such tests appear to be not so unrealistic.

Conclusion

We have illustrated a theoretical condition under which TR is violated. For certain types of decays this condition looks particularly simple and suitable for stating experimental tests for TRV, to be applied to non-leptonic two-body decays involving spinning particles. The tests may be realized by means of standard analyses of decay products. The condition found is rather general, independent of any specific model; therefore, in principle, comparing the results of the suggested tests with the SM predictions could help detecting NP effects. Lastly we observe that the condition we require for obtaining evidence for TRV is opposite to the one demanded for detecting CP violations, which needs interference among amplitudes, and therefore, necessarily, more terms in the sum (10) [43].

Acknowledgments

The authors are grateful to their friend J. Orloff for stimulating discussions. One of us (EDS) is also indebted to his friend A. Di Giacomo for useful suggestions.