Chandan Nandi

Ranjit Bag

Soumen Giri

Arindam Roy

Marie Cordier

Sundargopal Ghosh
email: sghosh@iitm.ac.in

Triple-decker complexes comprising heterocyclic middle-deck with coinage metals

Keywords: cobalt, rhodium, metallaheteroborane, triple-decker, copper

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

Introduction

Since the discovery of ferrocene in 1951, 1 the quest for novel sandwich complexes have been a significant area of study in the field of organometallic chemistry due to their fascinating physical and chemical characteristics as well as their distinctive structural features. 2 Syntheses of metallocenes with the increasing number of rings and metals led to triple-and multi-decker sandwich complexes which contain heterocyclic rings simultaneously bonded to two or more oppositely lying transition-metals. 3 Triple-decker complexes comprising a planar or bent middle ring are extremely versatile and exhibit significant unpaired electron delocalization between metal centers. [START_REF]ESI-MS (ESI + ): m/z calc. for [C20H32B2Co2S2CuBr-Br+CH3CN[END_REF][START_REF]ESI-MS (ESI + ): m/z calc. for [C20H32B2Co2Se2CuBr-Br+CH3CN[END_REF][6][7] The possibility of the existence of a triple-decker sandwich complex was first pointed out by Salzer and Werner in 1972 after the discovery of [Ni2Cp3] + (Cp = η 5 -C5H5). 8 Grimes and coworkers have isolated the first structurally characterized triple-decker complex, [(CpCo)2RC2B3H4] (R = H/Me) (Chart 1, I). 9 However, compared to triple-decker complexes with carbocyclic ligand systems, such complexes with solely inorganic ring systems or heterocyclic ligands are still comparatively understudied due to a lack of effective synthetic methods. In 1982, Rheingold and coworkers reported the first triple-decker complex, [(CpMo)2(μ,η 4 -As5)] (II) having cyclo-As5 moiety in middle-deck. 10 Later, in 1986, Scherer and coworkers isolated [(CpCr)2(μ,η 5 -P5)] (III), the first triple-decker complex in which a cyclo-P5 unit was stabilized as middle-deck. 11 Fehlner and coworkers have isolated a triple-decker complex comprising open-B5 moiety as middle-deck [(Cp*ReH)2B5Cl5] (IV), which is the first example of a dimetallaborane fully chlorinated at boron. 12 Kudinov and coworkers have reported a few triple-decker complexes of iron, ruthenium, and molybdenum, such as [Cp*Fe(μ,η 5 :η 5 -P5)MCp*] + (M = Fe, Ru) and [(C7H7)Mo(μ,η 5 :η 5 -P5)FeCp*] + . 13 Scheer and coworkers have isolated several unique neutral triple-decker sandwich complexes having functionalized cyclo-P5 moieties as middle-decks by using [Cp*Fe(η 5 -P5)] as starting material. 14 Recently, the same group has reported a new highyielding synthetic method for triple-decker complex [Cp*Fe(μ,η 5 :η 5 -P5)Mo(CO)3] (V). 15 Very recently, Roesky and coworkers have reported trinuclear triple-decker complexes with P5 or As5 middle-decks and a planar lead containing heterocycle as a lower deck that disclosed the first heterotrimetallic Pb-Fe-Li triple-decker polypnictogenides. 16 However, the development of such heterometallic systems that incorporate main group elements, semi-metals, and transition metals in the middle-deck are still relatively unexplored due to the lack of efficient synthetic routes. Although several triple-decker complexes having homocyclic ring composed of carbon, boron, phosphorus, and arsenic as middle-decks are known in the literature, heterocyclic ring systems as middle-decks are still limited. 17 As a result, syntheses of triple-decker complexes having coinage metal (i.e., Cu) in the middle-deck became of interest.

Coinage metal complexes have attracted a lot of attention due to their striking structural features and find prospective uses in various fields, particularly in nanotechnology. 18 Among several metallic nanoclusters, group 11 coinage metals have drawn significant interest and show different physio-chemical properties like luminescence, non-linear optical response, and molecular chirality. 19 (Chart 1. near here)

As part of our research on such complexes, [20][21][22] we have recently synthesized and characterized triple-decker complexes [(Cp*M)2{B2H2E2Pd(Cl)2}] (E = S and Se; M = Rh, Co, or Ir) comprising {PdCl2} moiety in the middle-deck. 22 Very recently, we have isolated a triple-decker sandwich complex of titanium [(Cp*Ti)2(μ-η 6 :η 6 -B6H6)(μ-H)6] containing a nearly planar [B6H6] ring sandwiched between two {Cp*Ti} fragments. 20c As a result, the isolation of triple-decker complexes comprising planar middle-deck following a high-yielding synthetic route became appealing. In this article, we have revisited the chemistry of group 9 transition metal complexes, [(Cp*M)2{μ-B2E2H2}] (M = Co, or Rh; E = S or Se) containing chalcogen atoms at the open site, with [CuBr(SMe2)]. Herein, we have reported syntheses, structures, and bonding of triple-decker complexes having a planar heterocyclic 5-membered middle-deck.

Results and discussion

Syntheses and characterization of triple-decker complexes, 4-6. Cluster expansion reactions of preformed metallaheteroboranes and metallaboranes by incorporating main group and metal fragments afforded clusters of both unusual shape and electron count. [23][24][25] However, syntheses of triple-decker complexes with planar middle-decks consisting of transition metals or main group elements are rare besides a few instances, including the first complex [(Cp*Re)2{μ-η 6 :η 6 -B4H4Co2(CO)5}]. 4b,25e As a result, we have studied the reactivity of open-core triple-decker

complexes nido-[(Cp*M)2{μ-B2H2E2}], 1-3 (1: E = S, M = Co; 2: E = Se, M = Co; 3: E = Se, M = Rh) with [CuBr(SMe2)].
As shown in Scheme 1, the reactions of 1-3 with [CuBr(SMe2)] at room temperature yielded 4-6, respectively. Immediately after the addition of [CuBr(SMe2)] with 1 and 2, the reaction mixtures changed to green due to the formation of 4 and 5. In contrast, the reaction mixture changed to red for 6. Note that all the complexes have been isolated as a single-product in good yields. The formation of 4-6 from 1-3 may be described as metallo-ligand approach, in which 1-3 act as a ligand to the {CuBr} unit. From the reaction mixtures, SMe2 was removed by washing with hexane, and products were characterized without further purification. Although the reactions of [Ag(PPh3)Cl] and[AuCl(SMe2)] with 1 and2 5), and C20H32B2Rh2Se2CuBr (6)]. All the spectroscopic data, along with mass spectrometric data, suggest comparable structures for 4-6. Finally, the single-crystal X-ray diffraction analyses were performed to determine the solid-state structures of 5 and 6.

(Fig. 1. near here)

The single-crystals of 5 and 6 were grown by slow evaporation of CH2Cl2 solution of the complexes at -5 °C. Unfortunately, we are not able to get good quality single crystals for 4. The solid-state structures of 5 and 6, shown in Fig. 1, correspond to triple-decker complexes where group 9 metals (Co/Rh) are sandwiched between five-membered [B2E2Cu] ring and Cp* ligands.

Middle-decks in 5 and 6 have slightly deviated from coplanarity with a dihedral tilt angle stretched from 1.95°-5.02° (Fig. 2), in which the deviation is maximum for 6. The bridging ligands in 5 and 6 contain a five-membered [B2E2Cu] ring in which Cu(I)Br unit bind through chalcogen atom with a bite angle of 96°. Interestingly, the bromine atom and chalcogen atoms in 5 and 6 are in a trigonal planar arrangement with Cu(I), while the axial group 9 metals are weakly coordinated with Cu(I). The B-B bond distance of 5 (1.69( 2) Å) and 6 (1.680(13) Å) are shorter than that of nido-2 (1.726( 7) Å) 26 and nido-3 (1.739(4) Å), 27 respectively (Table 1). While the average B-Se distances of 5 (1.993 Å) and 6 (2.001 Å) are longer than that of nido-2 (1.985( 6) Å], 16 and nido-3 (1.983(3) Å) 27 . The average Cu-Se distances of 5 (2.376 Å) and 6 (2.381 Å) are compared to that of {Cu[Se (2,4,6-i Pr3C6H2)]}6 (2.370(2) to 2.412(2) Å). 28 The average Co-Cu distance in 5

(2.676(2) Å) is considerably longer (0.3 Å) as compared to binuclear Cu-Co complex, [(tmed)CuCo(CO)4] (dCu-Co = 2.379 (1) Å; tmed = N,N,N',N'-tetramethylethylenediamine). 29 Similarly, the average Rh-Cu distance in 6 (2.777(2) Å) is longer as compared to Rh-Cu complex, [RhCu(PPh3)(cod)(η 5 -7,8-C2B9H9Me2)] (dRh-Cu = 2.633(2) Å, cod = cycloocta-1,5-diene). 30 Complexes 4-6 can be described as 60 cluster valance electrons (CVE) and 8 SEP: (Fig. 2. near here) (Table 1. near here)

The DFT analyses of 5 and 6 have been carried out using the Gaussian 16 program at the BP86/def2-TZVP level of theory to understand the structures and bonding of such triple-decker complexes. The optimized geometrical parameters of 5 and 6 are in accordance with their solidstate structural parameters (Table S1). Additionally, the GIAO-DFT (gauge-including atomic orbitals-density functional theory) calculated 11 B chemical shift values of 5 and 6 are also in good agreement with the values observed experimentally (Table 1). The frontier molecular orbital analyses of 5 and 6 display a lower HOMO-LUMO gap in 5 ( 1.269 eV) compared to 6 (1.379 eV). This may be due to the presence of different axial metals (Co or Rh) in their framework. The HOMOs of 5 and 6, shown in Fig. 3, are significantly concentrated over the axial metals (Co or Rh; d-orbitals), Se (p-orbitals), Cu (d-orbitals), and Br (p-orbital) atoms. In contrast, the LUMOs are mainly localized on the axial metals (d-orbitals) and Se atoms (p-orbitals). Further, the HOMO-12 of 5 and HOMO-14 of 6 (Fig. 3) show  bonding interactions between the axial metals and Cu the absence of a bonding interaction between two axial metal centers (Table S1). The natural charges obtained from NBO (natural bond orbital) analyses, shown in Table S3, display positive and negative charges on selenium and boron atoms, respectively, that indicate their donating nature towards metals.

(Fig. 3. near here)

The topological analyses of 5 and 6 were carried out using QTAIM method to reveal the interaction of the axial group 9 transition metals (Co/Rh) with the middle-deck. The positive value of ∇ 2 ρ(r) (Laplacian of the electron density) at the bond critical points of M-Se, Cu-Br, and M-B, reveal electron-donating nature of these ligands towards metals (Table 2). Additionally, the topological properties show that the Cu-Se bonds are more electrostatic than the Co-E or Rh-E bonds (E = S or Se). Furthermore, the electron localization function (ELF) values, negative ∇ 2 ρ(r), and negative H(r) values along the B-B and B-Se bonds suggest the pure covalent character of these bonds. These variations are observed in the contour map of the ∇ 2 ρ(r), shown in Fig. 4(a), in the [CuB2Se2] middle-deck for 5. For instance, the electron concentration is higher between Se1-B1-B2-Se2 bonds, while the electron density is lower between Cu-Se bonds. In addition, the topology analysis for 5, shown in Fig. 4 (b), suggests the 3c-2e interaction between the axial metals and B-B bond.

(Table 2. near here) (Fig. 4. near here)

Conclusion

In summary, we have isolated and structurally characterized triple-decker sandwich complexes of group 9 metals having planar 5-membered [CuB2Se2] ring in the middle-deck. The reported methodology is an efficient route for the generation of heterometallic triple-decker complexes with high yields. Interestingly, complexes 4-6 are the first examples of triple-decker complexes comprising coinage metal in the middle-deck. Further, theoretical calculations gave insight into the bonding of such complexes, in which the middle-deck is strongly bonded with the axial metals through the [B2E2] fragment. In contrast, a weak bonding interaction was observed between Cu and the axial group 9 metals. Besides the syntheses of such heterotrinuclear triple-decker complexes, the usefulness of these complexes is under investigation.

Experimental section

General procedures and instrumentation. All the syntheses were executed under Ar atmosphere by using standard Schlenk line techniques. Tertrahydofuran, toluene, DCM (dichloromethane) and hexane were distilled under an argon atmosphere. All reagents, such as Cp*H, RhCl3, CoCl2, n-BuLi in hexane, [LiBH4•THF] were purchased from Sigma Aldrich and used without purification.

[CuBr(SMe2)], 31 and 1-3 26,32 were synthesized by following the literature procedures. All 1 H, 11 B{ 1 H} and 13 C{ 1 H} NMR spectra for the synthesized complexes were recorded on 500 or 400 MHz Bruker FT-NMR spectrometers. The residual solvent protons (CDCl3, δ = 7.26 ppm) and carbon (δ = 77.1 ppm, CDCl3) were employed as a reference for the 1 H and 13 C{ 1 H} NMR spectra, respectively. 11 B{ 1 H} spectra of all compounds were processed with a backward linear prediction algorithm to remove the broad 11 B{ 1 H} background signal of the NMR tube. 33 Qtof Micro YA263 HRMS and 6545 Qtof LC/MS instruments were used to record the mass of the compounds. IR spectra were recorded on a JASCO FT/IR-1400 spectrometer in dichloromethane solvent.

Syntheses of 4 and 5:

In a flame-dried Schlenk tube, the deep blue solution of [(Cp*Co)2B2H2S2], 1 (0.03 g, 0.06 mmol) and [CuBr(SMe2)] (0.012 g, 0.06 mmol) in toluene (10 ml) was stirred at room temperature. The reaction mixture was changed from blue to green within 1 min and allowed to stir at room temperature for an additional 1 h. After filtration, the solvent was dried under vacuum and the residue was purified by washing with hexane, yielded a green solid, 4

[(Cp*Co)2B2H2S2CuBr] (yield: 90-92% for a set of experiments). Single crystals were grown by slow diffusion of dichloromethane solution of 4 in dichloromethane at -5 ºC.

6: ESI-MS (ESI +

): m/z calculated for [C20H32B2Rh2Se2CuBr-Br+CH3CN] + : 763.8702, found: 763.8659; 11 B{ 1 H} NMR (CDCl3, 160 MHz, 22 °C, ppm): δ = 17.9 (s, 2B); 1 H{ 11 B} NMR (CDCl3, 500 MHz, 22 °C, ppm): δ = 1.77 (s, 30H, C5Me5), 5.7 (s, 2BHt ); 13 C{ 1 H} NMR (CDCl3,125 MHz,22 °C,ppm): δ = 90.8 (s, C5Me5), 10.4 (s, C5Me5); IR (dichloromethane, cm -1 ): ṽexp= 2510 (BHt); ṽcal = 2538 (BHt).

Computational details: All the triple-decker complexes were optimized using the Gaussian 16 34 program at the BP86 35 functional along with def2-TZVP basis set from EMSL [START_REF]EMSL Basis Set Exchange Library[END_REF] Basis Set Exchange Library. The complexes were fully optimized starting from X-ray coordinates in the gaseous state (no solvent effect). Frequency calculations were carried out for the verification of the nature of the stationary state and to confirm the absence of any imaginary frequency, which eventually confirmed the minima on the potential energy hypersurface for all structures. NMR chemical shifts were computed by employing the gauge-including atomic orbitals (GIAOs) method 37 using the BP86/def2-TZVP optimized geometries. Chemical shifts corresponding to 11 B NMR were computed in relation to B2H6 (BP86 shielding constant for 11 B NMR: 80.9 ppm) and were then transformed to the standard [BF3.OEt2] scale by adding 16.6 ppm (the experimental δ ( 11 B) value of B2H6) to the computed values. 38 Wiberg bond indices (WBI) 39 were generated from natural bond orbital analysis (NBO) 40 . All the optimized geometries and orbital pictures were drawn by Chemcraft 41 visualization programs. Laplacian electronic distribution plots and twodimensional electron density were generated using the Multiwfn package. 42 X-ray structure determination analysis details: Suitable X-ray quality crystals for 5 and 6 were grown by slow diffusion of a hexane-CH2Cl2 solution at -5 ºC. The X-ray data were collected and integrated using a APEXII Bruker-AXS diffractometer for 5, Bruker D8 VENTURE for 6, and with graphite monochromated Mo-Kα (λ = 0.71073 Å) radiation at 150(2) K (for 5), 296(2) K (for Chart 1.

Triple-decker complexes having planar 5-membered middle-deck.

Scheme 1.

Syntheses of triple-decker complexes 4-6.

Fig.

  yielded almost immediately blue color species, they converted back to starting materials 1 and 2 within two hours, respectively. As a result, we are not able to isolate and structurally characterize the Ag and Au analogues of 4 and 5, respectively. (Scheme 1. near here) All complexes 4-6 were characterized by NMR, ESI-MS, and IR spectroscopy. The 1 H NMR spectra of 4-6 show singlet chemical shifts at δ = 1.60, 1.77, and 1.77 ppm, respectively which are due to Cp* ligands. The presence of BH terminal protons were confirmed by 1 H{ 11 B} NMR spectra that showed chemical shifts at δ = 4-6 ppm. The 11 B{ 1 H} NMR spectra show one boron peak at δ = 20.0 ppm for 4, 18.1 ppm for 5, and 17.9 ppm for 6. Further, the presence of Cp* ligands for 4-6 were also confirmed by 13 C{ 1 H} NMR spectra. The IR spectra of all these complexes display peak for BHt in the region of  2511-2500 cm -1 . The ESI-MS spectra show isotopic patterns at m/z = 580.0348 [M-Br+CH3CN] + for 4, 675.9225 [M-Br+CH3CN] + for 5, and 763.8659 [M-Br+CH3CN] + for 6 [M = C20H32B2Co2S2CuBr (4), C20H32B2Co2Se2CuBr (

  [2{Cp*M} + 2{BH}+ 2{E} + {CuBr}] = [2{2} + 2{2} + 2{4} +{0}]2 = 8 SEP] triple-decker complexes with a pentagonal bipyramidal geometry.

  through d-orbitals. The M-Cu Wiberg bond indices (WBI) for 5 and 6 (0.14) suggest weak bonding interaction between group 9 metals and Cu. Further, the WBI suggests a weaker Cu-Se (0.36) bond as compared to the Cu-Br (0.71) bond. The WBI of B-B and B-Se in 5 and 6 indicate strong bonding interactions. However, the M-M Wiberg bond indices (0.06) for 5 and 6 indicate

Fig. 2 .

 2 Fig. 2.Dihedral angles between three different planes in 5 (left) and 6 (right).

Fig. 3 .

 3 Fig. 3.Selected molecular orbitals of 5 and 6. Isosurfaces are plotted at an isovalue of ±0.045 (e/bohr 3 ) 1/2 .

Fig. 4 .

 4 Fig. 4.Contour line diagrams of the ∇ 2 ρ(r), Laplacian of the electron density, of (a) Co2-

Fig. 1 .Fig

 1 Fig. 1.

Fig

  Fig. 3.

Fig

  Fig. 4.

Table 1 .

 1 Selected bond parameters (angles in deg and distance in Å) and 11 B NMR of 4

-6.

Table 2 .

 2 ∇ 2 ρ(r) (Laplacian of the electron density), energy density (H(r)), and ELF (electron localization function) values at the bond critical points (BCPs) computed at the BP86/ def2-TZVP level of theory of selected bonds in 5 and 6.

Table 1 .

 1 Table1. Selected bond parameters (angles in deg and distance in Å) and 11 B NMR of 4-6. d[B-B] a d(M-Cu) a dihedral angle b Expt. Calc. Average distance in Å, b Cp* and B2E2Cu ring, c data not available, d M = Co, e M = Rh

	Triple-decker		Structural Parameters	11 B NMR (δ, ppm)
	Complexes d[B-E] 4 -c	-c	-c	-c	20.0	16.0
	5	1.993	1.690	2.676 d	3.96, 1.95	18.1	24.1
	6	2.001	1.680	2.771 e	5.02, 4.19	18.3	16.3
	a

a

Table 2 .

 2

			5			6
	bond	∇ 2 ρ(r)	H(r)	ELF	∇ 2 ρ(r)	H(r)	ELF
	B-B	-0.2280	-0.09167 0.9027	-0.2102	-0.0856 0.8992
	B-Se	-0.1467	-0.08170 0.7473	-0.1411	-0.0772 0.7588
	Se-Cu	0.1433	-0.0159 0.2531	0.1397	-0.0149 0.2497
	Cu-Br	0.2017	-0.0218 0.2291	0.2002	-0.0212 0.2272
	M-Cu a 0.0488	-0.0639 0.2324	0.0553	-0.0581 0.2127
	M-Se a	0.1250	-0.0146 0.2583	0.1234	-0.0128 0.2862
	B-B-M a 0.0906	-0.0230 0.4313	0.0781	-0.0237 0.4742

a M = Co for 5; M = Rh for 6

6

). The structures were solved using SHELXT-2014, SHELXS-97 43 , and refined using SHELXL-2017, SHELXL-2018, SHELXL-2014, 44 . X-ray structures were drawn using Olex2 45 . CCDC-2150288 (5) and2150289 (6 thanks the UGC for their research fellowships. The computational facility of IIT Madras is gratefully acknowledged.

Appendix A. Supplementary data

Supplementary data related to this article can be found at http://dx.doi.org.

Notes

The authors declare no competing financial interest.

Table of Contents Entry


Gérard Kerkyacharian

Thanh Mai Pham

Dominique Picard

LOCALIZED DECONVOLUTION ON THE SPHERE

Keywords: AMS 2000 subject classifications: Primary 62G05, 62G08, 62G20, 62G10 statistical inverse problems, minimax estimation, second-generation wavelets

We provide a new algorithm for the treatment of deconvolution on the sphere which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. We establish upper bounds for the behaviour of our procedure for any L p loss. It is important to emphasize the adaptation properties of our procedures with respect to the regularity (sparsity) of the object to recover as well as to inhomogeneous smoothness. We also perform a numerical study which proves that the procedure shows very promising properties in practice as well.

1. Introduction. The spherical deconvolution problem was first proposed by Rooij and Ruymgaart (1991) [START_REF] Arnoud | Regularized deconvolution on the circle and the sphere[END_REF] and subsequently solved in Healy et al. (1998) [START_REF] Dennis | Spherical deconvolution[END_REF]. [START_REF] Kim | Optimal spherical deconvolution[END_REF] [START_REF] Kim | Optimal spherical deconvolution[END_REF] established minimaxity for the L 2 -rate of convergence. The optimal procedures obtained there are using orthogonal series methods associated with spherical harmonics. One important problem arising with these procedures is their poor local performances due to the fact that spherical harmonics are spread all over the sphere. This explains for instance the fact that although they are optimal in the L 2 sense, they cease to be optimal for other losses, such as L p losses for instance.

In our approach we focus on two important points. We aim at a procedure of estimation which is efficient from a L 2 point of view, as well as it performs satisfactorily from a local point of view (for other L p losses for instance).

Deconvolution is an inverse problem and in such a problem there is a notable conflict between the inversion part which in presence of noise creates an instability reasonably handled by a Singular Value Decomposition (SVD) approach and the fact that the SVD basis very rarely is localized and capable of representing local features of images, which are especially important to recover. Our strategy is to follow the approach started in [START_REF] Kerkyacharian | Needlet algorithms for estimation in inverse problems[END_REF] [START_REF] Kerkyacharian | Needlet algorithms for estimation in inverse problems[END_REF] for the Wicksell case , [START_REF] Kerkyacharian | Inversion of noisy radon transform by svd based needlets[END_REF] [START_REF] Kerkyacharian | Inversion of noisy radon transform by svd based needlets[END_REF] for the Radon transform, which utilizes the construction borrowed from Narkowich Petrushev and Wald (2006) [START_REF] Narcowich | Local tight frames on spheres[END_REF], [START_REF] Narcowich | Decomposition of Besov and Triebel-Lizorkin spaces on the sphere[END_REF] of a tight frame (i.e. a redundant family) staying sufficiently close to the SVD decomposition but which enjoys at the same time enough localisation properties to be successfully used for statistical estimation (see for instance Baldi et al. (2009) [START_REF] Pietrobon | Spherical needlets for cmb data analysis[END_REF] [START_REF] Baldi | Adaptive density estimation for directional data using needlets[END_REF][START_REF] Baldi | Subsampling needlet coefficients on the sphere[END_REF][START_REF] Pietrobon | Spherical needlets for cmb data analysis[END_REF] for other types of applications). The construction [START_REF] Narcowich | Local tight frames on spheres[END_REF] produces a family of functions which very much resemble to wavelets, the needlets.

To achieve the goals presented above, and especially adaptation to different regularities and local inhomogeneous smoothness, we essentially use a projection method on the needlets (which enables a stable inversion of the devonvolution, due to the closeness to the SVD basis) with a fine tuning subsequent thresholding process.

This provides a reasonably simple algorithm with very good performances, both from a theoretical point of view and a numerical point of view. In effect, this new algorithm provides a much better spatial adaptation, as well as adaptation to wider classes of regularity. We give here upper bounds obtained by the procedure over a large class of Besov spaces and any L p losses.

It is important to notice that especially because we consider different L p losses, we provide rates of convergence of new types attained by our procedure, which, of course, coincide with the usual ones for L 2 losses.

Again, the problem of choosing appropriated spaces of regularity on the sphere in a serious question, and we decided to consider the spaces which may be the closest to our natural intuition: those which generalize to the sphere case the classical approximation properties of usual regularity spaces such as Holder spaces and include at the same time the Sobolev regularity spaces used in [START_REF] Kim | Optimal spherical deconvolution[END_REF] [START_REF] Kim | Optimal spherical deconvolution[END_REF].

Sphere deconvolution has a vast domain of application; our results are especially motivated by many recent developments in the area of observational astrophysics.

It is a common problem in astrophysics to analyse data sets consisting of a number of objects (such as galaxies of a particular type) or of events (such as cosmic rays or gamma ray bursts) distributed on the celestial sphere. In many cases, such objects trace an underlying probability distribution f on the sphere, which itself depends on the physics which governs the production of the objects and events.

The case for instance of ultra high energy cosmic rays (UHECR) iillustrates well the type of applications of our results. Ultra high energy cosmic rays are particles of unknown nature which arrive at the earth from apparently random directions of the sky. They could originate from long-lived relic particles from the Big Bang, about 13 billion years old. Alternatively, they could be generated by the acceleration of standard particles, such as protons, in extremely violent astrophysical phenomena, such as cluster shocks. They could also originate from Active Galactic Nuclei (AGN), or from neutron stars surrounded by extremely high magnetic fields.

Hence, in some hypotheses, the underlying probability distribution for the directions of incidences of observed UHECRs would be a finite sum of point-like sources -or near point like, taken into account the deflection of the cosmic rays by magnetic fields. In other hypotheses, the distribution could be uniform, or smooth and correlated with the local distribution of matter in the universe. The distribution could also be a superposition of the above. Identifying between these hypotheses is of primordial importance for understanding the origin and mechanism of production of UHECRs.

Of course, the observations of these events (X i 's in the sequel) are always most often perturbated by a secondary noise (ε i ) which leads to the deconvolution problem described below. Following Healy et al. (1998) [START_REF] Dennis | Spherical deconvolution[END_REF], [START_REF] Kim | Optimal spherical deconvolution[END_REF] [START_REF] Kim | Optimal spherical deconvolution[END_REF], the spherical deconvolution problem can be described as follows. Consider the situation where we observe Z 1 , . . . , Z N N i.i.d. observation with

Z i = ε i X i , (1)
where the ε i 's are i.i.d. random elements in SO(3) (the group of 3 ×3 rotation matrices) , and the Z i 's and X i 's are i.i.d. random elements of S 2 (twodimensional unit sphere of R 3 ) random elements, with ε i and X i assumed to be independent. We suppose that the distribution of resp. X, Z and ε are absolutely continuous with the Haar measure of resp. S 2 , S 2 and SO [START_REF] Dennis | Spherical deconvolution[END_REF] with the densities f Z , f ε , f X .

Then,

(2)

f Z = f ε * f X ,
where * denotes convolution and is defined below. In the sequel, f X will be denoted by f to emphasize the fact that it is the object to recover.

The following paragraph recall the necessary definitions. It is largely inspired by [START_REF] Kim | Optimal spherical deconvolution[END_REF] [START_REF] Kim | Optimal spherical deconvolution[END_REF] and and Healy et al. (1998) [START_REF] Dennis | Spherical deconvolution[END_REF].

2. Some preliminaries about harmonic analysis on SO(3) and S 2 . We will provide a brief overview of Fourier analysis on SO(3) and S 2 . Most of the material can be found in expanded form in Vilenkin (1969) [START_REF] Ja | Fonctions spéciales et théorie de la représentation des groupes[END_REF], [START_REF] Talman | Special functions: A group theoretic approach[END_REF] [START_REF] Talman | Special functions: A group theoretic approach[END_REF], [START_REF] Terras | Harmonic analysis on symmetric spaces and applications[END_REF] [START_REF] Terras | Harmonic analysis on symmetric spaces and applications[END_REF], [START_REF] Kim | Optimal spherical deconvolution[END_REF] [START_REF] Kim | Optimal spherical deconvolution[END_REF], and Healy et al. (1998) [START_REF] Dennis | Spherical deconvolution[END_REF]. Let

u(φ) =    cos φ -sin φ 0 sin φ cos φ 0 0 0 1    , a(θ) =    cos θ 0 sin θ 0 1 0 -sin θ 0 cos θ    ,
where , φ ∈ [0, 2π), θ ∈ [0, π). It is well known that any rotation matrix can be decomposed as a product of three elemental rotations, one around the z-axis first with angle ψ, followed by a rotation around the y-axis with angle θ, and finally another rotation again around the z-axis with angle φ. Indeed, the well known Euler angle decomposition says that any g ∈ SO(3) can almost surely be uniquely represented by three angles (φ, θ, ψ),with the following formula (see Healy et al. (1998) [START_REF] Dennis | Spherical deconvolution[END_REF] for details) :

(3)

g = u(φ)a(θ)u(ψ),
where φ ∈ [0, 2π), θ ∈ [0, π), ψ ∈ [0, 2π). Consider the functions, known as the rotational harmonics, (4) D l mn (φ, θ, ψ) = e -i(mφ+nψ) P l mn (cos θ), where the generalized Legendre associated functions P l mn for -l ≤ m, n ≤ l, l = 0, 1, . . . are fully described in Vilenkin (1969) [START_REF] Ja | Fonctions spéciales et théorie de la représentation des groupes[END_REF]. The functions D l mn for -l ≤ m, n ≤ l, l = 0, 1, . . . are the eigenfunctions of the Laplace Beltrami operator on SO(3), hence, √ 2l + 1D l mn , -l ≤ m, n ≤ l, l = 0, 1, . . . is a complete orthonormal basis for L 2 (SO(3)) with respect to the probability Haar measure. In addition, if we define the (2l + 1) × (2l + 1) matrices by [START_REF] Kerkyacharian | Inversion of noisy radon transform by svd based needlets[END_REF] D l (g) = [D l mn (g)], where for -l ≤ m, n ≤ l, l = 0, 1, . . . and g ∈ SO(3), they constitute the collection of inequivalent irreducible representations of SO(3) (for further details see [START_REF] Ja | Fonctions spéciales et théorie de la représentation des groupes[END_REF] [START_REF] Ja | Fonctions spéciales et théorie de la représentation des groupes[END_REF]).

Hence, for f ∈ L 2 (SO(3)), we define the rotational Fourier transform on SO(3) by [START_REF] Kerkyacharian | Needlet algorithms for estimation in inverse problems[END_REF] f l mn =

SO(3) f (g)D l mn (g)dg,
where again we think of (5) as the matrix entries of the (2l + 1) × (2l + 1) matrix f l = [ f l mn ] -l≤m, n≤l , l = 0, 1, . . . and dg is the probability Haar measure on SO [START_REF] Dennis | Spherical deconvolution[END_REF]. The rotational inversion can be obtained by

f (g) = l -l≤m, n≤l f l mn D l mn (g) = l -l≤m, n≤l
f l mn D l mn (g -1 ) (7) [START_REF] Kim | Optimal spherical deconvolution[END_REF] is to be understood in L 2 -sense although with additional smoothness conditions, it can hold pointwise.

A parallel spherical Fourier analysis is available on S 2 . Any point on S 2 can be represented by ω = (cosφ sin θ, sin φ sin θ, cos θ) t , with , φ ∈ [0, 2π), θ ∈ [0, π). We also define the functions :

(8) Y l m (ω) = Y l m (θ, φ) = (-1) m (2l + 1) 4π (l -m)! (l + m)! P l m (cos θ)e imϕ ,
for -l ≤ m ≤ l, l = 0, 1, . . ., φ ∈ [0, 2π), θ ∈ [0, π) and where P l m (cos θ) are the associated Legendre functions.

The functions

Y l m obey (9) Y l -m (θ, φ) = (-1) m Ȳ l m (θ, φ).
The set {Y l m , -l ≤ m ≤ l, l = 0, 1, . . .} is forming an orthonormal basis of L 2 (S 2 ), generally referred to as the spherical harmonic basis.

Again, as above, for f ∈ L 2 (S 2 ), we define the spherical Fourier transform on S 2 by [START_REF] Pietrobon | Spherical needlets for cmb data analysis[END_REF] f

l m = S 2 f (ω)Y l m (ω)dω,
where dω is the probability Haar measure on the sphere S 2 . The spherical inversion can be obtained by

f (ω) = l -l≤m≤l f l m Y l m (ω). ( 11
)
The bases detailed above are important because they realize a singular value decomposition of the convolution operator created by our model. In effect, we define for

f ε ∈ L 2 (SO(3)), f ∈ L 2 (S 2
) the convolution by the following formula:

f ε * f (ω) = SO(3) f ε (u)f (u -1 ω)du
and we have for all -l ≤ m ≤ l, l = 0, 1, . . .,

(12) ( f ε * f ) l m = l n=-l f l ε,mn f l n := ( f l ε f l ) m .
2.1. The SVD Method. The singular value method (see Healy et al. (1998) [START_REF] Dennis | Spherical deconvolution[END_REF] and Kim Koo (2002) [START_REF] Kim | Optimal spherical deconvolution[END_REF]) consists in expanding f in the spherical harmonics basis Y l m and estimating the spherical Fourier coefficients using the formula above [START_REF] Talman | Special functions: A group theoretic approach[END_REF]. We get the following estimator of the spherical Fourier transform of f :

f l,N m := 1 N N j=1 l n=-l f l ε -1 ,mn Ȳ l n (Z j ) (13) f l ε -1 := ( f l ε ) -1 ,
provided, of course, that these inverse matrices exist, and then the estimator of the distribution f is

f N (ω) = Ñ l=0 l m=-l f l,N m Y l m (ω), ( 14
)
where Ñ depending on the number of observations has to be properly selected.

3. Needlet construction. This construction is due to Narcowich et al. (2006) [START_REF] Narcowich | Decomposition of Besov and Triebel-Lizorkin spaces on the sphere[END_REF]. Its aim is essentially to build a very well localized tight frame constructed using spherical harmonics, as discussed below. It was recently extended to more general euclidean settings with fruitful statistical applications (see [START_REF] Kerkyacharian | Needlet algorithms for estimation in inverse problems[END_REF] [START_REF] Kerkyacharian | Needlet algorithms for estimation in inverse problems[END_REF]), Baldi et al. (2009) [START_REF] Pietrobon | Spherical needlets for cmb data analysis[END_REF] [START_REF] Baldi | Adaptive density estimation for directional data using needlets[END_REF][START_REF] Baldi | Subsampling needlet coefficients on the sphere[END_REF][START_REF] Pietrobon | Spherical needlets for cmb data analysis[END_REF].As described above, we have the following decomposition:

(15) L 2 (dx) = ∞ l=0 H l ,
where H l is the space of spherical harmonics of S 2 , of degree l (which dimension is 2l + 1). The orthogonal projector on H l can be written using the following the kernel operator

(16) ∀ f ∈ L 2 (dx), P H l f (x) = S d L l ( x, y )f (y) dy
where x, y is the standard scalar product of R 3 , and L l is the Gegenbauer polynomial with parameter 1 2 of degree l, defined on [-1, +1] and normalized so that

(17) 1 -1 L l (t)L k (t) = 2l + 1 8π 2
Let us point out the following reproducing property of the projection operators:

(18)

S d L l ( x, y )L k ( y, z ) dy = δ l,k L l ( x, z ) .
The following construction is based on two fundamental steps: Littlewood-Paley decomposition and discretization, which are summarized in the two following subsections.

3.1. Littlewood-Paley decomposition. Let φ be a C ∞ function on R, sym- metric and decreasing on R + supported in |ξ| ≤ 1, such that 1 ≥ φ(ξ) ≥ 0 and φ(ξ) = 1 if |ξ| ≤ 1 2 . b 2 (ξ) = φ( ξ 2 ) -φ(ξ) ≥ 0 so that (19) ∀|ξ| ≥ 1, j≥0 b 2 ( ξ 2 j ) = 1 . Remark that b(ξ) = 0 only if 1 2 ≤ |ξ| ≤ 2.
Let us now define the operator Λ j = l≥0 b 2 ( l 2 j )L l and the associated kernel

Λ j (x, y) = l≥0 b 2 ( l 2 j )L l ( x, y ) = 2 j-1 <l<2 j+1 b 2 ( l 2 j )L l ( x, y ) .
We obviously have:

(20) ∀f ∈ L 2 (S 2 ), f = lim J→∞ L 0 (f ) + J j=0 Λ j (f ) . and if M j (x, y) = l≥0 b( l 2 j )L l ( x, y ), then (21) Λ j (x, y) = M j (x, z)M j (z, y) dz .

Discretization and localization properties. Let us define

P l = l m=0 H m ,
the space of the restrictions to S 2 of the polynomials of degree less than l.

The following quadrature formula is true: for all l ∈ N there exists a finite subset X l of S 2 and positive real numbers λ η > 0, indexed by the elements η of X l , such that

(22) ∀f ∈ P l , S 2 f (x) dx = η∈X l λ η f (η) .
Then the operator M j defined in the subsection above is such that:

z → M j (x, z) ∈ P [2 j+1 ] , so that z → M j (x, z)M j (z, y) ∈ P [2 j+2 ] ,
and we can write:

Λ j (x, y) = M j (x, z)M j (z, y)dz = η∈X [2 j+2 ] λ η M j (x, η)M j (η, y) .
This implies:

Λ j f (x) = Λ j (x, y)f (y)dy = η∈X [2 j+2 ] λ η M j (x, η)M j (η, y)f (y)dy = η∈X [2 j+2 ] λ η M j (x, η) λ η M j (y, η)f (y)dy .
We denote

X [2 j+2 ] = Z j , ψ j,η (x) := λ η M j (x, η) for η ∈ Z j ,
It can also be proved that the set of cubature points X l can be chosen so that:

(23)

1 c 2 2j ≤ #Z j ≤ c 2 2j
for some c > 0. Actually in the simulations of §5 we make use of some sets of cubature points such that #Z j = 12.2 2j exactly. It holds, using (20)

f = L 0 (f ) + j η∈Z j f, ψ j,η L 2 (S d ) ψ j,η .
The main result of Narcowich et al. (2006) [START_REF] Narcowich | Decomposition of Besov and Triebel-Lizorkin spaces on the sphere[END_REF] is the following localization property of the ψ j,η , called needlets: for any k ∈ N there exists a constant c k such that, for every ξ ∈ S 2 :

(24)

|ψ j,η (ξ)| ≤ c k 2 j (1 + 2 j d(η, ξ)) k •
where d is the natural geodesic distance on the sphere (d(ξ, η) = arccos η, ξ ). In other words needlets are almost exponentially localized around their associated cubature point, which motivates their name.

A major consequence of this localization property can be summarized in the following properties which will play an essential role in the sequel.

For any 1 ≤ p < ∞, there exist positive constants c p , C p , c, C and D p such that

c p 2 2j( p 2 -1) ≤ ψ jη p p ≤ C p 2 2j( p 2 -1) , (25) c2 j ≤ ψ jη ∞ ≤ C2 j , ( 26
)
η∈Z j λ η ψ j,η π ≤ c2 2j( 1 2 -1 π ) η∈Z j |λ η | π 1/π (27) η∈Z j λ η ψ jη p p ≤ D p η∈Z j |λ η | p ψ jη p p (28) η∈Z j u η ψ jη ∞ ≤ C sup η∈Z j |u η |2 j (29)
To conclude this section, let us give a graphic representation of a spherical needlet in the spherical coordinates in order to illustrate the above theory. In the following graphic, we chose j = 3 and η = 250.

3.3.

Besov spaces on the sphere. The problem of choosing appropriated spaces of regularity on the sphere in a serious question, and we decided to consider the spaces which may be the closest to our natural intuition: those which generalize to the sphere case the classical approximation properties used to define for instance Sobolev spaces. In this section we summarize the main properties of Besov spaces which will be used in the sequel, as established in [START_REF] Narcowich | Decomposition of Besov and Triebel-Lizorkin spaces on the sphere[END_REF]. Let f : S 2 → R be a measurable function. We define

E k (f, π) = inf P ∈P k f -P π ,
the L π distance between f and the space of polynomials of degree k. The Besov space B s π,r is defined as the space of functions such that

f ∈ L π and ∞ k=0 (k s E k (f, π)) r 1 k 1/r < +∞. Remarking that k → E k (f, π) is decreasing, by a standard condensation argument this is equivalent to f ∈ L π and ∞ j=0 (2 js E 2 j (f, π)) r 1/r < +∞.
and the following theorem states that as it is the case for Besov spaces in R d , the needlet coefficients are good indicators of the regularity and in fact Besov spaces of S 2 are Besov bodies, when expressed using the needlet expansion.

Theorem 1. Let 1 ≤ π ≤ +∞, s > 0, 0 ≤ r ≤ +∞. Let f a measurable function and define f, ψ j,η = S d f (x)ψ j,η (x) dx def = β j,η ,
provided the integrals exist. Then f belongs to B s π,r if and only if, for every j = 1, 2, . . . ,

η∈X j (|β j,η | ψ j,η π ) π 1/π = 2 -js δ j ,
where (δ j ) j ∈ r .

As has been seen above,

c2 2j( 1 2 -1 π ) ≤ ψ j,η π ≤ C2 2j( 1 2 -1 π ) ,
for some positive constants c, C, the Besov space B s π,r turns out to be a Banach space associate to the norm

(30) f B s π,r := (2 j[s+2( 1 2 -1 π )] (β jη ) η∈Z j π ) j≥0 r < ∞, and
Using standard arguments (reducing to comparisons of l q norms), it is easy to prove the following embeddings:

B s π,r ⊂ B s p,r for p ≤ π B s π,r ⊂ B s-2( 1 π -1 p ) p,r for π ≤ p and s > 2( 1 π - 1 p ). (31)
Moreover, it is also true that for s > 2 π , if f belongs to B s π,r , then it is continuous, and as a consequence bounded.

In the sequel we shall denote by B s π,r (M ) the ball of radius M of the Besov space B s π,r .

4. Needlet algorithm: thresholding needlet coefficients. The first step is to construct a needlet system (frame) {ψ jη : η ∈ Z j , j ≥ -1} as described in section 3.

The needlet decomposition of any f ∈ L 2 (S 2 ) takes the form

f = j η∈Z j (f, ψ jη ) L 2 (S 2 ) ψ jη .
Using Parseval's identity, we have

β jη = (f, ψ jη ) L 2 (S 2 ) = lm f l m ψ lm jη with f l m = (f, Y l m ) and ψ lm jη = (ψ jη , Y l m ). Thus (32) βjη = lm f l,N m ψ lm jη ,
is an unbiased estimate of β jη . We recall that f l,N m has been defined in [START_REF] Terras | Harmonic analysis on symmetric spaces and applications[END_REF]. Notice that from the needlet construction (see the previous section) it follows that the sum above is finite. More precisely, ψ lm jη = 0 only for 2 j-1 < l < 2 j+1 .

Let us consider the following estimate of f :

f = J j=-1 η∈Z j t( βjη )ψ jη ,
where t is a thresholding operator defined by

t( βjη ) = βjη I{| βjη | ≥ κt N |σ j |} with (33) t N = log N N , ( 34
)
σ 2 j = M 2 ln | m ψ lm jη f l ε -1 mn | 2 . ( 35
)
Here κ is a tuning parameter of the method which will be properly selected later on. M is such that f ∞ ≤ M . Notice that the thresholding depends on the resolution level j through the constant σ j which will also be specified later on, and the same with regard to the upper level of details J.

Performances of the procedure.

Theorem 2. Let 1 ≤ p < ∞, ν > 0, and let us assume that

(36) σ 2 j := M 2 ln | m ψ lm jη f l ε -1 mn | 2 ≤ C2 2jν , ∀ j ≥ 0.
Let us take κ 2 ≥ 16p and

2 J = d[t N ] -1 (ν+1) with t N as in (34) and d is a positive constant. Then if π ≥ 1, s > 2/π, r ≥ 1 (with the restriction r ≤ π if s = (ν + 1)( p π -1)
), there exists a constant C such that:

(37) sup

f ∈B s π,r (M ) E f -f p p ≤ C(log(N )) p-1 [N -1/2 log(N )] µp , where µ = s s + ν + 1 , if s ≥ (ν + 1)( p π -1) µ = s -2/π + 2/p s + ν -2/π + 1 , if 2 π < s < (ν + 1)( p π -1).
The proof of this theorem is given in section 6.

Remarks 1. The rates of convergence found here are standard in inverse problems. They can be related to rates found in [START_REF] Kim | Optimal spherical deconvolution[END_REF] in the same deconvolution problem, with a L 2 loss and constraints on the spaces comparable to B 2 s2 (M ). In the deconvolution problem on the interval, similar rates are found even for L p losses (with standard modifications since the dimension here is 2 instead of 1): see for instance Johnstone et al. ( 2004) [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF]. These results are proved to be minimax (see [START_REF] Kim | Optimal spherical deconvolution[END_REF] [START_REF] Kim | Optimal spherical deconvolution[END_REF]) up to logarithmic factors, for the case p = 2 with a B 2 s2 (M ) constraint on the object to estimate. 2. It is worthwhile to notice that the procedure is adaptive, meaning that it does not require a priori knowledge on the regularity (or sparsity) of the function. It also adapt to non homogeneous smoothness of the function. The logarithmic factor is a standard price to pay for adaptation. 3. The parameter ν appearing here is often called degree of ill-posedness of the problem (DIP). It appears here through condition (36) which is essential in this problem. In [START_REF] Kim | Optimal spherical deconvolution[END_REF] for instance, and very often in diverse inverse problems, this DIP parameter is introduced with the help of the eigenvalues of the operator (i.e. here the discrepancy of the coefficients of f ε in its expansion along the spherical harmonics). In the following subsection, we prove that (36) is in fact a consequence of the standard 'ordinary smooth' condition. 36) and the smoothness of f ε . Following Kim and Koo (2002) [START_REF] Kim | Optimal spherical deconvolution[END_REF] (condition 3.6), we can define the smoothness of f ε spectrally. We place ourselves in the 'ordinary smooth' case

Condition (

(38) ( f l ε ) -1 op ≤ d 0 l ν and f l ε op ≤ d 1 l -ν as l → ∞,
for some positive constants d 0 , d 1 and nonnegative constant ν, and where the operator norm of the rotational Fourier transform f l ε is defined as

f l ε op = sup h =0, h∈E l f l ε h 2 h 2 ,
E l being the (2l+1)-dimensional vector space spanned by {Y l m : -l ≤ m ≤ l}. The following proposition states that condition (35) is satisfied in the ordinary smooth case by the needlets system.

Proposition 1. If f l ε op ≤ d 0 l -ν , then |σ j | 2 := M 2 ln | m ψ lm jη f l ε -1 mn | 2 ≤ C2 2jν , ∀ j ≥ 0.
Proof. Since ψ lm jη = 0 only for 2 j-1 < l < 2 j+1 ,

2 j+1 l=2 j-1 l n=-l | m ψ lm jη f l ε -1 mn | 2 = 2 j+1 l=2 j-1 ( f l ε ) -1 ψ jη 2 2 ≤ 2 j+1 l=2 j-1 ( f l ε ) -1 2 op ψ jη 2 2
which proves the result using inequality (25).

We now give a brief review of some examples of smooth distributions which are discussed in depth in Healy et al (1998) [START_REF] Dennis | Spherical deconvolution[END_REF] and [START_REF] Kim | Optimal spherical deconvolution[END_REF] [START_REF] Kim | Optimal spherical deconvolution[END_REF].

4.2.1. Rotational Laplace distribution. This distribution can be viewed as an exact analogy on SO(3) of the Laplace distribution on R. Spectrally, for some ρ 2 > 0, this distribution is characterized by

(39) f l ε,mn = (1 + ρ 2 l(l + 1)) -1 δ mn ,
for -l ≤ m, n ≤ l and l = 0, 1, ..., and where δ mn = 1 if m = n and 0 otherwise.

The Rosenthal distribution.

This distribution has its origin in random walks in groups (for details see [START_REF] Rosenthal | Random rotations: characters and random walks on SO(N )[END_REF] [START_REF] Rosenthal | Random rotations: characters and random walks on SO(N )[END_REF]).

If one considers the situation where f ε is a p-fold convolution product of conjugate invariant random for a fixed axis, then, [START_REF] Rosenthal | Random rotations: characters and random walks on SO(N )[END_REF] [START_REF] Rosenthal | Random rotations: characters and random walks on SO(N )[END_REF] (p.407) showed that f l ε,mn = sin(l + 1/2)θ (2l + 1) sin θ/2 p δ mn , for -l ≤ m, n ≤ l and l = 0, 1, ... and where 0 < θ ≤ π and p > 0.

5. Practical performances. In this section we produce the results of numerical experiments on the sphere S 2 . The sets of cubature points ξ jη in the simulations that follow have been generated with the HEALPix pixelisation. For each resolution level j, the HEALPix pixelisation gives 12.2 2j cubature points. In the two examples below we considered samples of cardinality N = 1500. The maximal resolution level J is taken such that J = (1/2) log 2 N log N . In order not to have more cubature points than observations we set J = 3 for N = 1500. We recall the expression of the estimate of the needlets coefficients of the density of interest:

(40) βjη = 1 N λ jη 2 j+1 l=2 j-1 b(l/2 j ) l m=-l Y l m (ξ jη ) l n=-l f l ε -1 ,mn N u=1 Ȳ l n (Z u ),
where the cubature weight is equal to λ jη = 4π/(12.2 2j ). We replace the rotational Fourier transform ( f l ε ) mn := f l ε,mn (defined in ( 6)) by its empirical version. Indeed, we suppose that the noise is unknown, this is a situation which is very likely to occur for instance in the context of astrophysics. We precise again that f l ε -1 ,mn denotes the (m, n) element of the matrix ( f l ε ) -1 := f l ε -1 which is the inverse of the (2l + 1) × (2l + 1) matrix ( f l ε ). In order to get the empirical version f l,N ε -1 ,mn of f l ε -1 ,mn , we have first to compute the empirical matrix ( f l,N ε ) then to inverse it to get the matrix

( f l,N ε ) -1 := f l,N ε -1 . The (m, n) entry of the matrix ( f l,N ε
) is given by the formula:

f l,N ε,mn = 1/N N j=1 D l m,n (ε j ),
where the rotational harmonics D l m,n have been defined in [START_REF] Johnstone | Wavelet deconvolution in a periodic setting[END_REF]. The ε j 's are i.i.d realizations of the variable ε ∈ SO [START_REF] Dennis | Spherical deconvolution[END_REF]. For the generation of the random variable ε ∈ SO(3), we chose the Oz as the rotation axis and an angle φ following a uniform law. Various cases are investigated for the angle φ, that is to say a uniform law with different supports such as [0, π/8], [0, π/4], [0, π/2]. The more the support of the uniform law is large, the more the level noise will be intense. This particular choice of rotation matrix entails in the decomposition of an element of SO(3) (see formula (3)) the angles ψ and θ to be both equal to zero. For this specific setting of perturbation, we deduce the following form for the rotational harmonics:

D l m,n (ε j ) = P l mn (1)e -iφ j = δ mn e -iφ j ,
where φ j ∼ U [0, a] and a is a positive constant which will be specified later.

For the reconstruction of the density f , we recall that

f := 1 |S 2 | + J j=0 12.2 2j η=1 βjη ψ jη I {| βjη |≥κt n σ j } ,
the expression of σ j is given in (35), and t N = log(N )/N . The constant M which appears in the expression of σ j is such that f ∞ ≤ M .

Example 1. In this first example, we consider the case of the uniform density f = 1 4π . It is easy to verify that β jη = f, ψ jη L 2 = 0 for every j and every η. Following Baldi et al. (2009) [START_REF] Baldi | Adaptive density estimation for directional data using needlets[END_REF], a simple way of assessing the performance of the procedure is to count the number of coefficients surviving thresholding.

j = 0 j = 1 j = 2 j = 3 κ0 = 0.08 2 1 75 337 κ 0 = 0.29 0 0 2 8 κ 0 = 0.34 0 0 0 2

Table 1 number of coefficients surviving thresholding for various values of κ

0 , φ ∼ U [0, π/8]. j = 0 j = 1 j = 2 j = 3 κ0 = 0.05 4 35 135 556 κ 0 = 0.17 0 4 24 103 κ0 = 0.30 0 0 1 5 κ 0 = 0.34 0 0 0 3

Table 2 number of coefficients surviving thresholding for various values of κ

0 , φ ∼ U [0, π].
We precise that both in the cases of an angle following a law U [0, π/8] or U [0, π], all the coefficients are killed for κ 0 = 0.38. Accordingly, we can conclude that the thresholding procedure based on spherical needlets is very efficient.

Example 2. We will now deal with the example of a density of the form f (ω) = ce -4|ω-ω 1 | 2 , with ω 1 = (0, 1, 0) and c = 1/0.7854. With this choice of density, it turns out that f ∞ = 1/0.7854 = 1.2732. Hence we can set M such that M = 1.2732. The graph of f in the spherical coordinates

(Φ, Θ) (Φ = longitude, 0 ≤ Φ ≤ 2π, Θ = colatitude, 0 ≤ Θ ≤ π) is given in Fig 2.
We also plot the noisy observations for different cases of perturbations. We remark that for rather big rotation angles such that φ ∼ U [0, π/4] or φ ∼ U [0, π/2], the observations tend to be spread over a large region on the sphere and not being concentrated in a specific region any more. Consequently, denoising might prove to be difficult. In the context of the deconvolution on the sphere, a large amount of noise corresponds to a rotation around the Oz axis with a large angle. A fundamental issue in the field of astrophysics is to detect the place of the peak of the bell which in the present density case is localized in (Θ = π/2, Φ = π/2). For each case of noise, we plot the observations both on the sphere and on the flattened sphere and give the reconstructed density in the spherical coordinates. At a closer inspection, we notice that the position of the peak of the estimated bell is pretty well localized whatever the amount of noise. Nevertheless in the case of the law U [0, π/2], the longitude coordinate of the peak tends to slightly move, its colatitude coordinate remaining well localized. Therefore even if in the case of rather big rotations such that φ ∼ U [0, π/4] and φ ∼ U [0, π/2], our estimation procedure allows us to detect the position of the peak. Of course, one remarks that the base of the bell tends to become a bit larger when the noise increases, this is due to the fact that the observations are not concentrated in a specific region any longer, but the genuine form of the density is well preserved.

6. Proof of Theorem 2 . In this proof, C will denote an absolute constant which may change from one line to the other.

We begin with the following proposition Proposition 2. For any p ≥ 1,

E| βjη -β jη | p ≤ c[ σ 2 j N ] p 2 (41)
P[| βjη -β jη | ≥ σ j κt N ] ≤ cN -γ(κ) (42) γ(κ) = -κ 2 /4 (43)
To prove (41), we will use Rosenthal inequality:

βjη -β jη = 1 N N i=1 G(Z i ) -EG(Z i ) G(x) = lm ψ lm jη n f l ε -1 mn Y l n (x).
Hence,

V arG(Z) ≤ S 2 |G(ω)| 2 f Z (ω)dω ≤ M S 2 |G(ω)| 2 dω, since f Z ∞ ≤ M (
this is a consequence of the fact that s > 2/π which in turns implies that f is continuous, and so is f Z on the sphere). Then using Parseval:

S 2 |G(ω)| 2 dω = ln | m ψ lm jη f l ε -1 mn | 2 = σ 2 j .
Moreover, using Cauchy Schwarz inequality, and noticing again that ψ lq jη = 0 only for 2 j-1 ≤ l ≤ 2 j+1 :

|G(ω)| ≤ [ ln | m ψ lm jη f l ε -1 mn | 2 ] 1/2 [ lm I{2 j-1 ≤ l ≤ 2 j+1 }|Y l m (ω)| 2 ] 1/2 ≤ Cσ j 2 j/2 , since m |Y l m (ω)| 2 = 2l+1
4π because of the addition formula ( see [START_REF] Kim | Optimal spherical deconvolution[END_REF] [START_REF] Kim | Optimal spherical deconvolution[END_REF]) inequality (6.1)). Then using Rosenthal inequality, we get (we recall that 2 j ≤ N 1/2 for j ≤ J):

E| 1 N N i=1 G(Z i )-EG(Z i )| p ≤ C[ 1 N p σ 2 j (σ j 2 j/2 ) p-2 + 1 N p (N σ 2 j ) p 2 ] ≤ Cσ p j N -p/2 .
To prove (42), we use Bernstein inequality:

P[| βjη -β jη | ≥ σ j κt N ] ≤ 2 exp - κ 2 σ 2 j log N 2(σ 2 j + 1 3 κσ j t N σ j 2 j/2 π -1/2 ) ≤ 2N -κ 2 /4 .
This ends up the proof of the proposition. Now, to get the result of Theorem 2, we begin by the following decomposition :

E f -f p p ≤ 2 p-1 {E J j=-1 η∈Z j (t( βjη ) -β jη )ψ jη p p + j>J η∈Z j β jη ψ jη p p } =: I + II
The term II is easy to analyse, as follows. We observe first that since B s π,r (M ) ⊂ B s p,r (M ) for π ≥ p, this case will be assimilated to the case π = p and from now on, we will only consider π ≤ p. Since f belongs to B s π,r (M ), using the embedding results recalled above in (31), we have that f also belongs to B s-( 2π -2 p ) p,r (M ), for some constant M and for π ≤ p. Hence j>J η∈Z j

β jη ψ jη p ≤ C2 -J[s-2( 1 π -1 p )] .
Then we only need to verify that

s-2( 1 π -1 p ) ν+1
is always larger that µ, which is not difficult. Indeed, on the first zone s ≥ (ν + 1)(p/π -1). So, s + ν + 1 ≥ (ν + 1) p π which entails that

s (s+ν+1) ≤ s (ν+1) p π . We need to check that s -2( 1 π -1 p ) ≥ sπ p . We have that s -2 π + 2 p -sπ p = 2( sπ 2 -1)( 1 π -1 p ) ≥ 0 since s ≥ 2
π and p ≥ π. On the second zone, we obviously have that

s-2( 1 π -1 p ) ν+1 is always larger that µ = s-2/π+2/p s+ν-2/π+1 .
Bounding the term I is more involved. Using the triangular inequality together with Hölder inequality, and property (28) for the second line, we get

I ≤ 2 p-1 J p-1 J j=-1 E η∈Z j (t( βjη ) -β jη )ψ jη p p ≤ 2 p-1 J p-1 C J j=-1 η∈Z j E|t( βjη ) -β jη | p ψ jη p p .
Now, we separate four cases :

J j=-1 η∈Z j E|t( βjη ) -β jη | p ψ jη p p = J j=-1 η∈Z j E|t( βjη ) -β jη | p ψ jη p p I{| βjη | ≥ κt N σ j } +I{| βjη | < κt N σ j } ≤ J j=-1 η∈Z j E| βjη -β jη | p ψ jη p p I{| βjη | ≥ κt N σ j } I{|β jη | ≥ κ 2 t N σ j } + I{|β jη | < κ 2 t N σ j } +|β jη | p ψ jη p p I{| βjη | < κt N σ j } I{|β jη | ≥ 2κt N σ j } +I{|β jη | < 2κt N σ j } = : Bb + Bs + Sb + Ss.
Using Proposition 2, we have

Bb ≤ C J j=-1 η∈Z j σ p j N -p/2 ψ jη p p I{|β jη | ≥ κ 2 t N σ j } Ss ≤ J j=-1 η∈Z j |β jη | p ψ jη p p I{|β jη | < 2κt N σ j }.
And, using in addition ( 25) and ( 36)

Bs ≤ J j=-1 η∈Z j [E| βjη -β jη | 2p ] 1/2 [P{| βjη -β jη | ≥ κ 2 t N σ j }] 1/2 ψ jη p p I{|β jη | < κ 2 t N σ j } ≤ C J j=-1 η∈Z j σ p j N -p/2 2 1/2 N -κ 2 /32 ψ jη p p I{|β jη | < κ 2 t N σ j } ≤ C J j=-1 2 jp(ν+1) N -p/2 N -κ 2 /32 ≤ CN -κ 2 /32 .
Now, since f belongs to B s π,r (M ), using again Proposition 2 and (31), we have

Sb ≤ J j=-1 η∈Z j |β jη | p ψ jη p p P{| βjη -β jη | ≥ 2κt N σ j }I{|β jη | ≥ 2κt N σ j } ≤ J j=-1 η∈Z j |β jη | p ψ jη p p 2N -κ 2 I{|β jη | ≥ 2κt N σ j } ≤ C J j=-1 2 -jp(s-2( 1 π -1 p ) + ) N -κ 2 ≤ CN -κ 2 2 2J ≤ CN -κ 2 + 1 (ν+1) .
It is easy to check that in any cases if κ ≥ 1 -p the terms Bs and Sb are smaller than the rates announced in the theorem.

We have using Proposition 2, (25) and condition (36) for any p ≥ z ≥ 0:

Bb ≤ CN -p/2 J j=-1 σ p j 2 j(p-2) η∈Z j I{|β jη | ≥ κ 2 t N σ j } ≤ CN -p/2 J j=-1 σ p j 2 j(p-2) η∈Z j |β jη | z [t N σ j ] -z ≤ Ct N p-z J j=-1 2 j[ν(p-z)+p-2] η∈Z j |β jη | z .
Also, for any p ≥ z ≥ 0

Ss ≤ C J j=-1 2 j(p-2) η∈Z j |β jη | z σ p-z j [t N ] p-z ≤ C[t N ] p-z J j=-1 2 j(ν(p-z)+p-2) η∈Z j |β jη | z .
So in both cases we have the same bound to investigate. We will write this bound on the following form (forgetting the constant) :

A+B := t N p-z 1 [ j 0 j=-1 2 j[ν(p-z 1 )+p-2] η∈Z j |β jη | z 1 ]+t N p-z 2 [ J j=j 0 +1 2 j[ν(p-z 2 )+p-2] η∈Z j |β jη | z 2 ].
The constants z i and j 0 will be chosen depending on the cases, with the only constraint p ≥ z i ≥ 0.

We recall that we only need to investigate the case p ≥ π, since when p ≤ π, B s πr (M ) ⊂ B s pr (M ). Let us first consider the case where s ≥ (ν + 1)( p π -1), put q = p(ν + 1) s + ν + 1 , and observe that on the considered domain, q ≤ π and p > q. In the sequel it will be useful to observe that we have s = (ν + 1)( p q -1). Now, taking z 2 = π, we get:

B ≤ t N p-π [ J j=j 0 +1
2 j[ν(p-π)+p -2] η∈Z j

|β jη | π ]. Now, as p q - 2 π + ν( p q -1) = s + 1 - 2 π and η∈Z j |β jη | π = 2 -jπ(s+1-2 π ) τ π j ,
with (τ j ) j ∈ l r (this last thing is a consequence of the fact that f ∈ B s π,r (M )), we can write :

B ≤ t N p-π j=j 0 +1 2 jp(1-π q )(ν+1) τ π j ≤ Ct N p-π 2 j 0 p(1-π q )(ν+1) .
The last inequality is true for any r ≥ 1 if π > q and for r ≤ π if π = q.

Notice that π = q is equivalent to s = (ν + 1)( p π -1). Now if we choose j 0 such that 2 j 0 p q (ν+1) ∼ t N -1 we get the bound t N p-q , which exactly gives the rate announced in the theorem for this case. As for the first part of the sum (before j 0 ), we have, taking now z 1 = q, with q ≤ π (and also q ≤ p since we investigate the case p ≥ π), so that [

1 2 2j η∈Z j |β jη | q] 1 q ≤ [ 1 2 2j η∈Z j |β jη | π ]
1 π , we get

A ≤ t N p-q [ j 0 -1
2 j[ν(p-q)+p-2] η∈Z j

|β jη | q] ≤ t N p-q [ j 0 -1 2 j[ν(p-q)+p-2q π ] [ η∈Z j |β jη | π ] q π ] ≤ t N p-q j 0 -1 2 j[(ν+1)p(1-q q )] τ q j ≤ Ct N p-q 2 j 0 p[(ν+1)(1-q q )] ≤ Ct N p-q .
The last two lines are valid if q is chosen strictly smaller than q (this is possible since π ≥ q). Let us now consider the case where s < (ν + 1)( p π -1), and choose now

q = p ν + 1 -2 p s + ν -2 π + 1 .
In such a way that we easily verify that p -q = p s-2/π+2/p 1+ν+s-2/π , q -π = (p-π)(1+ν)-πs s+ν-2 π +1

> 0. Furthermore we also have s + 1 -2 π = p q -2 q + ν( p q -1). Hence taking z 1 = π and using again the fact that f belongs to B s π,r (M ),

A ≤ t N p-π [ j 0 -1 2 j[ν(p-π)+p-2] η∈Z j |β jη | π ] ≤ t N p-π j 0 -1 2 j[(ν+1-2 p ) p q (q-π)] τ π j ≤ Ct N p-π 2 j 0 [(ν+1-2 p ) p q (q-π)] .
This is true since ν + 1 -2 p is also strictly positive since ν + 1 > s p π -1 ≥ 2 p-π ≥ 2 p . If we now take 2 j 0 p q (ν+1-2 p ) ∼ t N -1 we get the bound

t N p-q ,
which is the rate announced in the theorem for this case. Again, for B, we have, taking now z 2 = q > q(> π)

B ≤ t N p-q [ J j=j 0 +1
2 j[ν(p-q)+p-2] η∈Z j

|β jη | q].
But η∈Z j |β jη | q ≤ C2 -j q(s+1-2 π ) τ q j , and s + 1 -2 π = p q -2 q + ν( p q -1), hence

B ≤ Ct N p-q j=j 0 +1
2 j[(ν+1-2 p ) p q (q-q)] τ q j ≤ Ct N p-q 2 j 0 [(ν+1-2 p ) p q (q-q)] ≤ Ct N p-q , which completes the proof of Theorem 2.

Fig 1 .

 1 Fig 1. A spherical needlet.

Fig 2 .Fig 3 .Fig 4 .

 234 Fig 2. The target density

Fig 5 .Fig 6 .

 56 Fig 5. Observations φ ∼ U [0, π/4].

Fig 7 .Fig 8 .

 78 Fig 7. Observations φ ∼ U [0, π/2].

Acknowledgements. The authors would like to thank Erwan Le Pennec for many helpful discussions and suggestions concerning the simulations.


Benjamin Simon
email: simon@rm.mce.uec.ac.jp

Ryuki Sato
email: sator@rm.mce.uec.ac.jp

Jean-Yves Choley
email: jean-yves.choley@supmeca.fr

Aiguo Ming
email: ming@mce.uec.ac.jp

Development of a Bio-inspired Flexible Tail System

Keywords: Robot, Flexible tail, Bio-inspired, Cable-driven mechanism, Jumping

In the animal kingdom, many mammals are using their tails to assure their stability while running, jumping, falling, etc. Researches have been performed in order to add tails on wheeled or legged robots in order to improve their stability. However, those tails are often far from the biological structure and resumed to a simple rigid tube with a mass on the extremity. On the other hand the biological tail is composed of many bones moving one on the other and actuated with an important number of muscles. Therefore replicating this structure would result in big and heavy systems using many actuators that would reduce the physical performances of the robot. In this paper, we developed a new bio-inspired cable-driven tail system capable of swinging a flexible tail from base to tip around the pitching axis using a single actuator. First we designed a new flexible tail mimicking the structure of the animal tail. Then we developed a new actuation system using the combination of cams and drive wheels to realize the actuation of 8 cables using a single actuator. Finally, we measured the efforts generated by the swinging motion of the tail and perform experiments to observe its effects on the jumping motion.

I. INTRODUCTION

Animals are good sources of inspiration for mechanisms with their bodies built over years of evolution. They are capable of fast running and high jumping but over that they are capable of staying balanced during those performances. The source of their stability can often be found in the use of a tail to change the orientation of their body and stay balanced. For example, cheetahs use their tails to stay stable and perform quick turns while running at high speed, kangaroos use their tails as counterweight while hopping, lizards and cats use their tails to reposition their bodies to be in the optimal position for landing after a jump or a fall, the cats being also able to use their tails to stay balanced in case of lateral disturbances during the walk [START_REF] Dawson | Muscular Anatomy of The Tail of the Western Grey Kangaroo, Macropus Fuliginosus[END_REF]- [START_REF] Liu | A Bio-Inspired Hopping Kangaroo Robot with an Active Tail[END_REF].

Some studies were therefore conducted to implement tail systems on robots and use their effects for stability or motion improvement [START_REF] Briggs | Tails in Biomimetic Design: Analysis, Simulation, and Experiment[END_REF]- [START_REF] Wada | Anatomical Structure and Action of the Tail Muscles in the Cat[END_REF]. However, the structures of the tails used in those studies usually consist in a simple rigid tube with a mass placed on its extremity and actuated on a single axis, either the pitching or the rolling axis. On the other hand, the biological tail is composed of many small vertebrae linked with ligaments and surrounded by many muscles and tendons resulting into a very flexible limb that can be moved in three dimensions [START_REF] Dawson | Muscular Anatomy of The Tail of the Western Grey Kangaroo, Macropus Fuliginosus[END_REF] [12]- [START_REF] Chang-Siu | A Lizard-Inspired Active Tail Enables Rapid Maneuvers and Dynamic Stabilization in a Terrestrial Robot[END_REF]. Using a rigid structure helps in simplifying the calculations but also generates less inertia than a multi-body structure would do with a decomposition of the motion. Indeed, with the vertebrae sliding one on the other, the animal tail can generate a whip-like motion from the base to the tip to generate more inertia just like golfers or pitchers would decompose their motion from the torso to the hands to transfer the maximum of energy.

In our previous study [START_REF] Sato | Development of a Flexible Tail for Legged Robot[END_REF], a flexible tail was built in order to improve our biped robot's jumping capacity using the advantage of its flexibility. This flexible tail was composed of multiple segments linked with tension springs and was actuated with a single DC motor placed on its base. Since only the first block of the tail was actuated, there were some delay between the rotation of the actuator and the end of the swinging motion of the tail due to the lack of active control of the deformation that also result in an important imbalance of the robot. To avoid this issue, common flexible tails mechanisms use cables to actively control the deformation of the structure [START_REF] Rone | Static Modeling of a Multi-Segment Serpentine Robotic Tail[END_REF] [START_REF] Rone | Continuum Robotic Tail Loading Analysis for Mobile Robot Stabilization and Maneuvering[END_REF]. However, those cables are usually attached to the tip of the tail, which implies that the motion comes from the tip to the base. Therefore the tail would lose the advantage of the whip-like motion. To avoid that, additional cables can be attached to middle parts to decompose the tail in many sections and have a succession of bending motions going from base to tip. But increasing the number of cables also results in an increase of the number of actuators required thus an increase of the mass and the electric consumption that would reduce the performances of the robot on which the system is mounted.

In this study, we are aiming at the development of a new flexible tail system using a bio-inspired structure. This new system should be able to realize a bending of the tail from base to tip with an active control of the deformation while using the minimum of actuator possible. The system is designed to be mounted on the biped robot previously designed in our group [START_REF] Ming | Development of Robot Leg Composed of Parallel Linkage and Elastic Spring for Dynamic Locomotion[END_REF]. This paper is organized as follow. The Section II details the design of the bio-inspired flexible tail. Section III describes the new actuation system for the tail. Section IV shows the efforts generated by the swinging of the tail and the use of the swinging motion for jumping.

II. THE FLEXIBLE TAIL

A. Design of the flexible tail

We decided to base the design of our flexible tail on the biological structure of the tails of kangaroos and cats. For these two animals, the structure of the tail is similar: a large number of vertebrae placed in series, connected to each other with ligaments and surrounded by several muscles that can be divided into 4 groups: dorsal and ventral for the vertical motion, lateral left and lateral right for the horizontal motion as shown in Fig. 1. Although the tail is composed of a large number of vertebrae, they do not move individually one after another but by group since only one muscle can be linked to several of them [START_REF] Wada | Anatomical Structure and Action of the Tail Muscles in the Cat[END_REF].

For our tail shown in Fig. 2a we decided not to replicate the skeletal structure of the animal tail by using a large number of individually movable 'vertebra' parts but rather to copy the movement of the vertebrae moving together. To do this we use an accordion shape that uses the combination of several small discrete deformations to obtain a more global and continuous deformation, see Fig. 2b. This accordion shape rotate along the tail in order to obtain the same deformation in the horizontal plane only in the vertical plane. However, this form allows a compression of the tail in its length, which differs from the rigid behavior of the vertebrae and poses problems for the tension of the cables used to actuate the tail. Therefore we added a rigid core to the tail to prevent any compression of the tail in its length.

We also decided to copy the muscle structure of the animal tail for the actuation of our tail. Where common systems are using a 3-wire configuration for deformation in 3 dimensions, we use 4 cables placed on the 4 sides of the tail. As a result, we can can use these cables in pairs and place them in opposition to separate the vertical and horizontal movements of the tail. In order to have a better control of the deformation of the tail, we divided it into 4 identical sections, each actuated by the 4 cables fixed to it. We therefore use a total of 16 cables for the actuation of the tail. For this study we limit ourselves to the movement of the tail in the vertical plane and thus are using only 8 cables, 4 to bend the tail upwards and 4 to bend it downwards.

B. Properties of the tail

The dimensions of the designed tail are based on the dimension of our previous tail [START_REF] Sato | Development of a Flexible Tail for Legged Robot[END_REF] as well as on the dimensions of our previously developed biped robot [START_REF] Ming | Development of Robot Leg Composed of Parallel Linkage and Elastic Spring for Dynamic Locomotion[END_REF] on which the tail is mounted on to be able to have the tail bent down with flexed legs without it touching the ground. The tail is 210 mm long with four 23x23x50 mm accordion shape sections and five 30x30x2 mm rings, four to attach the cable and one to fix the tail to its actuation system. The tail is 3D printed using Nylon 12 powder melt with laser using the Multi-Jet Fusion method to allow a good solidity of the part and the same deformation in all direction. Once the tail printed we could confirm that the deformation is the same for the four sections and that the tail can reach a 90 degrees deformation. The mass of the tail is 47 g which is half the mass of the previous spring-tail and represent 2% of the mass of the biped robot. Thanks to the accordion shape, the mass of the tail is distributed along the tail and not concentrate around the 5 disks. The tail is lighter in order to reduce the mass of the system added to the robot and thanks to the active control of its deformation a higher speed can be obtained to compensate this loss in the generated inertia. The lengths of cable that needs to be pulled and released to bend one section can be calculated using the formula

l pull = l 0 -2 ( d φ -h ) sin ( φ 2
) ( 1
)
l release = hφ (2)
with l 0 the length of cable with the section not bent, d the length of the central core between the middle of a ring and the next one, φ the desired bending angle of the section in radians and h the distance between the center of the tail and the hole where the cable is attached. With l 0 = d = 52 mm, h = 13.5 mm, to reach a deformation of φ = π/8 for one section, 5.6 mm of cable need to be pulled and 5.3 mm need to be released on the opposite side. However because of the accordion shape, the shape of the bent tail is not a perfect circle arc and the length of cable that needed to be pulled when checking the deformation was 6.5 mm. This is the length that was used for the measurement of the tensions of the cables and the design of the actuation system. We used a force sensor placed on a linear guide to measure the pulling force that need to be applied on the cables to bend the sections of the tail. The required tensions are going from 5.8 N for the first section at the base to 3.6 N for the fourth section on the tip of the tail. We design the actuation system using the tension of the first cable for the four sections to keep a margin.

III. THE ACTUATION SYSTEM

The second step of this study consists in the designing of the actuation system for the tail. Usually, cable systems use three cables per section to control the deformation in three dimensions with one actuator for each cable [START_REF] Rone | Static Modeling of a Multi-Segment Serpentine Robotic Tail[END_REF] [START_REF] Rone | Continuum Robotic Tail Loading Analysis for Mobile Robot Stabilization and Maneuvering[END_REF]. For our tail, since we are using 16 cables that would mean using 16 actuators for the 4 sections to have a deformation in three dimensions. Since we are using the cables in pairs we can reduce the number of actuator to 4 for the vertical motion and 4 for the horizontal motion. This represents an important addition of mass and an important electric consumption due to the fact that the motors need to stay turned on to prevent the release of the cables and keep the sections bent in the desired position. Therefore we had to find a solution to reduce the number of actuators required for the actuation of the tail.

A. Design of the tail's actuation system

The main difficulty comes from the sequence that we want to use for the bending of the tail. Indeed to bend the tail from the tip to the base is an easy task that can be realized with two cables attached to the tip of the tail. When pulling a cable, the section on the extremity would bend first until it reaches its maximum of deformation. Then the previous sections would bend, followed by the next one up to the first section at the Fig. 4. Close look at the pulley -drive wheel system. The pulleys are placed between the the brake beneath it and the drive wheels above. The cables are rolled around the pulleys and fixed to it.

base. However, bending the tail from base to tip is a much more difficult challenge. When pulling the cable attached to the first section at the base, we also need to pull the cables of all the other sections in order to avoid the cables to be loosened with the compression of the first section. Therefore we need to be able to pull all the cables at once then, when the first section reaches its maximum of deformation, stop pulling the first cable while continuing to pull the others to continue the bending motion of the tail. Then stop pulling the second cable when its section reaches its maximum of deformation while continuing to pull the other, and repeat again the operation up to the tip. To have a smooth motion of the tail it is better not to stop the bending motion when we stop pulling a cable. Therefore, we need to be able to engage or disengage the motion of each individual pair of cable without stopping the complete system.

The transmission system that corresponds to this criteria is the drive wheel system. Using drive wheels, we can easily engage or disengage the motion by placing the wheels to actuate in contact with the drive wheels and use the friction for the transmission of the rotation. In our case, the wheels to actuate are the pulleys on which the cables are rolled on. As shown in Fig. 3 we placed 4 drive wheels fixed on a single axis and placed the 4 pulleys beneath it. Using this system, we can easily engage or disengage the motion of the cables rolled on the pulleys by pressing the pulleys in contact with the drive wheels or placing the pulleys away from the drive wheels.

In order to move the pulleys to place them in contact with the drive wheels or place it in contact with the brake to prevent their rotation we are using a cam system. Using the rotation of the cam, the cam follower is moved from a down position to an up position that can correspond to our two states, active rotation of the pulley or locked rotation of the pulley. In order to avoid using two actuators for the rotation of the drive wheels axis and the rotation of the cams axis we synchronized the two using gears, allowing us to use the same initial rotation and therefore use a single actuator. However, we need the pulleys to stay in the active position during the motion of the cables. Therefore we cannot use simple eccentric cams since the up state is too short. Instead we decided to use centered cylinders with steps to move the cam-followers from a low step position to a top step position and keep a state during a desired period. Yet in order to stay in the up position during the required period for the pulling of the necessary length of cable for the bending of a section, we would need to have cams with diameters which we cannot afford due to the dimensions of our system. Therefore, to overcome this issue the 4 cams were combined into a single horizontal disk with steps on and called the sequencing disk. Using the rotation of the disk, the camfollowers are moved from the low step where the pulley is pressed against the brake using a spring to the top step with the required length where the pulleys are pressed against the drive wheels as shown in Fig. 4.

Using this configuration a single actuator is required instead of two, one for the power transmission and one for the sequencing of the motion. First, the rotation of the actuator is transmitted to the sequencing disk in order to realize the sequencing of the deformation of the tail by moving the cam followers in the top steps in the desired order. Then using two gears stages, first with two bevel gears with a gear ratio of 1 then with two spur gears with a gear ratio of 2, the rotation of the sequencing disk axis is transmitted to the drive wheels axis. The drive wheels will then transmit the rotation to the pulleys using the pressure and friction between the two parts, the pulleys having a silicone surface and the drive wheels having a rubber surface. Moreover in order to prevent the slipping of the cables on the pulley due to the tension created by the bending of the tail the cables are attached to the pulleys. In addition, to prevent the slipping of the pulleys against the aluminum brake in the locked state, we placed a rubber band on the brake to increase the friction in addition to the pinching of the cables realized with the springs.

B. The sequencing disk

The sequencing disk (Fig. 5) is the central part of this actuation system. Everything can be linked to the rotation of the sequencing disk. The bending of a section of the tail depends on the length of cable pulled which also corresponds of the length of the steps on the sequencing disk as shown in this equation

L = 2πR d r 1 r 2 θ step 360 (3)
with L the length of cable pulled, R d the radius of the drive wheel, r 1 and r 2 the gear ratio of the bevel gears and spur gears respectively, θ step the length of the step in degree of rotation of the sequencing disk. Therefore, with the radius of the drive wheel fixed to 6 mm and the gears ratio fixed to 1 and 2 we need to have a 32 degree long step to pull the 6.5 mm of cables required for the complete bending of one section of the tail. Since the position of the tail is decided by the bending of each section (i.e. the length of cable pulled for each section) we can also link the position of the tail to the angle of the sequencing disk, 0 being the straight position of the tail, +128 degrees the complete bending up of the tail and -128 degrees the complete bending down of the tail (Fig. 6). The speed of the bending motion is also directly proportional to the speed of the sequencing disk rotation. Therefore to bend the tail in a desired position at a desired speed, we only need to input the angle of the sequencing disk and the rotational speed corresponding. More importantly, it is on this part that the sequence of the bending of the tail's sections is inputted. Due to the dimensions of the robot on which the system will be mounted, a maximum of 4 paths could be placed on the sequencing disk, thus the previous division of the tail into 4 sections. In order to realize the whip-like motion of the tail, the sequence placed on the sequencing disk corresponds to the bending of the sections of the tail from its base to the tip, from the straight position of the tail to its full bending up or down as illustrated in Fig. 7. Indeed, moving the tail from tip to base would make it rolls on itself which has two undesired consequences: first the length of the tail is at its minimum when the first section is moved which generated lower inertia as it is linked to the length of the rotating object. Second, since the motion is going from tip to base, the first section is not moving in a first time which also reduce the generated inertia. In the other hand, if we move the tail from base to tip then when the first section moves, the tail is in the maximal extension and the full tail is moved. Then with the bending going from base to tip, the decreasing length of the part of the tail that is moved is compensated with an increasing bending speed.

IV. EXPERIMENTS

A. Measurement of the generated efforts

In order to measure the efforts generated by the swinging motion of the tail we placed the complete system on top of a 6-axis force sensor, as shown in Fig. 8. On the six efforts that we measured, the three that interested us are the force F y that pulls the robot backward with the extension of the tail when it reaches the straight position or in the contrary push the robot forward at the end of the bending motion, the force F z that pulls the robot up or pushes it down depending on the direction of the motion, and the momentum M x created by those two forces and that acts on the pitching rotation of the robot.

Those efforts were measured for different bending patterns of the tail that are listed in Table-I. The results of the measurement are shown in Fig. 9. The two patterns that generates the maximum of efforts are the extension of the tail from the bent down position to the straight position and from the bent up position to the straight position. This can be explained by the increased of the length of the distance from the base of the tail to its center of mass, thus generating more inertia as well with the fact that the straight position is the rest position of the tail.

B. Jumping experiment

The developed tail system was mounted on the biped robot as shown in Fig. 10. Since the addition of the tail system must not have an important negative impact on the robot's jumping capacity we performed a jumping experiment to measure the impact of the new tail system on the robot. We performed several jumps with different patterns to use the previously measure force F z to help to compensate the mass of the actuation system. Using the flexible tail during the jumps, the robot could reach a maximum height of 310 mm at the hip joint which is inferior to the 330 mm reachable without the tail system, see Fig. 11. Moreover, the same maximum height was reached without actuating the tail. The flexible tail is too light to have a real impact on the robot's jump height and cannot help compensating the addition of the actuation system to the robot.

V. CONCLUSION

In this paper we presented our new tail system composed of a bio-inspired flexible tail and its actuation system capable of moving 4 pairs of cables according to a desired sequence using a single actuator with the use of the combination of drive wheels and a cam system. With this actuation system, a single sequence can be placed on the sequencing disk at a time but it is enough for our desired use of the system as a simple swinging tail. However the mass of the tail system would need to be further reduced to improve the effect of the tail of the robot. The next step of this research will be to use the tail system for the stabilization of the robot in case of perturbations not predicted by the motion planning. The tail will then be used for the static stabilization with the displacement of the center of mass of the system as well as the dynamic stabilization of the robot using the generated momentum . Afterwards, the use of a second actuation system would allows the tail to have a deformation in the horizontal plane.

Fig. 1 .

 1 Fig. 1. Muscles of the kangaroo tail at different sections with the 4 categories, dorsal on top, ventral at the bottom and laterals left an right on the sides [1].

Fig. 2 .

 2 Fig. 2. (a) The 3D printed tail using the accordion shape with a central core. (b) The 3D model of the tail fully bent with C i the cables attached to the section i.

Fig. 3 .

 3 Fig. 3. 3D model of the actuation system.

Fig. 5 .

 5 Fig. 5. 3D model of the sequencing disk with the 4 paths for the 4 sections.

Fig. 6 .

 6 Fig. 6. Positions of the cam followers on the sequencing disk (in red) and the positions of the tail corresponding as the sequencing disk turns clockwise.

Fig. 7 .

 7 Fig. 7. Sequence of the bending of the tail from the first section to the fourth section with the position of the stems during the rotation of the sequencing disk from -128 to +128 degrees.

Fig. 8 .

 8 Fig. 8. Setup for the measurement of the generated efforts. 1: 6 axis force sensor. 2: Power source. 3: Mbed micro-controller. 4: Motor driver. 5: Actuation system. 6: Flexible tail. 7: Wires' guide from the actuation system to the tail.

Fig. 9 .

 9 Fig. 9. Values of the measured efforts with D-Down position, S-Straight position, U-Up position, f-fast, s-slow. The graphs do not start at 0 due to the moving average filter.

Fig. 10 .

 10 Fig.10. The biped robot with the tail system mounted on.

Fig. 11 .

 11 Fig. 11. Comparison on the maximum high reached (a) without the tail system, (b) with the bending of the tail, (c) without the bending of the tail.

TABLE I

 I

	BENDING PATTERNS
			Sequencing disk
	Patterns	Time (s)	angle (degree)
	Straight to Up (slow)	0.30	0 to 128
	Straight to Up (fast)	0.15	0 to 128
	Straight to Down	0.25	0 to -128
	Straight to Down to Up	0.40	0 to -128 to 128
	Down to Straight	0.15	-128 to 0
	Down to Up	0.25	-128 to 128
	Up to Straight	0.15	128 to 0
	Up to Down	0.25	128 to -128