Anne Le Cunuder

Artyom Petrosyan

Georges Palasantzas

Vitaly Svetovoy

Sergio Ciliberto

Measurement of the Casimir force in a gas and in a liquid

We present detailed measurements of the Casimir-Lifshitz force between two gold surfaces, performed for the rst time in both gas (nitrogen) and liquid (ethanol) environments with the same apparatus and on the same spot of the sample. Furthermore, we study the role of double-layer forces in the liquid, and we show that these electrostatic eects are important. The latter contributions are subtracted to recover the genuine Casimir force, and the experimental results are compared with calculations using Lifshitz theory. Our measurements demonstrate that carefully accounting for the actual optical properties of the surfaces is necessary for an accurate comparison with the Lifshitz theory predictions at distances smaller than 200nm.

Introduction. As devices enter the submicron range, Casimir forces [112] between neutral bodies at close proximity become increasingly important. As Casimir rst understood in 1948 [START_REF] Casimir | On the Attraction Between Two Perfectly Conducting Plates[END_REF], these forces are due to the connement of quantum uctuations of the electromagnetic (EM) eld. Indeed Casimir proved that when two parallel, perfectly reecting plates, are introduced in vacuum, they impose, on the EM eld, boundary conditions which select only the uctuations compatible with them. As a result, an attractive force between the plates is produced, which depends only on fundamental constants, on the distance d between the surfaces and on their area A:

F c (d) = - π 2 A c 240 d 4 (1)
with the Planck constant and c the speed of light. Following Casimir's calculation [START_REF] Casimir | On the Attraction Between Two Perfectly Conducting Plates[END_REF], Lifshitz and co-workers in the 50's [START_REF] Lifshitz | The theory of molecular attractive forces between solids[END_REF] considered the more general case of real dielectric plates by exploiting the uctuation-dissipation theorem, which relates the dissipative properties of the plates and EM uctuations at equilibrium. Furthermore, for real surfaces, roughness and material optical properties can strongly alter the Casimir force [START_REF] Svetovoy | Inuence of surface roughness on dispersion forces[END_REF][START_REF] Broer | Roughness correction to the casimir force at short separations: Contact distance and extreme value statistics[END_REF].

Lifshitz formalism describes the Casimir force in a general case, where the medium between the plates need not be vaccum. According to this formalism, the force can be tuned from attractive to repulsive with a suitable choice of the interacting materials. These predictions boosted Casimir experiments to test the possibility of repulsive forces [START_REF] Munday | Measured long-range repulsive casimirlifshitz forces[END_REF]. In liquids, the determination of the Casimir force is more complex than in a gas because of the presence of additionnal eects, as e.g. the Debye screening. The Casimir-Lifshitz force has been measured between two gold surfaces immersed in ethanol [START_REF] Munday | Precision measurement of the casimir-lifshitz force in a uid[END_REF]; in this experiment, electrostatic forces are found to be negligible as sodium iodide (NaI) was added to ethanol, decreasing the Debye screening length. However, the role of electrostatic forces and their screening by the Debye layer are impor-É DES S URFACES e ux s urfa ce s s 'e xe rce n t à une ha ute ur m o ye nne co m pris e e n t de s a s pé rité s . Il e s t do nc indis pe ns a ble de dé te rm ine r ce tte co nna ître la dis ta nce de s é pa ra tio n e ff e ctive pe nda n t la m s s urfa ce s rug ue us e s né ce s s ite une a pproc he s ta tis tique . L a o n ta ct d 0 e s t dé te rm iné e à pa rtir de la dis tributio n e n ha Thi s con di t i on can be con si der ed as an equ at i on for t he h ei gh t of t h e h ei ght asper i t y d ue t o a shar p exp on en t i al beh avi or of t he di st r i but i on tilever, the reference one is reected by the static base and the second one by the cantilever free end. When the cantilever is bended the optical path dierence δ between the two beams is measured through an interferometer [START_REF] Bellon | Exploring nano-mechanics through thermal uctuations. Accreditation to supervise research[END_REF].

tant and one has to consider carefully their contributions during force measurements in liquids [START_REF] Geyer | Comment on precision measurement of the casimir-lifshitz force in a uid[END_REF][START_REF] Munday | Reply to comment on `precision measurement of the casimir-lifshitz force in a uid[END_REF]. In order to clarify the interplay of the Casimir force and additionnal eects in liquids, we have performed measurements of the Casimir force in a nitrogen atmosphere in the rst place, and then, using the same system and sample, in ethanol. The contact area is the same in both measurements. We observe that electrostatic forces, screened over the Debye length, are of the same magnitude as the Casimir force, in the 50-200nm distance range. After subtracting the electrostatic force, we obtain a Casimir force in quantitative agreement with Lifshitz theory [START_REF] Lifshitz | The theory of molecular attractive forces between solids[END_REF]. Furthermore, the accuracy of our measurement allows us to highlight the importance of accurately characterizing the optical properties of the samples before any meaningful comparison with theory. Experimental setup. We use an atomic force microscope (AFM) to measure the Casimir force between metallic surfaces. In order to measure the force with a good accuracy, the cantilever displacement is measured with a home-made quadrature phase interferometer [START_REF] Bellon | Exploring nano-mechanics through thermal uctuations. Accreditation to supervise research[END_REF], whose operating principle is sketched on Fig. 1.

The experiment is performed in a sphere-plane geometry to avoid the need to maintain two at plates perfectly parallel. Thus, a polystyrene sphere of radius R = (75.00 ± 0.25)µm (Sigma-Aldrich) is mounted on the tip of the cantilever with a conductive glue and then the whole probe is coated by a gold lm whose thickness is about 100nm. The plates have been gold coated using cathodic sputtering by ACM, at the LMA-CNRS. The diameter of the sphere has been determined from Scanning Electron Microscopy. We use a cantilever (size 500µm × 30µm × 2.7µm, NanoAndMore) of stiness κ = 0.57 ± 0.03N/m. The precise value of κ is determined using equipartition, ie. < δ 2 >= k B T κ , where k B is the Boltzmann constant and T the temperature. The resonance frequency of the sphere-cantilever ensemble is

f o = 2271Hz in vacuum.
The sphere faces a glass at plate which is coated by a gold lm of a thickness of about 100nm [20]. According to [START_REF] Lisanti | Observation of the skin-depth eect on the casimir force between metallic surfaces[END_REF], the layer is suciently thick to be considered as a bulk-like lm. The plate is mounted on a piezo actuator (PZ38, Piezojena) which allows us to control the plane-sphere distance. During the experiment, the plate is moved continuously towards the sphere and the induced deexion of the cantilever is detected by the interferometer.

In air, because of water vapor, the capillary force far exceeds the Casimir force. Therefore, our measurement in a gas was performed after lling our cell with nitrogen.

Calibrations. The total force between the surfaces is the sum of the Casimir force F cas (d) and additionnal con- tributions:

F total = F cas (d) + F el (d) + F H (d, v) . (2)
Electrostatic forces F el (d) are due to a potential dier- ence between surfaces, owing to dierences between the work functions of the materials used, and the possible presence of trapped charges [START_REF] Speake | Forces between conducting surfaces due to spatial variations of surface potential[END_REF]. Hydrodynamic forces F H (d, v) are due to the motion of the uid during the approach of the plate towards the sphere, and depend on their relative velocity v [START_REF] Brenner | The slow motion of a sphere through a viscous uid towards a plane surface[END_REF]. These hydrodynamic effects are negligible in a nitrogen atmosphere, where the viscosity is γ = 1.76 10 -6 Pa s, but have to be considered in ethanol where the viscosity is 1000 times higher (γ = 1.2 10 -3 Pa s).

There are two main requirements for a precise determination of the Casimir force. Firstly, additionnal forces must be measured with accuracy and subtracted from the total measured force. Secondly, because the force has a strong dependance on the distance between surfaces, an independant measure of the distance is necessary, which becomes dicult when the separation approaches nanometer scales. The diculty originates principally from surface roughness: when the two surfaces come into contact, the highest asperities of each surface touch each other and the surfaces are still separated by a distance upon contact d 0 [START_REF] Van Zwol | Distance upon contact: Determination from roughness prole[END_REF].

The piezo actuator includes a position sensor which gives us the displacement of the plate: d piezo . We de- ne the origin of d piezo as the position of contact of the highest peak of the sphere roughness with the surface of the plate, as the sphere is much rougher than the plate. The eective separation distance which appears in the expression of the force can be written as (see Fig. 1):

d = d piezo + d 0 -δ (3)
where d 0 is the distance upon contact due to surface roughness and δ is an additional correction which results from the static deexion of the cantilever in response to the total force F total .

We determined the separation upon contact d 0 from hydrodynamic calibration, performed in ethanol. Immediatly after measuring the Casimir force in nitrogen, we injected carefully ethanol into the cell, and we performed calibrations and measurements of the Casimir force in ethanol. As the horizontal drift of our system is negligible, the contact area and the separation distance upon contact d 0 are the same in each measurement. This assumption is further justied a posteriori: our experimental curves all superimpose on top of each other and on top of the theoretical curve after shifting the distance by the same value of d 0 . The hydrodynamic calibration is presented in next section, while topographic analysis is presented in [20]. The value of d 0 obtained from hydrodynamic calibration is comparable with the value obtained from roughness analysis.

Hydrodynamic calibration in ethanol. The theoretical expression of the hydrodynamic force, for non-slip boundary conditions, is given by [START_REF] Brenner | The slow motion of a sphere through a viscous uid towards a plane surface[END_REF]:

F H = - 6πηR 2 d v ( 4
)
where η is the uid viscosity, R is the radius of the sphere, and v = ∂d ∂t is the relative velocity between the plate and the sphere. Indeed, in our case, the slippage can be neglected as the mean free path is in the order of intermolecular distances (e.g a few Angstroms), and negligible in comparison with the roughness of the surface (e.g a few tens of nanometers) [START_REF] Renã© | Hydrodynamic force measurements under precisely controlled conditions: Correlation of slip parameters with the mean free path[END_REF] As is clear from Eq. ( 2), among the dierent forces occuring between the surfaces, the hydrodynamic force is the only one which depends on velocity. Thus we performed two force measurements, moving continuously the plate towards the sphere: a rst one at velocity v 1 = 348nm s -1 , and a second one at a velocity v 2 = 5109nm s -1 . By taking the dierence between these two measurements, we cancelled all the velocity-independent forces and from Eq. ( 4) we obtained

F H measured at v = v 2 -v 1 = 4742nm s -1 : F total (d, v 2 ) -F total (d, v 1 ) = F H (d, v) . (5)
Here, v 2 and v 1 are the relative velocities between the sphere and the sample, which are not exactly the piezo velocities v 2 and v 1 because the cantilever is deected when the plate is moved towards the sphere. v 2 and v 1 were determined precisely measuring the deection of the cantilever. Measurements of the hydrodynamic force are presented in [20]. Comparing the measured hydrodynamic force with the theoretical expression of Eq. ( 4), we determined the separation distance upon contact d 0 = (31 ± 2)nm.

Electrostatic forces. Even if the surfaces are as clean as possible, there always remain electrostatic forces between them. First, an electrostatic potential dierence V c still exists between clean, grounded, metallic surfaces owing to dierences between the work functions of the materials used [START_REF] Cc Speake | Forces between conducting surfaces due to spatial variations of surface potential[END_REF]. Second, electrostatic forces can remain due to the presence of trapped charges. In liquids, these trapped charges induce double-layer forces, due to the rearrangement of ions in solution, screening the electrostatic interactions.

When d << R, the expression of the electrostatic force is [START_REF] Laurent | Casimir force measurements at low temperature[END_REF]:

F e = - π 0 d R λ D V 2 c exp(-d/λ D ) (6)
The term

V 2 c
d is the contribution of the contact potential V c between the surfaces and the term exp(-d/λ D ) represents the double-layer force, screened over a distance λ D (the Debye length) [START_REF] Hans | Surface and interfacial forces[END_REF].

As there is no free charge in nitrogen, the Debye length is innite and the electrostatic interaction is not screened, consequently there are no double-layer forces. In nitrogen, the contact potential was calibrated to V c = (87 ± 2)mV, and was compensated by an applied voltage dierence during the measurement of the Casimir force.

In contrast, in ethanol, the contact potential is strongly screened by the ions constituting the Debye layer. Moreover, applying an electrostatic potential in a polar liquid can yield a transcient current [START_REF] Fort | Measurement of contact potential dierence between metals in liquid environments[END_REF] and consequently, charges accumulate on surfaces, preventing us from applying the method suggested in [START_REF] Ether | Double-layer force suppression between charged microspheres[END_REF] to cancel the Debye force. Therefore, we simply subtracted the contribution of electrostatic forces from force measurements, after determining λ D = (46 ± 6)nm and V c = (63 ± 13)mV [20]. In ethanol V c is lowered because the dissociation of molecules at the surface leads to the formation of a rst very thin screening layer of a few nm.

Measurement of the Casimir force in nitrogen. Static measurements of the Casimir force were carried out in a nitrogen atmosphere, between a gold-coated sphere and a gold-coated plate. We subtracted the vertical thermal drift by tting linearly each force curve measurement between 300nm and 1µm and subtracting it from each measured curve. All force curves were shifted in distance corresponding to the separation upon contact d 0 = 31nm.

The measured Casimir force is shown on Fig. 2(a) for separations ranging from 90nm to 370nm, averaging 28 independent measurements. For theoretical calculations, thermal corrections are negligible as the thermal energy where blue cross correspond to the mean measured force. The orange solid line corresponds to the Lifshitz theory where the dielectric function is evaluated from measured optical data of a real gold lm [START_REF] Svetovoy | Optical properties of gold lms and the casimir force[END_REF]; for the green dashed-dotted line, is evaluated using the handbook optical data [START_REF] Edward | Handbook of optical constants of solids[END_REF].

k B T is too small to populate the mode of lowest energy c/λ T , as the separation distance d satises

d < 370nm λ T = c k B T ≈ 7 µm , at 300 K . (7)
We compared our experimental result with theoretical predictions of the Casimir force, based on optical properties of Au taken from: 1) handbook of tabulated data (green dashed-dotted line) [START_REF] Edward | Handbook of Optical Constants of Solids[END_REF], and 2) measurements on Au samples presenting the same roughness and preparation conditions as ours (orange solid line) [START_REF] Svetovoy | Optical properties of gold lms and the casimir force[END_REF].

The deviation from Lifshitz theory based on dielectric properties of real samples is less than 5 pN at closest separations, while it reaches 10 pN at closest separations for calculations based on data from handbook. As the signal to noise ratio degrades with increasing distance, the deviation of the measurements from Lifshitz theory increases at larger distances. However, they remain compatible within the error bars, including systematic and statistical errors (dominant at large distances). This demonstrates that surfaces must be carefully characterized for high precision measurements of the Casimir force. In order to make this argument more quantitative, we present the dierence (F exp -F th )/F th on Fig. 2(b), showing the dierences between the theoretical and experimental Casimir forces, for calculations based on data from handbook and calculations based on dielectric properties measured on lms with the same morphology as our lms.

Measurement of the Casimir force in ethanol. In a liquid, the scenario is richer than in a gas because of the presence of additional eects, namely the hydrodynamic force and the Debye screening of the electrostatic interactions.

Measurements in ethanol were performed with the same apparatus, immediatly after the measurement in nitrogen, so that the contact are be the same, as explained previously. During the measurement of the Casimir force, the approach velocity was chosen in order to compromise between the hydrodynamic force F H we wanted to minimize and vertical drift, limiting the time of measurement. The results presented in this paper were obtained with 100nm s -1 .

In order to average the data collected from consecutive runs, 8 data sets were acquired. First, all force curves were shifted in distance corresponding to the separation distance upon contact d 0 = 31nm. Second, the hydrodynamic force F H was subtracted from each measurement independently. As the F H dependence on distance is accurately known [20], it can be safely subtracted from the measured force. Third, to remove vertical thermal drift from force measurement, each force curve was tted linearly between 300nm and 600nm. Fig. 3 (a) shows a single force measurement in ethanol after subtracting the hydrodynamic force. Blue points correspond to the raw measurement, and the red line corresponds to the linear t of the thermal drift, which is then subtracted from each measurement. In Fig. 3 (b), we represent both a single force measurement (blue) and the average measured force (red), after subtracting the thermal drift and the hydrodynamic force. The remaining force shows the presence of repulsive forces at separation distances larger than 60nm. These repulsive forces are attributed to the presence of ions in solution and on metallic surfaces. Then, the remaining curve corresponding to F cas + F Debye was averaged over 8 measurements and the mean curve was tted by an exponential function A exp(-d/λ D ) in the distance range d > 120nm.

A and λ D are adjustable parameters. In this distance range, we assume that the Casimir force is negligible in comparison to the double-layer force (as predictions of the Lifshitz theory indicate). We obtained a Debye length λ D = (46 ± 6)nm consistent with measurements reported by [START_REF] Pj Van Zwol | Weak dispersive forces between glass and gold macroscopic surfaces in alcohols[END_REF] and [START_REF] Munday | Measurements of the casimirlifshitz force in uids: The eect of electrostatic forces and debye screening[END_REF]. The exponential t is also used to determine the electrostatic potential at the gold surface ψ 0 from the expression of the double-layer force in a sphere-plane geometry: F = 4π 0 Rψ 2 0 /λ D e -d/λ D . We evaluated the surface potential ψ 0 adj = (63 ± 13)mV. After subtracting the measured double-layer force from each force measurement, the mesured Casimir force is obtained.

The measured Casimir force is presented on Fig. 3(b). The experimental data are compared to Lifshitz's theory for a gold sphere of radius R = 75µm and a gold plate separated from a distance d in ethanol. Finally the dierences between the theoretical predictions and the measured data are plotted in Fig. 3(c). In spite of the rather large error bars we can distinguish the two theoretical predictions: Casimir force measurements are in better agreement with Lifshitz theory based on optical properties of real Au lms, presenting the same morphology as ours.

Conclusion. In conclusion, we have presented measurements of the Casimir force performed both in gas (nitrogen) and liquid (ethanol) environments with the same apparatus and on the same spot of the sample. The force measurements yield experimental evidence of the importance of electrostatic eects in ethanol. These eects were properly measured and subtracted, in order to determine the genuine Casimir force. Furthermore, these measurements demonstrate that the Casimir force is sensitive to changes in the optical properties of gold at distances of less than 200nm mostly in the gas enviroment where the force is the strongest. Notably, to the best of our knowledge, this is the rst time that this inuence was measured experimentally at this range of separations. Our measurements are of signicant interest given the fundamental implications of the Casimir force in the search of new hypothetical forces, technology applications of Casimir forces for micro/nano device actuation [START_REF] Genet | Electromagnetic vacuum uctuations, casimir and van der waals forces[END_REF][START_REF] Svetovoy | Do the precise measurements of the casimir force agree with the expectations?[END_REF], and the very timely nature of our measurements [START_REF] Garrett | Measurement of the casimir force between two spheres[END_REF] .

APPENDIX

This work has been supported by the ERC contract Outeucoop. We thank Irénée Frérot for helpful discussions.

ROUGHNESS ANALYSIS

Because the Casimir force is sensitive to optical properties of gold lms [START_REF] Svetovoy | Optical properties of gold lms and the casimir force[END_REF], we characterized carefully the topography of the surfaces. Indeed, it is commonly accepted that these properties can be taken from the handbooks tabulated data. In fact, optical properties of deposited lms depend on the method of preparation. An interesting study [START_REF] Svetovoy | Optical properties of gold lms and the casimir force[END_REF] reported a signicant variation of 5 -15% in Casimir force calculations, due to changes in optical properties of Au lms.

After coating, the surface morphology of both the sphere and the plate was determined using a commercial AFM (Bruker). It is important to stress that these analysis were performed directly on the surfaces used in the measurement of the Casimir forces. The AFM im- The rms roughness of the plate is

w p = (1.3 ± 0.2)nm.
This morphology analysis is then used in order to compare our force measurements with computations where optical properties are taken from real lms, with a similar topography.

From morphology analysis, we also evaluated approximatively the separation upon contact d 0,rough . However, a surface of 1 × 1 µm is not large enough to determine precisely d 0 . This analysis just helps us to check that we nd a separation distance upon contact d 0,rough of the same order of magnitude as d 0 obtained from hydrodynamic calibration. The AFM images 4 indicate that the sphere is much rougher than the plate. Consequently, the separation distance upon contact can be evaluated as: d 0,rough = d 0,sph + w p , where w p is the rms roughness of the plate and d 0,sph is the highest peak of the sphere within the contact area [START_REF] Sedighi | Casimir force measurements from silicon carbide surfaces[END_REF]. One estimates that on a surface of 1µm 2 , there is in average only one asperity larger than 2.8w sph and less than one with height larger than 3w sph , which is statistically coherent with the image of g.4a) and the tails of the distribution in g.5. Thus, in a contact area of about 1 µm 2 one expects to nd d 0,sph = 2.8w sph = (33.6 ± 2.4)nm. Notice that this value is statistically signicant because the area involved in the force measurements are of the order of 1 µm 2 ) at d < 100nm (see ref. [START_REF] Van Zwol | Distance upon contact: Determination from roughness prole[END_REF]). For the plate, the rms roughness is w p = (1.3 ± 0.2)nm. Thus from the topography analysis we evaluate that the maximum is d 0 < (34.9 ± 2.4)nm on an area of 1 µm 2 ). This value is, within error bars, statistically coherent with the hydrodynamic calibration discussed in section 2. It is important to stress that the hydrodynamic calibration is performed directly on the surfaces used in the measurement of the Casimir forces. Indeed, calibration were performed in ethanol immediatly after measurements of the Casimir force. As the liquid was introduced very carrefully in the cell after the measurement in nitrogen and as the horizontal drift of the sample is negligible, the contact are is the same during both measurements and calibration. This assumption is further justied a posteriori: our experimental curves all superimpose on top of each others and on top of the theoretical curves after shifting the distance by the same value of d 0 . In contrast the topography study is done on surfaces with the same statistical properties but not on the same position of the contact area used in the experiment. Thus we use for d o the value obtained from the hydrodynamic calibration, i.e. d = (31 ± 2)nm.

HYDRODYNAMIC CALIBRATION

The measured hydrodynamic force is plotted as a function of d in g.6 a) where it is compared to the theoretical force expressed in eq.( 4

CALIBRATION OF THE CONTACT POTENTIAL DIFFERENCE

An electrostatic potential dierence V c exists between the sphere and the plate, even if the surfaces are coated with gold and they are both electrically grounded. Indeed, there can exist a large potential dierence between clean, grounded, metallic surfaces owing to dierences between the work functions of the materials used and the cables used to ground the metal surfaces [START_REF] Cc Speake | Forces between conducting surfaces due to spatial variations of surface potential[END_REF]. A small potential dierence around ten mV is sucient to overhelm the Casimir force so the contact potential dierence has to be measured and the experiment has to be carried out with a compensating voltage present at all times.

Following a procedure described in [START_REF] De Man | Halving the casimir force with conductive oxides: Experimental details[END_REF], we measure the contact potential dierence V c between the sphere and the plate by applying an oscillating potential V = V 1 cos ω 1 t + V 2 to the plate, keeping the sphere grounded. When d << R, the expression of the electrostatic force induced by the voltage potential dierence V and V c can be approximated by:

F e = - π 0 d R d (V 1 cos(ω 1 t) + V 2 -V c ) 2 = - π 0 d R 2d [V 1 cos(2ω 1 t) + + 4V 1 (V 2 -V c ) cos(ω 1 t) + 2(V 2 -V c ) 2 + V 1 2 ] (8)
Because of the existence of a contact potential dierence V c , the system oscillates both at 2ω 1 and at ω 1 . We determine V c adding a constant potential V 2 until the excitation at the frequency ω 1 disappears. Indeed, when V 2 = V c , the system no longer oscillates at the frequency ω 1 (see eq.8) . We measure the contact potential V c as a function of d between 1µm and 110nm. In practice, we move the plate towards the sphere by discrete displacements. At each separation distance d n , we measure the potential V 2 which minimizes the amplitude of the oscillation at ω 1 . The result of the measurement is plotted in g. [START_REF] Chang | Reduction of the casimir force from indium tin oxide lm by uv treatment[END_REF], where we see that in our experiment V c is constant as it is theoretically expected.

FIG. 1 .

 1 FIG. 1. Experimental setup. A gold-coated polystyrene bead is glued on the cantilever tip which measures the sphereplane interaction force at distance d. The deection of the cantilever is detected by an interferometric technique: two lasers beams, orthogonally polarized, are focused on the can-

FIG. 2 .

 2 FIG.2. Measurement of the Casimir force between two Au-surfaces in a nitrogen atmosphere. (a) Blue points correspond to the mean measured force. Blue circles correspond to a single measurement of the Casimir force. Orange solid curve corresponds to the Lifshitz theory in which the dielectric function is evaluated from measured optical data of a real gold lm[START_REF] Svetovoy | Optical properties of gold lms and the casimir force[END_REF]. For the green dashed-dotted line, is evaluated using the handbook optical data[START_REF] Edward | Handbook of optical constants of solids[END_REF]. Red dashed line corresponds to the theory in the case of ideal conductors. (b) Relative dierence between the theoretical and experimental Casimir forces.Same colors as in (a). The error bars include the statistical and the systematic errors due to uncertainties on the separation upon contact d0 = (31 ± 2)nm (using the hydrodynamic estimation), on the stiness κ = (0.57 ± 0.03)N/m and on the diameter of the sphere D = (150.0 ± 0.5)µm.

FIG. 3 .

 3 FIG.3. Measurement of the Casimir force in ethanol. (a) Single force measurement (blue points) after subtracting the hydrodynamic force. The distance has been shifted by d0 = 31nm. The red line corresponds to the linear t of the thermal drift in the range 300nm -600nm. (b) Single force measurement (red circles) and mean force measurement (blue points) (averaged over 8 measurements) of Fcas + F Debye obtained after subtraction of the thermal drift and the hydrodynamic force. The mean measured force is tted by an exponential in the range d > 120nm. In this distance range, double-layer forces remove largely the Casimir force. The blue line corresponds to the exponential t, yielding the Debye length λD = (46 ± 6)nm and the surface potential ψ0 adj = (63 ± 13)mV through Eq. (6). (c) Measurement of the Casimir force between two Au-surfaces in a ethanol,

FIG. 4

 4 FIG. 4. a -AFM image of 1 × 1 µm 2 sample of the sphere surface after the deposition of a 100nm thick gold lm. b AFM image 1 × 1 µm of the surface of the glass plate with a 100nm thick gold lm

FIG. 5

 5 FIG. 5. a) Height distribution of the sphere roughness b) Height-height correlation function of the sphere surface in loglog scale. The height-height correlation function is dened as: H(r) =< [h(r) -h(0)] 2 >, where h(r) is the surface height. The roughness exponent α = 0.940 ± 0.001 is extracted from the slope of the linear t and the correlation lenght ξ = 50nm is determined from the intersection between the linear t and the saturation line.

  FIG. 6. a) Hydrodynamic force in ethanol measured at v = 4.742µm s -1 .b) Inverse of the hydrodynamic force as a function of d.

FIG. 7 .

 7 FIG.7. Value of the potential V2 which minimizes the excitation at frequency ω1. We measure a constant contact potential dierence between 110nm et 1.1µm.


Alain Guez

Lydia Gély

Alexandre Boudjeloud

Blansché

Brieuc Conan-Guez
email: brieuc.conan-guez@univ-lorraine.fr

Alain Gély
email: alain.gely@univ-lorraine.fr

Lydia Boudjeloud-Assala
email: lydia.boudjeloud-assala@univ-lorraine.fr

Alexandre Blansché
email: alexandre.blansche@univ-lorraine.fr

K-Spectral Centroid : Extension and Optimizations

come

Extension of K-Spectral Centroid

Introduction

Unsupervised classification of time series has been receiving a great deal of attention for many years. In this work, we focus on a specific model, K-Spectral Centroid (KSC) [START_REF] Yang | Patterns of temporal variation in online media[END_REF], a K-means-like algorithm devised for time series clustering. KSC distinguishes itself from other classic partitioning models by relying on a specific shape-based dissimilarity measure: this measure is invariant to Y-scaling and global time shifting (contrary to the classic DTW measure which applies a non uniform transformation of the time axis). Invariance to scaling (magnitude in Y values) is classically addressed through a Y-normalization, whereas invariance to shift-lag is obtained by testing all possible time translations of one time series with respect to the other time series. This shape based measure is well adapted to time series for which it is difficult to define a time origin.

Even if KSC has been succesfully applied to several real world problems (e.g. the temporal evolution of hashtags on Twitter [START_REF] Yang | Patterns of temporal variation in online media[END_REF]), this model suffers from two downsides: firstly its dissimilarity measure can only handle non negative time series. Secondly, KSC has a running time complexity which is cubic in the time series length. Processing massive data is therefore very time consuming.

In this work, we address these two downsides. We show that KSC can naturally be extended to cope with time series of arbitrary signs by generalizing its dissimilarity measure. We denote EKSC this extension of KSC. Moreover, we prove that the new dissimilarity measure is a metric distance (the triangle inequality). In the second part of this work, we reduce the complexity and the running time of EKSC thanks to four exact optimizations. In particular, we apply Elkan's algorithm [START_REF] Elkan | Using the triangle inequality to accelerate k-means[END_REF] which relies on the triangle inequality to cut off a lot of distance evaluations.

In the experimental part, we focus on evaluating the solution quality of EKSC. K-Shape [START_REF] Paparrizos | Fast and accurate time-series clustering[END_REF], which is a similar model, is used as comparison. Finally, EKSC accelerations obtained thanks to the four optimizations are reported.

K-Spectral Centroid for arbitrary time series

Let x and y be two discrete one-dimensional time series. Time series lengths, L(x) and L(y), can be different. Adding two time series is carried out thanks to a suitable null padding. We recall the expression of d + , the shape based dissimilarity measure used in KSC to compare non negative time series:

d + (x, y) = min o∈Z,α∈R x -α τ o (y)
x

. is the Euclidean norm, and τ o is the time shift operator of parameter o (a signed integer). Minimization with respect to α (a real number) addresses the problem of magnitude scale, whereas minimization with respect to o provides a time series alignment which is time-shift invariant.

In [START_REF] Yang | Patterns of temporal variation in online media[END_REF], Yang and Leskovec show that this dissimilarity measure is symmetric thanks to a simple reformulation of d + . If we denote g(c) = √ 1 -c 2 and x • y the scalar product (with null padding), we have d + (x, y) = min o∈Z g x•τo (y) x y

.

The measure d + is only defined for non-negative time series (the notation uses subscript + for this purpose). Indeed, d + collates a time series x and its opposite (α = -1 as α is not constraint to be non negative), which makes no sense in many real case applications. In this work, we propose to define a new measure d, an extension of d + to time series of arbitrary signs. We denote d(x, y) = g max o∈Z (x•τo(y))

x y . d is symmetric, and we have d + (x, y) ≤ d(x, y). We have an equality for non negative time series.

We show now that d verifies the triangle inequality : d is a metric distance. Let θ x,y be the angle between x and y: θ x,y = arccos x•y

x y (we denote acos in the proof). We denote sin + (x) the function such that sin + (x) = sin(x) on [0, π/2] and sin

+ (x) = 1 on [π/2, π]. We can remark that sin + (acos(c)) = √ 1 -c 2 = g(c) for c ∈ [0, 1]
. Finally, as time series have finite support, max o∈Z x • τ o (y) is always non negative. We use now the fact that the angular distance is metric:

θ τo(x),τ o (y) ≤ θ τo(x),z + θ z,τ o (y) min o,o θ τo(x),τ o (y) ≤ min o θ τo(x),z + min o θ z,τ o (y) min o,o acos τ o (x) • τ o (y) x y ≤ min o acos τ o (x) • z x z + min o acos z • τ o (y) z y acos max o,o τ o (x) • τ o (y) x y ≤ acos max o τ o (x) • z x z + acos max o z • τ o (y) z y
This last inequality is obtained because arccos is non increasing. The next inequality uses the fact that sin + is non decreasing and subadditive (see [START_REF] Dongen | Metric distances derived from cosine similarity and pearson and spearman correlations[END_REF]).

sin

+ acos max o,o τ o (x)•τ o (y) x y ≤ sin + acos max o τ o (x) • z x z + sin + acos max o z • τ o (y) z y g max o x • τ o (y) x y ≤ g max o x•τ o (z) x z + g max o z•τ o (y) z y d(x, y) ≤ d(x, z) + d(z, y)
We consider now N time series {x i } 1≤i≤N . We denote L the average length of these time series. Let K be the number of clusters, denoted C k , and µ k be the barycenter of cluster C k . As explained in the introduction, EKSC, which uses d as dissimilarity measure, is a variant of the well-known K-means clustering model. It seeks to minimize the within-cluster sum of squares:

W CSS = K k=1 xi∈C k d 2 (µ k , x i )
EKSC proceeds by alternating between two steps: the assignment step which produces a new partition thanks to the metric distance d, and the update step which computes the new centroids (cluster barycenters). Contrary to the Euclidean case, computing a cluster barycenter is more technical when d (or d + ) is used as dissimilarity measure. Barycenter µ k is obtained by minimizing µ k = arg min µ xi∈C k d 2 (µ, x i ). Yang and Leskovec [START_REF] Yang | Patterns of temporal variation in online media[END_REF] assume that the optimal time shifts o i have already been found during the previous assignment step. This allows for an exact extraction of µ k thanks to the minimization of the following Rayleigh quotient:

µ k = arg min µ µ t T k µ µ 2 with T k = xi∈C k Id - τi τ t i τi 2
, where τ i = τ oi (x i ). µ k is therefore an eigenvector associated to the smallest eigenvalue of matrix T k . In [START_REF] Yang | Patterns of temporal variation in online media[END_REF], a complete diagonalization of T k is carried out to obtain the desired eigenvector. This approach is very costly (cubic complexity in L).

Two remarks can be made concerning the barycenter extraction procedure. We recall that if µ is an eigenvector, -µ is also an eigenvector. In the case of non negative time series, we can prove that the barycenter can be chosen as non negative. Moreover, in the case of arbitrary time series, the extraction procedure provides µ and -µ as solutions. But only one makes sense with respect to dissimilarity d. In order to choose between µ and its reverse counterpart, we memorize in each cluster the closest time series x i to the barycenter. x i is some kind of medoid. If d(µ k , x i ) ≤ d(-µ k , x i ), we choose µ k as the barycenter, otherwise we choose -µ k .

Fast Implementation

Yang and Leskovec [START_REF] Yang | Patterns of temporal variation in online media[END_REF] point out the scalability issue of the KSC algorithm due mainly to the cubic complexity of the update step. In this section, we propose four exact optimizations of EKSC (these optimizations can be applied to the original KSC as well). Thanks to these optimizations, the running time complexity of EKSC is reduced, which leads to large acceleration factors.

Computation of measure d thanks to Fourier transforms: As explained in [START_REF] Paparrizos | Fast and accurate time-series clustering[END_REF], we can avoid testing all possible time shifts during the evaluation of measure d (this brute force approach has quadratic complexity in L). Indeed, we seek to maximize the cross-correlation function:

CC(o) = x • τ o (y).
We denote x 0 the vector x right padded with zeros. We denote 0 y, the vector y left padded with zeros. We set L(x 0 ) = L( 0 y) = L(x) + L(y) -1. It is well known that CC ≡ F -1 F(x 0 ) F * ( 0 y) , where F (resp. F -1 ) is the Fourier transform (resp. the inverse Fourier transform), and * is the conjugate operator. Each evaluation of measure d therefore involves three Fourier transforms. With a fast algorithm (for instance, in the specific case of padded vectors, for which lengths are power of 2), the running time complexity is reduced from O(L 2 ) to O(L ln(L)). One interesting element which doesn't appear in [START_REF] Paparrizos | Fast and accurate time-series clustering[END_REF] is that even if time series haven't got the same length, it is possible to precompute spectra once and for all before running the EKSC algorithm (initialization step). Thanks to this precomputation, during EKSC iterations, the evaluation of measure d involves only the computation of the inverse transform F -1 . This remark leads to quite an important speed up.

Elkan's algorithm[2]: This algorithm, initially devised for K-Means, avoids a lot of measure evaluations thanks to the triangle inequality. It uses two cut off strategies. Let µ(x) be the currently assigned barycenter to the time series x. The first strategy is based on the fact that if another barycenter µ is located far enough from µ(x), then x can't be reassigned to µ. And therefore, the evaluation of d(µ, x) is avoided. The second strategy leverages the temporal continuity of the K-Means process: if the new position of the barycenter is close enough to the previous one, there is no reason for a time series to leave its current cluster. Once again, distance evaluations are avoided.

Power method: Firstly we can remark that the problem of a barycenter extraction is equivalent to this new maximization problem:

µ k = arg max µ µ t S k µ µ 2 , where S k = xi∈C k τi τ t i τi 2 .
The eigenvector associated to the largest eigenvalue of this Rayleigh problem is the solution. We can prove that all eigenvalues are non negative, and therefore, we can use the power method which extracts the eigenvalue of largest magnitude. Therefore complete diagonalization of matrices S k can be avoided, which is very efficient (diagonalization has cubic complexity). The power method is an iterative algorithm, which starts with an estimate of the desired eigenvector, and which produces a better approximation of the solution after each iteration. Each iteration simply involves a matrix vector product: S k µ. As a barycenter tends to move slowly during the end of the clustering process, the power method only has to carry out a few iterations to converge towards the updated barycenter.

Incremental computation of matrices S k : At each iteration of EKSC, computing the K matrices S k from scratch is time consuming. When the number of cluster modifications or shift modifications is low, it is more efficient to store the matrices S k in the memory, and to maintain them thanks to an incremental strategy. If the shift o i or the assigned cluster of x i is modified during the assignment step, the previous contribution τi τ t i τi 2 is removed from the previous matrix and the new contribution is added to the new matrix. In the case of a shift modification, the previous and the new matrices are the same ones. A broad rule to decide which of both computation strategies (incremental or from scratch) has to be prefered at each EKSC iteration can be easily devised: for instance we can compute the number of addition operations implied by both methods.

Centering time series

For the original KSC, the question of data centering was not addressed in [START_REF] Yang | Patterns of temporal variation in online media[END_REF]. Indeed, time series were assumed to be non negative, and therefore a representative barycenter has to be non negative. In this work, as we extend KSC to arbitrary time series, we may want to apply centering to the data. And moreover, we may wish to obtain centered barycenters. We follow the same path as in [START_REF] Paparrizos | Fast and accurate time-series clustering[END_REF]. We consider the centering matrix Q = Id -1 L 1. Id is the identity matrix and 1 is the square matrix with only 1 as values. Q is symmetric and idempotent (Q 2 = Q). We maximize the Rayleigh quotient under the constraint Qµ = µ. We have

µ k = arg max µ µ t Q t S k Qµ µ 2
. This time, the solution is obtained as an eigenvector associated to the largest eigenvalue of matrix Q t S k Q. As Q t S k Q is symmetric and positive semidefinite, the power method can be used again (the largest eigenvalue is the largest eigenvalue in magnitude). Moreover, the incremental computation of matrices S k can also be conserved if we center µ and S k µ c at each iteration of the power method (µ c is µ after centering). In this case, no product with matrix Q has to be computed.

We show now that the solution µ k is centered. We have Q t S k Qµ k = λ max µ k . By multiplying by Q, we obtain QQ t S k Qµ k = λ max Qµ k . And as

QQ t S k Qµ k = Q t SQµ k , we obtain λ max µ k = λ max Qµ k . This implies µ k = Qµ k : µ k is centered.

Experiments

In these experiments, we compare EKSC to a similar clustering model: K-shape [START_REF] Paparrizos | Fast and accurate time-series clustering[END_REF]. K-Shape (KS) is based on a distinct dissimilarity measure d KS (x, y) = 1 -maxo x•τo (y) x y

. EKSC and K-Shape share the same update step. For EKSC, we present acceleration factors obtained thanks to the four optimizations. These optimizations are applied identically to K-Shape apart from Elkan's cut off. Indeed, the K-Shape dissimilarity d KS is not metric [START_REF] Dongen | Metric distances derived from cosine similarity and pearson and spearman correlations[END_REF].

We use five datasets, each with a ground truth partition (see the UCR archive www.cs.ucr.edu/ ˜eamonn/time_series_data): ECG5000 (E), WordSynonyms (S), Fish (F), Non-Invasive Fetal ECG Thorax1 (N), Haptics (H). Table 1 reports dataset information. In all cases, the centered EKSC and KS are carried out ten times with random initializations. Parameter K is set to the number of classes of the ground truth partition. The stopping criterion is a threshold on the relative inertia gain. As EKSC and KS don't have the same dissimilarity measure, inertia can't be compared. Therefore, we evaluate both models thanks to the Rand-Index measure (ground truth vs produced partitions). RI results are quite similar for both models, but thanks to Elkan, EKSC is significantly faster.

Conclusion

In this work, we present a natural extension of K-Spectral Centroid (EKSC) to arbitrary time series. We show that the new dissimilarity measure is a metric distance. Finally, we propose to speed up EKSC thanks to four exact optimizations, and show that this model is significantly faster than K-Shape.

Table 1 :

 1 Table 2 reports the acceleration factors for the three steps of EKSC: assignment, matrix computation, barycenter extraction. For the fourth dataset, the fully optimized EKSC is 405 times faster than EKSC with no optimization. Size: nb of time series -Len: time series length -Cl : nb of classes -RI: Rand-Index -It: cumulated nb of iterations (10 runs) -Elkan: optimized EKSC -Elkan : optimized EKSC with Elkan disabled -Base: EKSC with no optimization -OKS : optimized KS -Elkan, Elkan, Base, OKS : cumulated running times in seconds

	OKS

Table 2 :

 2 Quotient: acceleration factor compared to Base or Elkan Steps: assignment (ass.), matrix computation (mat.), barycenter extraction (bar.)


Keywords: Deep Brain Stimulation, Essential Tremor, Simulations, Intraoperative Visualization

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Learning more about the optimal anatomical position for deep brain stimulation in essential tremor patients: 3d visualisation of intraoperative stimulation test results

Ashesh Shah, Fabiola Alonso, Jean-Jacques Lemaire, Daniela Pison, Jerome

  Coste, Karin K Wårdell, Erik Schkommodau, Simone Hemm-Ode To cite this version: Ashesh Shah, Fabiola Alonso, Jean-Jacques Lemaire, Daniela Pison, Jerome Coste, et al.. Learning more about the optimal anatomical position for deep brain stimulation in essential tremor patients: 3d visualisation of intraoperative stimulation test results. 17th Quadrennial Meeting of the World Society for Stereotactic and Functional Neurosurgery, WSSFN, Jun 2017, Berlin, Germany. pp.186

Abstract:

Introduction: The outcome of deep brain stimulation (DBS) depends heavily on the position of the implanted lead. After a preoperative anatomical planning, most groups collect numerous intraoperative data such as therapeutic effects induced by stimulation tests. To choose the final implant position, physicians "mentally" visualise all available data. The aim of the present work was to develop a method visualising intraoperative stimulation test results, patient's images, electric field (EF) simulations for the patientspecific stimulation conditions and the corresponding therapeutic effects quantitatively evaluated by accelerometry. The application to five essential tremor (ET) patients should give a first idea about the optimal target position.

Methods:

In Clermont-Ferrand University Hospital the anatomic target structure and the neighbouring structures were manually outlined, a target and a trajectory defined and two parallel trajectories per hemisphere intraoperatively evaluated. Stimulation tests were performed at 7 to 8 positions per trajectory and several stimulation current amplitudes. The therapeutic effect was evaluated using a previously published method based on accelerometry. Finite element models and simulations were performed for up to three stimulation amplitudes per position and EF isosurfaces (0.2V/mm) were extracted. For the 3D visualization of the numerous overlapping isosurfaces, we generated "improvement maps" by assigning to each voxel within the isosurfaces the highest tremor improvement. Those maps were visualized together with anatomical images, delineated structures and trajectories (Paraview, Kitware Inc). The method was applied to 5 ET patients implanted in the ventro-intermediate nucleus of the thalamus (VIM). Results were analysed by the neurosurgeon regarding the optimal implant position. Results: The clinical teams were able to identify the optimal implant position for all patients with more ease and in less time compared to the routine discussion using pen and paper. Additionally, for 7 of the 9 improvement maps, the highest improvement region was found to be in the posterior subthalamic area, inferior and posterior to the VIM.

Conclusion:

Improvement maps assist the clinicians in determining the optimal implant location of the chronic DBS lead. Results support findings of other studies that the fibre tracts in the posterior subthalamic area like prelemniscal radiations may be responsible for alleviating tremor in ET patients.


This paper presents the control of ARCHI, a 3-dof over-actuated parallel mechanism, designed as a subpart of a 5-axis hybrid machining robot. Control strategies of the parallel sub-mechanism (3 dof / 4 actuators) managing the internal constraint on the mechanism are proposed and experimental results are discussed.

I. INTRODUCTION

The concept of parallel robots yet demonstrated its efficiency in terms of mechanism stiffness and dynamic performances. At the present time, these robots are used for various applications such as pick-and-place (Delta [START_REF] Clavel | Delta, a fast robot with parallel geometry[END_REF] or H4 [START_REF] Company | A new 3T-1R parallel robot[END_REF], [START_REF] Pierrot | H4 parallel robot: modeling, design and preliminary experiments[END_REF] robots can reach accelerations higher than 10 g for carrying objects, allowing high production rates), high-speed machining where high stiffness and velocities are essential (for example the Renault Automation 3-dof machining center UraneSx [START_REF] Company | Modelling and preliminary design issues of a 3-axis parallel machine-tool[END_REF] able to reach 3.5 g or Toyoda 6-dof machine-tool HexaM [START_REF] Pierrot | From Hexa to HexaM[END_REF]). However, if the efficiency of theses machining centers is not to be demonstrated limitations subsist. In fact, on one hand, UraneSx machine has only 3 dof -insufficient to complete all kind of machining tasks-and, on the other hand, 6dof machines have a limited tilting angle (± 20 degrees orientation capability for the HexaM spindle) that restricts their possibility of use. To overcome these limitations, a solution consists in designing mechanical amplification devices such as Twice [START_REF] Krut | Twice: A Tilting Angle Amplification System for Parallel Robots[END_REF]. Another solution is to design over-actuated robots that eliminate the singular positions from the workspace. In this

CONTROL OF A 3-DOF OVER-ACTUATED PARALLEL MECHANISM

Frédéric Marquet

Olivier Company

Sébastien Krut Olivier Gascuel François Pierrot

LIRMM -UMR 5506 CNRS / UM2 161, rue Ada -34392 Montpellier cedex 5, France URL: www.lirmm.fr/rdc/ marquet@lirmm.fr way, relatively few over-actuated parallel robots have been yet studied (however, interesting works concerning the control of a one and a two-dof redundant robots described in [START_REF] Boulet | Robust control of a robot joint with hydraulic actuator redundancy, ISR[END_REF] and [START_REF] Kock | A mixed elastic and rigid body dynamic model of an actuation redundant parallel robot with high-reduction gears[END_REF] have to be mentioned) . In order to obtain 5-axis machines, a possibility is to design hybrid machines associating a serial to a parallel over-actuated mechanism as proposed by Seoul University with the efficient (but complex) Eclipse machine [9][10].

Combining both concepts, ARCHI robot presented in this paper is a 5-axis hybrid over-actuated machine. The study of the control strategies of the redundant parallel 3-dof submechanism is the aim of this paper. After recalling ARCHI design and modeling, several control strategies managing the internals constraints on the over-actuated robot are presented ("two-arm like" control, minimization of the Euclidian and infinite norm of actuators forces) and tested on a prototype.

II. ARCHI PARALLEL REDUNDANT ROBOT

II.1. ARCHI concept

ARCHI design is depicted in details in [START_REF] Marquet | a Redundant Mechanism for Machining with Unlimited Rotation Capacities[END_REF]. ARCHI robot consists of the association of a hybrid parallel mechanism actuated by 4 linear drives providing two translations and one rotation dof and a serial mechanism allowing a rotation and a translation (see Figure 1 -P representing a Prismatic joint, R a Revolute, U an Universal and S a Spherical joint-). The total number of dof is then equal to 5. The parallel part can be considered as the association of two 2-dof robots that "cooperate" to displace the nacelle (Figure 2). It has been demonstrated that thanks to the redundancy there is no limitation on the nacelle rotation (no overmobility-type singular position).

II.2.1. Position equations

Basic models of ARCHI robot have been yet described in [START_REF] Marquet | ARCHI: a New Redundant Parallel Mechanism : Modeling, Control and First Results[END_REF]. It has been pointed out that the inverse kinematic model is unique and easy to compute (analytic solution). Concerning the forward kinematic model (FKM), if the robot is perfect, the solution doesn't depend on which set of 3 arms among 4 is selected. Of course, in reality, as the robot is not perfect (errors on arms lengths, offset errors on drives positions…), the solution to the FKM will depend on which non-redundant 3arm subsystem is considered.

II.2.2. Velocity equation

If q = [q 1 ,q 2 ,q 3 ,q 4 ] T is the vector of drives positions, x = [x,y,θ] T the nacelle configuration described by the position of point C and the nacelle orientation, (x 12 , y 12 ) and (x 34 ,y 34 ) the coordinates of B 12 and B 34 , (Figure 3), the velocity equation can be classically written as:

[ ] T y x y x 34 34 12 12 , , , & & & & = v q J J q v & = v with A s v (G
), the relation between and can be expressed as:

4 4Ấ Î q v J J , . x & q & x G& = 3 4Ấ Î q J x J q & & = x with (
), and if is not singular:

G J J v x = 3 4Ấ Î x J q J x J q m & & = (1)
where:

q q q q q q q q m J ² ² 12 1 y L s - = , ² ² 34 2 y L s - =
For given drives positions eq. ( 1) is an over-determined linear system of unknown which least square better solution is:

x

& q J x m & & + = ( 2
)
(+ denotes the pseudo-inversion operator)

y x L L L L q1 q4 q3 q2 B 12 B 34 q A 1 A 4 A 2 A 3 d C

Figure 3: ARCHI parameterization

III. INTERNAL CONSTRAINT MANAGEMENT

Let's consider a m-dof robot actuated by n drives ( ) and f the linear application from  to ( ) as:

m n ³ , m n  n m  ) ( f  ΠL ) ( d nac F F f = (3)
where is the force acting on the nacelle and drives forces. If is f null space and its range,

nac F Ker d F ) ( f ) ( f R n = + ) dim( ) dim( R(f) Ker(f)
and as , the dimension of f kernel is then:

m = ) dim(R(f) m n p - = = ) dim(Ker(f)
For a non-redundant robot (m = n: f endomorphism), Ker(f) = {0} and "

.

nac d nac F F F F = Â Î $ Â Î ) ( / , d n m f ! Otherwise eq. (3)
+ = (4)
For any over-actuated mechanism, the relation between F d and F nac can be expressed by:

d m d nac F J F F T f = = ) ( , m nẤ Î m J (5)
And the solutions of the underdetermined linear system (5) are:

]F J J [I F J F m m nac m d T T T + + - + = ( 6
)
where I is the n-by-n identity matrix and F is any force belonging to (the vector [ F corresponds to its projection onto the null-space).

n  ] T T m m J J I + - For ARCHI robot (n = 4, m = 3): 4 3Ấ Î T m J 3 ) ( ) dim( = = m rank J R(f) dim( 1 ) = Ker(f)
and consequently eq. ( 4) can be rewritten as:

ker ker S S d F F F l + = , Â Î l (7)
Any value of will lead to drives forces reacting against each other and generating internal constraints on the mechanism. It is then necessary to solve eq. ( 3) taking into account the internal forces acting on the nacelle. The following sections present 3 approaches for drives forces computation managing this internal constraint.

l

III.1. "Two-arm like" control

In order to solve the underdetermined system, an intuitive idea consists in adding one supplementary equation to the linear system (the dimension of the jacobian kernel is equal to one) that, in static, cancels the compression force acting on the nacelle (Figure 4): As demonstrated in [START_REF] Marquet | ARCHI: a New Redundant Parallel Mechanism : Modeling, Control and First Results[END_REF], the constraint expressed by eq. ( 7) leads to the solution:

0 = • - n f f 2 1 ) ( (8)
ú û ù ê ë é = 0 nac e d F J F (9)
where:

4 4 T - - Â Î = 1 f q 1 x e J J J J
] [ and:

ú ú ú ú û ù ê ê ê ê ë é - - - - =
q q q q q q q q sin cos sin cos cos sin cos sin

1 0 1 0 0 1 0 1 d d d d f J
Of course, it is also possible to set the mechanical constraint acting on the nacelle to a given non-null value in order to eliminate the backlash phenomenon [START_REF] Boulet | Robust control of a robot joint with hydraulic actuator redundancy, ISR[END_REF]: the computation method keeps unchanged. Basically, this approach is similar to the one proposed by Dauchez [START_REF] Dauchez | Hybrid control of a two-arm robot handling firmly a single rigid object[END_REF] for two-arms robots control.

III.2. Minimization of drives Euclidian norm

The force F d solution of eq. ( 5) lying into J m T kernel is the unique force that minimizes the Euclidian norm of actuators forces:

nac m d F J F + = T (10)

III.3. Minimization of the maximal drive force

The minimization of the maximal drive force is ensured if the infinite norm of actuators forces is minimized. It has been demonstrated that for ARCHI robot the general solution of eq. ( 5) can be written: . As the solution corresponding to the minimal infinite norm has necessarily two components equals in absolute value [START_REF] Boulet | Robust control of a robot joint with hydraulic actuator redundancy, ISR[END_REF]:

ker S F { }, , / ) , ( 3 4 minimal T ¥ = Â Ấ Î " d nac d m nac d F F F J F F { } ) ( ) ( / ², 4 ,..., 1 ) , ( j F i F j i j i d d = ¹ Î $ (j) F (i) F d d ± = (i.e. )
the set of solutions of the under-determined linear system (5) can then be represented on Figure 5 where each drive force component , , is plotted as a function of λ (as well as -to find the points corresponding to the condition ).

) (k F d ) (k (j) d { 4 ,..., 1 Î k } F d F ± (i) F d = n y q x 2 f 1 f z m f , z Figure 5

. Computation of the minimal drive forces

The candidate solutions to the minimization problem, depicted on the graph by • (points ), correspond to the intersection of two straight lines. Thanks to the symmetry of the graph, only 12 are to be known and considering that the minimal solution can't correspond to the intersection of two lines having slopes of the same sign, only 6 are to be computed. The solution to the minimization problem can then be found among this set of candidate solutions by checking the value of the infinite norm of each one and choosing the minimal. It has to be noticed that this way of solving the minimization problem may be extended to the case of a multi-dimensional null-space.

l 2 , ij l + - - + - - + + ij ij ij ij P , P , P , P )) ( ) ( k k + l ( ) ( ker F k F S d ± = ker F S : ) ( ) ( ) ( ) ( ker ker 1 , j F i F j F i F S S d d ij - + = l , ) ( ) ( ) ( ) ( ker ker 2 , j F i F j F i F S S d d ij - - = l
4-Test of this set of candidate solutions to find the one minimizing the maximal drive force.

(j) F (i) F j i j i d d ± = ¹ Î $ , ², 4 ,..., 1 ) , ( Necessary Minimization Condition 1 , ij l , ) ( 1 , ij l d F 2 , ij l , ) ( 2 , ij l d F (j) F (j) F (j)λ F (i) F (i) F (i)λ F d S ij
0 ker = = ) ( F S T ker S m F J General Solution { }² 4 ,..., 1 ) , ( Î j i (j) F (j) F (j)λ F (i) F (i) F (i)λ F d S ij, S d S ij, S = - - = +

IV.1. Experimental setup

Robot control task is a RTX (Real Time eXtension) process (implemented in C language) that generates a periodical control task sampled at 2 kHz. User's orders come from the G.U.I (Graphical User Interface, implemented in Visual C++) and are transmitted to the control task through shared memory. An I/O board is a plugged into the computer PCI bus (260 MHz, 64 Mb, running WinNT). The control is ensured by 2 controllers: the first one is a force controller built in the amplifier and the second one a Cartesian Space controller associating a PD controller to one of the drives forces computation algorithms above mentioned. The Figure 7 sums up the robot control:

IV.2. Prototype

A prototype of the parallel over-actuated sub-mechanism has been built (Figure 8). Its main dimensions and characteristics are:

IV.3. Experimental results

The experimental results have been obtained for a 400 ms displacement from the start point x i = [0.95 m; -0. 5 m; π/2 rad] to the end point x f = [1.35 m; -0.65 m; 0 rad] (sine/ramp trajectory generation). The total of moving masses was about 15 Kg. During that motion tracking errors on x and y remain smaller than 2.5 mm and the maximal angular tracking error was lower than ½ degree (Figure 9)1 . The maximal linear drives velocity was above 2 m/s and the corresponding nacelle velocity on x and y was 1.5 m/s whereas the rotation velocity was close to 2 revolutions per second (Figure 10). The nacelle reached a 2 g acceleration (Figure 11).

• Cartesian tracking errors (pseudo-inversion algorithm)

Errors on x and y (m)

Error on θ (rad) The following figures describe the evolution of drives forces for each control strategy. During the displacement specified above, the maximal drive force corresponding to the "2-arm like" control was equal to 152 N (Figure 12) whereas in the case of the Euclidian norm minimization it only reached 141 N (Figure 13) and 120 N (Figure 14) when minimizing the infinite norm. It is then clear that for any robot the control minimizing the infinite norm of drives forces permits to reach higher accelerations and to take the best advantage of drives potentialities (even if it generates drives forces that have no effect on the nacelle force (Figure 15)).

Velocities on x and y (m/s)

• Actuators forces when selecting the " 2-arm like control"

Velocity on θ (rad/s)

V. CONCLUSION

In this paper, several control strategies of a 3-dof overactuated mechanism have been proposed and tested on a prototype. The "two arm like control" seems to be a good and simple method to cancel out the internal force acting on the nacelle. Methods based on the minimization of the Euclidian norm or the infinite norm of drives forces also proved their efficiency, especially the second one that permits to take the best advantage of actuators potentialities. According to the theory, experimental results have clearly demonstrated the validity and the efficiency of these approaches for controlling over-actuated robots.

Figure 1 :Figure 2 :

 12 Figure 1: Kinematic representation of ARCHI

Figure 4 :

 4 Figure 4: Forces acting on the nacelle

  solving the system setting one component of to a given value (for example ).

Figure 6 .

 6 Figure 6. Maximal drive force minimization algorithm

Figure 7 .

 7 Figure 7. Robot control

•Figure 8

 8 Figure 8. Robot prototype

Figure 9 :

 9 Figure 9: Cartesian errors

Figure 10 :

 10 Figure 10: Cartesian velocities

Figure 11 :

 11 Figure 11: Accelerations on x and y (m/s)

Figure 12 .

 12 Figure 12. "Two-arm like" control: drives forces (N)

Figure 13 .Figure 14 .FFigure 15 .

 131415 Figure 13. Minimization of the Euclidian norm : drives forces (N)

In figures 9 to 15, horizontal axis represents Time in (s)