Keywords: Molybdenum, Tungsten, Alkyl ligands, Cyclopentadienyl ligands, Oxido ligands, Aqueous organometallic chemistry

The chemistry of organomolybdenum and -tungsten compounds in high oxidation states (mostly V and VI) is reviewed, with special attention on the behaviour in protic media and with respect to protic reagents. The high metal-carbon bond covalency imparts a great degree of inertness to these bonds relative to protonolysis. Hence, extensive chemistry in a wide variety of media and with protic reagents is possible without affecting the metal-(hydrocarbyl ligand) interactions. This article does not cover metallocene [i.e. bis(cyclopentadienyl)], alkylidene and alkylidyne derivatives, that have been extensively covered in other recent reviews.

Introduction

Although organometallic chemistry is mostly developed in the low oxidation states, high oxidation state organometallics are also common. They comprise homoleptic compounds such as [W(CH3)6] [1] and [Re(CH3)8] 2- [2], and complexes that also contain other electronegative ligands, such as oxido (e.g. [Cp*ReO3] [3] and [CH3ReO3] [4]), halido (e.g.

[Cp*WF5] [5]), imido (e.g. [(tBuN)3Re(Ar)] with Ar = o-tol, xylyl and mes [6]), nitrido (e.g.

[Mo(N)(CH2CMe3)3] [7]), etc. The elevated Lewis acidity of high oxidation state metals is not fully satisfied by the metal-carbon bonds alone, thus the homoleptic compounds are typically quite unstable and reactive. For instance, compounds [W(CH3)6] [1] and [Re(CH3)6]

[2] thermally decompose at room temperature or below and react rapidly with water and other compounds containing acidic hydrogens to give methane. On the other hand, -donor (e.g. O-, N-based) ancillary ligands, even though poorer  donors because of their greater electronegativity relative to carbon, impart greater stability to the resulting complexes rendering them less susceptible to decomposition or oxidative/hydrolytic reactions. Since metal-carbon bonds with X-type [8] organic ligands (X: alkyl, aryl; XL: allyl; XL2: pentadienyl, cyclopentadienyl) enjoy a high degree of covalency when the metal is in a high oxidation state, they become less susceptible to hydrolytic decomposition and more resistant to oxidative degradation in air. Conversely, M-L bonds (L: olefins, L2: dienes, L3, arenes), which largely rely on back bonding for stabilization, are less and less compatible with the higher oxidation states, so that high oxidation state organometallic chemistry is more or less restricted to X-type ligands. In combination with ancillary ligands that resist hydrolytic decomposition, the compounds may become stable in an aqueous environment. As an illustrative example from the early literature, compound [CpMoCl4] is rapidly hydrolyzed to [CpMoO2]2 which is stable in water [9]. Thus, the reagent Mo V -Cl bonds are hydrolytically sensitive, but less so than the Mo V -Cp bond, whereas all bonds in the final product withstand an aqueous environment.

From the above arguments, it can be concluded that organometallic compounds containing metal carbon bonds of M-X type (e.g. alkyl, aryl, allyl, pentadienyl ligands) may be compatible with protic reagents and solvents (e.g. water or aqueous solvents) in the high oxidation states. Furthermore, injecting electrons into these systems increases the basicity of ancillary ligands, favouring their protonation. For the particular case of oxido ligands, their conversion to hydroxido and aqua ligands (see Scheme 1) This review covers high oxidation state organometallic chemistry with particular emphasis on processes where protic reagents or solvents are used and is limited to the Group 6 elements molybdenum and tungsten. Although examples of organometallic oxido derivatives of Cr V and Cr VI have been described (e.g. [Cp*CrO2]2 [10], [Cp*CrO(CH3)2] and [Cp*CrO2(CH3)] [11]), their chemistry has been much less developed and first principles suggest that they will be less compatible with protic reagents than their Mo and W analogues.

Within Mo and W chemistry, most studies have been focused on mono-and bis(cyclopentadienyl) derivatives.

The aqueous chemistry of bis(cyclopentadienyl)molybdenum derivatives (molybdocenes) has been developed extensively, particularly by D.

R. Tyler, and a review of this work has recently appeared in this journal [12], whereas related tungsten systems have been much less investigated in protic environments [13][14][15]. This family of materials will be excluded from the present review. We just wish to underline here the hydrolytic stability of the [Cp2Mo] 2+ moiety, no Cp loss being detected at neutral pH over a period of several weeks. The facile dissociation of chlorides from the readily available [Cp2MoCl2] [16] and the acidity of the resulting molybdocene ion in water [17], giving the charged complexes [Cp2Mo(OH)(H2O)] + and [(Cp2Mo)2(OH)2] 2+ as the main species at physiological pH [18], makes this system attractive for biomedical applications [19]. Another reason for excluding these compounds is that the presence of two soft Cp ligands renders the highest oxidation states (V and VI) less accessible [20]. There are no examples to the best of our knowledge of molybdenocenes in these oxidation states, whereas rare examples of tungsten compounds have been obtained by the oxidation of [Cp2W(OCH3)(CH3)] with Cp2Fe + yielding the W V products [(Cp2W(CH3))2(-O)] 2+ [21], which then further disproportionates in MeCN to yield the W VI complex [Cp2W(O)(CH3)] + and solvent-trapped [Cp2W(MeCN)(CH3)] + [22]. Compounds [Cp2WCl(CH3)] + and [Cp2W( 2 -O2)(CH3)] + have also been described in the same report [21]. Another W VI complex, [Cp2W(O)(C2H5)] + , was prepared by aerial oxidation of, or more conveniently by oxygen atom transfer from Me2SO

or propylene oxide to, [Cp2W(C3H4)H] + [23].

Another class of compounds that will be excluded from this review are those featuring alkylidenes and alkylidynes, the main members of these family being mononuclear species of type [M(=CHR)(NAr)X2] that catalyze olefin metathesis processes (Schrock catalysts).

Contrary to another family of olefin metathesis catalysts based on ruthenium (Grubbs catalysts), these compounds are typically incompatible with protic functional groups and with aqueous media. Furthermore, they have already been covered in other reviews [24][25][26][27]. We only wish to mention three trinuclear alkylidyne-capped Mo compounds, [Mo3(CCH3)2-(O2CCH3)6(H2O)3] n+ (n = 1, 2) [28,29] and [Mo3O(CCH3)(O2CCH3)6(H2O)3] + [30], reported several years ago by Cotton and Bino, which are fully compatible with an aqueous environment. The [Mo3(CCH3)2(O2CCH3)6(H2O)3] 2+ complex spontaneously couples the two alkylidyne moieties in aqueous solution at room temperature to yield 2-butyne [31]. A model of the intermediate in this transformation, [Mo3Br7(O2CCH3)(H2O)2(CH3C≡CCH3)], has been isolated by carrying out the reaction in the presence of HBr, and crystallographically characterized [31].

This article provides a general overview of molybdenum and tungsten chemistry in the highest oxidation states (mostly VI and V) with X-type hydrocarbyl ligands with respect to protic reagents and solvents. It is organized in terms of the type of hydrocarbyl ligand, first alkyls and aryls, then -allyls, and finally cyclopentadienyls. Many of the investigations in this area, especially the most recent ones, find justification in the potential application of these materials to oxidation catalysis, particularly olefin epoxidation, although recent work has also shown potential in reductive-type processes, namely the ketone hydrosilylation [32]. These catalytic aspects have been recently reviewed and will not be covered here [33]. Other recent relevant reviews have appeared on organometallic oxides [34] and hydroxides [35] of the transition elements, as well as on dioxygen activation by organometallics of the early transition metals [36]. A concept article and a special report highlighting advances from our own group in aqueous Cp*Mo chemistry have also been published a few years ago [37,[START_REF] Collange | Perspectives in Organometallic Chemistry[END_REF].

Finally, a recent review by Kühn and collaborators highlights the synthesis, reactivity and catalytic applications of mononuclear organomolybdenum(VI) dioxido complexes [START_REF] Kühn | [END_REF]. As such, there is a partial overlaps with this article, although the present review provides a more general coverage of high oxidation state Mo and W chemistry and has greater focus on chemical processes involving protic substrates and media.

Alkyl and Aryl Compounds

Beside the homoleptic compound [W(CH3)6] already mentioned in the introduction [1],

a variety of alkyl-and aryl-molybdeum(VI) and tungsten(VI) complexes have been described. based chelating ligand such as a bipyridine [44][45][46][47][48] or a 1,4-diazadiene [44,49]), WO(CH3)4(bipy) [46], and (XL2)MO2R (M = Mo, W; XL2 =  3 -HB(3,5-Me2pz)3 [50,51], 2-N-(2-pyridylmethyl)aminophenolato [52]). Compounds containing both alkyl and cyclopentadienyl derivatives, e.g. CpWO2R and CpWOR3 [53], will be analyzed later in section 4.

In many cases, it is mentioned that the compounds are quite stable and can be handled in air, at least for short periods of time. Their thermal decomposition has been studied in detail [48], but their stability or reactivity in water has not been systematically tested. A notable exception is the alkaline hydrolysis of compounds [MoO2(R)Br(bipy)] (R = Me, Et, CH2tBu) [54,55]. The hydrolysis occurs rapidly, but at room temperature and neutral pH is not accompanied by Mo-C cleavage and leads to [RMoO3] -intermediates that only slowly decompose to [MoO4] 2-and RH. The X-ray structure of [PPh4][(Mes)MoO3], obtained by an alternative non-aqueous route, has been reported [56]. The hydrolysis process is faster in the presence of reductants, such as V 2+ . The [(tBuCH2)MoO3] -anion decays with t1/2 = ca. 15 days at 23°C and 282 min at 70°C at pH 11 [55].

Compounds [MoO2R2(bipy)] (R = Me, CH2tBu) are quite resistant to hydrolysis, a fact that is attributed to their hydrophobic nature. Quantitative yields of methane or neopentane, respectively, requires fusion with KOH, while only 13% and 0.2% decomposition occurs upon heating in glycerol at 90°C for 1 h. Their half life is reduced to 30 min in glacial acetic acid at 90°C and to 5 min in concentrated H2SO4 at room temperature. However, the Mo-C bonds become more sensitive to cleavage upon reduction [57]. The hydrolysis of one member of the Tp*MO2R family, specifically Tp*WO2(CHPPh3), was reported to occur readily, leading to evolution of methane and formation of [Ph3POH][Tp*WO3] [50].

Most synthetic efforts in high oxidation state alkylmolybdenum and tungsten chemistry have been carried out in search for better catalysts for the olefin epoxidation reaction. High oxidation state oxidomolybdenum and tungsten species are commonly used in this reaction, with the preferred oxygen transfer agents being the protic substances tert-butylhydroperoxide (TBHP). Most reactions are investigated in apolar solvents, but the use of aqueous media is gaining interest [58]. Dialkyl compounds of type MoO2R2L2 (R = Me, Et; L2 = substituted diazadiene or bipyridine) were found to be active catalysts when using TBHP, but inactive when using H2O2 [START_REF] Kühn | [END_REF]. A spectroscopic study of the interaction between [MoO2(CH3)2L2] (L2 = 4,4'-dihexyl-2,2'-bipyridine) and TBHP, of relevance to catalytic cycle, suggests that the TBHP hydrogen atom is transferred to one of the terminal oxido ligands, rather than leading to CH4 elimination or protonation of the Lewis base ligand, see Scheme 2 [59].

-Allyl Compounds

The chemistry of allylmolybdenum and allyltungsten derivatives in high oxidation states has been very little developed.

A rare example of an allyl derivative is Tp*WO2(CH2CMe=CH2) [60], but since the allyl ligand is only -bonded to the transition metal, this compound qualifies as a simple alkyl complex. Related -bonded complexes might be accessible by replacing Tp* with a bidentate (XL-type) ligand. A -allyl oxido tungsten derivative has been proposed as an intermediate in an allyl alcohol deoxygenation process induced by compound [WCl4(PMe3)2] [61, 62] and a chloride derivative, [WO( 3 -C3H5)Cl(PMe3)2] has been isolated and structurally characterized [63]. The behaviour of these compounds in water or in the presence of protic reagents does not appear to have been explored. A number of carbonyl-based allyl derivatives containing also hydroxido or aqua ligands are known in lower oxidation states [64][65][66], but oxidative processes leading to higher oxidation state derivatives have not so far been reported.

Cyclopentadienyl Compounds

Complexes containing "open" pentadienyl ligands have received a certain degree of attention for early transition metals [67], but no such compounds have been described to date for molybdenum and tungsten, to the best of our knowledge. On the other hand, investigations of derivatives with the ubiquitous cyclopentadienyl ligand, especially the fully methyl-substituted version (Cp*), have been extensive.

Synthetic aspects

Two basic approaches have mainly been used: (1) the direct oxidation of suitable lowvalent (typically carbonyl) precursors; (2) hydrolysis processes from easily accessible highvalent precursors.

Oxido and mixed halido-oxido derivatives

Most early synthetic efforts leading to high oxidation state organomolybdenum and tungsten derivatives have been restricted to anhydrous conditions and non protic reagents, e.g.

the oxidation of low-valent carbonyl precursors with O2, or the replacement of a Mo-Cl bond with a Cp -delivering agent. However, wet routes have also been developed relatively early and have recently found to be quite convenient, especially for the substituted ring systems.

The systems containing the unsubstituted Cp ligand were the first ones to be described.

Early [68], and to red tetranuclear [CpMoO2]4 (I) by treatment with excess H2O [9]. The latter compound is unstable and isomerizes to orange dinuclear [Cp2Mo2O4], the structure of which has been determined [74]. The precise structure of intermediate I, on the other hand, is still unknown.

An analogous hydrolytic behaviour was also observed for the tetrabromido derivative,

[CpMoBr4] [9], as well as for  5 -C5H4R derivatives (R = Me, iPr, tBu) [71], see Scheme 3.

Since compound [CpMoCl4] has later become more readily available in high yields by oxidation of [Cp2Mo2(CO)6] or [CpMo(CO)3(CH3)] and PCl5 or PhICl2 [75][76][START_REF] Gordon | Synthetic Methods of Organometallic and Inorganic Chemistry[END_REF], the hydrolytic synthesis of the oxido derivatives has been further optimized [START_REF] Saurenz | [END_REF]. The procedure is in fact a convenient entry to the Mo VI derivative [Cp2Mo2O5], which is obtained by further oxidation of the CpMo V oxido products. While orange dinuclear [Cp2Mo2O4] is only oxidized with difficulty [71] (probably because of the presence of a strong Mo-Mo bond), the red tetranuclear intermediate is rapidly oxidized by PhIO [START_REF] Saurenz | [END_REF].

Compound [Cp2Mo2O5] is not indefinitely stable in water [79]. On the other hand, the Cp* analogue appears to resist hydrolytic decomposition in water in the entire pH range (more details in section 4.2) [80], although a very slow decomposition occurs under extreme acidic conditions as shown in section 4.4. This is the main reason for which greater effort has been dedicated to the study of the aqueous chemistry of the Cp* Mo and W system. Bz) [88]. The reactions are fast at room temperature and high-yielding. The protic reagent (the acidity of TBHP is greater than that of water) does not appear to perturb the cyclopentadienyl-metal interaction. This protocol is also effective when using higheroxidation state starting materials, e.g. compound [(C5Bz5)MoO2Cl] is obtained from

- O 2 , OH - OH - O 2 OH - [Cp*MoCl 4 ] [Cp*MoO 2 Cl] [Cp*MoO 2 ] 2 [Cp*MoOCl 2 ] [Cp* 2 Mo 2 O 5 ] [Cp*MoO 3 ] - H + slow 2OH - Scheme 4
Although the yields of the above synthetic procedures starting from [Cp*MCl4] or The same oxidation process occurs more rapidly when the [Cp'M(NO)R2] precursors are treated with H2O2 [107], although the stoichiometry of the oxidizing agent must be carefully controlled to avoid further conversion. In fact, these alkyl derivatives react rather easily with excess H2O2 or other hydroperoxide reagents to yield the corresponding peroxido derivatives (see section 4.4) [101,103,107].

More recently, it has been shown that dioxidoalkyl compounds may be obtained by oxidation of readily accessible tricarbonyl alkyl derivatives with TBHP or H2O2. However, the reaction easily proceeds further to the peroxide complex, even when the stoichiometric amount of oxidant is used. Thus, it is best to convert [Cp'M(CO)3R] directly to TBHP [98]. Since these dioxido alkyl complexes have been shown to efficiently catalyze the olefin epoxidation reaction, which requires TBHP as a stoichiometric oxidant, the more easily accessible tricarbonyl derivatives can be used as pre-catalysts, generating the active dioxido complexes in situ [98,110,111].

Oxido-phosphine complexes

A particular type of phosphine-containing cyclopentadienyl oxido derivatives of molybdenum have been obtained by treatment of hydride complexes with water. By analogy with the Cp2Mo 2+ system, the presence of soft phosphine donor ligands destabilizes the highest oxidation states, these derivatives existing only in oxidation states up to IV.

Thus, complexes [CpMoO(dmpe)] + and [CpMoO(PMe3)2] + have been obtained from the [CpMo(dmpe)H4] + [112] and [CpMo(PMe3)3H] + [113] precursors upon reaction with water.

In the latter case, DFT studies and the isolation of intermediates indicates a complex Higher oxidation state versions of these compounds have not been described. On the other hand, a 1-electron reduced version of [CpMoO(PMe3)2] + , the neutral Mo III compound [CpMoO(PMe3)2], has been spectroscopically identified in solution [113].

Imido complexes

M(V) imido compounds have been obtained from cyclopentadienyl containing precursors by direct exchange of two M-Cl bonds with a primary amine, for instance

[(C5H4R)M(NR')Cl2] from [(C5H4R)MCl4
] and three equivalents of R'NH2 [91,117]. The same procedure starting from oxido functionalities is unsuccessful. However, oxido ligands may be transformed into imido ligands by use of isocyanates (see section 4.4). M(VI) derivatives are more easily prepared by ion metathesis, e.g. [CpM(NtBu)2Cl] (M = Mo, W) from [Mo(NtBu)2Cl2] or [W(NtBu)2Cl2(py)2] and CpLi or CpNa [118].

Sulfido and other chalcogenido complexes

Speciation studies

Investigations in this area have so far been limited to the Cp*Mo VI system. Compound whereas it maintains its neutral molecular form in organic solvents. Several X-ray diffraction studies of the molecular form have been published [81][82][83], and the structure of the trioxide anion (as the sodium salt) has also been recently reported [START_REF] Dinoi | [END_REF]. A similar ionic dissociation was also demonstrated for the chlorido derivative, Scheme 8 [80]. or unsolvated ([Cp*MoO2] + ) species remained ambiguous after the initial study [80].

Evidence gathered in the subsequent study [123], which was backed up by DFT calculations, confirmed the stabilization of the species by H2O coordination. Re3 17+ , 4-electron cluster), which was analyzed by theoretical calculations at the extended Huckel and SCF X-SW levels [134,135]. The electronic structure of this system can actually be easily understood starting from that of the individual four-legged piano stool "Cp*MO4" fragments, which leaves two metal-based orbitals available for metal-metal bonding: a lower energy xy orbital and a higher-energy z 2 orbital [136]. Assembly of three such units in a D3h arrangement generates the metal-metal interactions illustrated in Figure 2.

The most important interaction is the mixed /-type overlap of the three z According to the electronic structure analysis given above and to the previous [133], this [(Cp*Mo)3(-OH)4(-O)2] 2+ cluster should exhibit a broad NMR signal and no EPR signal. However, contrary to this prediction, the cluster (in both isolated salts) shows a sharp EPR spectrum in isotropic solution at room temperature and no NMR signal, neither broad nor sharp [141]. The EPR spectrum (g = 1.962, aMo = 26.2 G) is very similar to that shown [133] for compound [(Cp*Mo)3(-OH)n(-O)6-n]Cl2 (g = 1.962, aMo = not reported). The measured magnetic moment is 1.66 B (relatively constant in the 4-60K range), near the expected value for a S = ½ system. Thus, these two salts do not undergo complex redox equilibria, differently from the behaviour was said to indicate coupling to three equivalent nuclei on the basis of the satellite intensities relative to the central resonance [133]. On the other hand, the simulated spectra for the two

[(Cp*Mo)3(-OH)n(-O)6-n] 2+ report
[(Cp*Mo)3(-OH)4(-O)2]
2+ clusters had too strong satellites, relative to the experimental spectrum, when using a coupling model to three equivalent Mo atoms, and too low when considering coupling to only a single Mo atom. The simulation perfectly fits the experimental spectrum when considering coupling to two equivalent Mo atoms [141]. A rationalization of this phenomenon was possible on the basis of the DFT electronic structure calculations.

Because of the asymmetric nature of the bridge system, the point group symmetry is lowered from D3h to C2 with consequent splitting of the e(xy) degeneracy into separate e1(xy) and e2(xy) orbitals (becoming respectively a' and a" in C2), the shapes of which remain close to those expected in D3h symmetry (see Figure 2). The orbital ordering and occupation shown by the calculation is a'(z 2 ) 2 a'(xy) 2 a"(xy) 1 , with the unpaired electron in the non-degenerate a"(xy) [formerly e2(xy)] orbital, rationalizing the sharpness of the spectrum. This orbital has a nodal plane that contains one of the three Mo atoms, imposing a zero contribution of this atom's xy orbital, even in the lower C2 symmetry, and therefore rationalizing the observed coupling to only two Mo atoms [141].

A different cluster, [(Cp*Mo)3(3-O)(-O)3(-O2CCF3)3] + (crystallized as a salt of the [Zn2(O2CCF3)6] 2-ion) was also obtained from the same Zn reduction experiment described above, under different crystallization conditions and in small amounts, sufficient for an X-ray analysis but not for a thorough spectroscopic study except for an ESI-MS [142]. This is a Mo3 15+ , 3-electron cluster, namely less reduced relative to the 5-electron [(Cp*Mo)3(-OH)4(-O)2] 2+ cluster described above. Possibly, the reduction process for this particular batch was incomplete. This result, together with the rich reductive electrochemistry shown by the coupled mass spectrometric analysis (vide supra), suggests that this family of clusters should undergo facile transformations involving addition/removal of protons, electrons, and water molecules. More detailed investigations of such transformations have not yet been carried out.

Reductive processes of Cp*W VI starting materials have been much less investigated, perhaps because their synthesis was less convenient than that of their Mo counterparts. The recently developed aqueous route to [Cp*2W2O5] in high yields [START_REF] Dinoi | [END_REF] (see section 4.1) has opened the way to more detailed investigations of this system in our laboratory. The electrochemical study of [Cp*2W2O5] by flow-through electrochemistry coupled to on-line ESI-MS detection was not as successful as the corresponding Mo study (vide supra). A more negative electrode potential was necessary to start reducing the complex, as expected from the greater stability of W vs. Mo in the highest oxidation states. At this point, an insoluble film deposited on the electrode, blocking any further electrochemical process, and no reduced products could be detected by ESI-MS. Thus, it seems that any primary reduction product is unstable under these highly reducing conditions and proceeds further to yield insoluble products.

Greater success was achieved by chemical reduction with Zn/HA in MeOH-H2O (HA = CF3COOH, CF3SO3H), i.e. the same protocol used previously for the Mo system [143]

Other reactions

The vast majority of other reactivity studies on cyclopentadienyl molybdenum and tungsten derivatives involve the "inorganic" part of the molecule. They are carried out in many cases under anhydrous conditions. When protic reagents or solvents are used, the transformation generally leaves the cyclopentadienylmetal moiety (and any M-alkyl bonds that may be present) intact.

M-oxido exchange with water.

This reaction has been investigated in detail for compound [Cp*MoO2Cl] using H2 18 O. but the formation of mononuclear oxido-sulfido complexes was not discernible [122].

Electrophilic addition to M=O bonds

The

Conclusion

The heavier group 6 metals have an extensive and rich organometallic chemistry in protic media. The more extensively investigated Cp*Mo and Cp*W system have demonstrated a particular robustness of the Cp*M bond and its compatibility with water in a variety of oxidation states and with proton donors of different strengths (down to pH 0 in water for the Cp*Mo VI system). Alkyl and aryl derivatives (in cyclopentadienyl-free systems)

have been much less investigated and -allyl systems are completely unexplored, but a rich chemistry may be anticipated for these systems in protic media. The majority of the transformations observed for these organometallic compounds concern the rest of the coordination sphere, the M-C bonds remaining intact, especially for the higher-hapticity cyclopentadienyl derivatives. This feature allows the use of organic substituents for compatibilizing high oxidation state metal centers with organic media and for blocking a water [141,143]. The reduced metal centres in these clusters assemble in a triangular fashion in order to share electrons and bridging ligands, thereby saturating their coordination sphere, but the M(-OH)M bridge opening process (perhaps after an additional protonation) appears feasible by a number of substrates. The interaction of these reduced clusters with different substrates and a study of their activation and transformation, possibly leading to catalytic and electrocatalytic reduction processes, seems a promising research avenue for further investigations.

Acknowledgement

  Scheme 1

  The first documented examples of mixed alkyl or aryl-oxido derivatives appear to be [MoO2(Mes)2] [40] and [MoO(Mes)3] [41] (Mes = 2,4,6-C6Me3H2), later followed by [MO(CH2tBu)3Cl] [42], [MO(CH2tBu)4] [42], and [Mg(THF)4][MO(CH3)4]2 [43] (M = Mo, W). Derivatives where the alkyl-or aryl-oxido scaffold is supported by additional Lewis bases (mostly N-donor ligands) are also known, such as [MO2(CH3)2L2] (M = Mo, W; L2 = N-

  Scheme 2

  Scheme 3

[

  (C5Bz5)MoCl3(COMe)] (the reaction takes place with the intermediate formation of [(C5Bz5)MoOCl2]) and [(C5Bz5)WO2Cl] from [(C5Bz5)WCl4] [89]. The first reported wet route involved the treatment of [Cp*MoCl4] with either Na2MoO4, Na2WO4, or NaVO3 in water, yielding [Cp*2Mo2O5] after acidification with HCl [90], whereas the simple addition of water without inorganic oxidants gave poorly characterized materials, and the same reaction carried out in THF led to a Mo V product, [Cp*2Mo2O3Cl2]. A subsequent study revealed that the basic (NaOH) hydrolysis of [Cp*MoCl4] in the absence of air yields [Cp*MoOCl2], whereas the same procedure in air yields [Cp*MoO2Cl] or [Cp*2Mo2O5], depending on the amount of base used [91]. In the case of tungsten, aerial oxidation also leads to the chloridodioxido product, [Cp*WO2Cl], whereas in the absence of air the reaction allows the identification of two intermediates, formulated as [Cp*WCl3(OH)] and [Cp*WCl2(OH)2] on the basis of spectroscopic evidence [91]. The alternative hydrolysis using tBuNH2 yields [tBuNH3][Cp*MoO3], from which [Cp*2Mo2O5] may be recovered by acidification (the tungsten analogue was similarly prepared via the [tBuNH3][Cp*WO3] intermediates) [91]. Compound [Cp*MoO2Cl] was also shown to be conveniently prepared (86% yield) by a wet oxidative route, from [Cp*2Mo2(CO)4] and excess (> 10:1) 30% aqueous H2O2 in chloroform which was further acidified with HCl [92]. Interestingly, this procedure does not yield the peroxido derivative [Cp*MoO(O2)Cl], which is easily formed from [Cp*MoO2Cl] and TBHP in organic media (vide infra). It is possible that the peroxido product is unstable in the strongly acidic medium, although this has not been tested to the best of our knowledge. The hydrolysis of [(Cp*MoOCl)2(-O)] in the presence of NEt3 gives a high yield of [Cp*MoO(-O)]2 [92]. Later, the synthesis of [Cp*2Mo2O5] was further optimized and extended to the preparation of compounds [Cp'2Mo2O5] with Cp' = C5HiPr4 and 1,2,4-C5H2tBu3 [78]. It was found that the anaerobic hydrolysis of [Cp*MoCl4] ultimately leads to [Cp*2Mo2O4] in the presence of excess NaOH, whereas the sodium salt, Na[Cp*MoO3], is obtained in the presence of air and excess NaOH, see Scheme 4. Since the dinuclear Mo V compound is more resistant than [Cp*MoOCl2] to aerial oxidation, the aerobic hydrolysis of [Cp*MoCl4] in the presence of excess base always leads to mixtures of Na[Cp*MoO3] and [Cp*2Mo2O4], lowering the final [Cp*2Mo2O5] yield after acidification. Formation of the Mo V by-product is minimized by first carrying out the aerial hydrolysis with only 2 equivalents of NaOH, and then completing the hydrolysis with additional base after most of the material has been converted to [Cp*MoO2Cl]. An alternative hydrolytic approach involves protonolysis of [Cp*M(NtBu)2Cl] (M = Mo, W) with aqueous HCl [93]. The starting materials used for this approach were obtained from Mo(NtBu)2Cl2 or W(NtBu)2Cl2(py)2 and Cp*Li.

2OH

[ 5

 5 Scheme 5

[

  Cp'MO(O2)R] and then reduce the peroxide function to oxido by treatment with an oxygen atom acceptor such as PPh3. Using this strategy, the acetylide complexes [Cp*WO(O2)(C≡CR)] (R = Ph, CH2OMe, nPr and C(Me)=CH2) were obtained in good yields from the corresponding [Cp*W(CO)3(C≡CR)] and acidic solutions of H2O2 at room temperature and then converted to [Cp*WO2(C≡CR)] [108]. The same strategy gives [Cp*WO2(C≡C-C≡CH)] from [Cp*W(CO)3(C≡C-C≡CH)] and [Cp*O2W-C≡C-C≡C-WO2Cp*] from [Cp*(CO)3W-C≡C-C≡C-W(CO)3Cp*] via the corresponding peroxide complexes [109]. Analogously, [CpMoO2(CH3)] was obtained from [CpMo(CO)3(CH3)] and

  mechanism involving water-induced disproportionation of the paramagnetic (17-electron, formally Mo III ) hydride to give [CpMo(PMe3)3(H2O)H] 2+ , followed by double deprotonation to [CpMo(PMe3)3(OH)] (which is in equilibrium with isolated [CpMo(PMe3)2(OH)] and free PMe3), subsequent oxidation to [CpMo(PMe3)3(OH)] + (which was isolated and structurally characterized[114]), and further oxidation, deprotonation and PMe3 decoordination to complete the process[113]. Oxidation of compounds [(C5R5)Mo(dppe)H3] (R = Me, Et) in the presence of water, on the other hand, yields the neutral bis(hydroxido) derivatives[(C5R5)Mo(dppe)(OH)2][115], the structure of which was confirmed by X-ray diffraction for R = Et.An analogous system containing a phosphine-functionalized cyclopentadienyl ring that acts as a tripodal ligand, [CH3C(CH2- 5 -C5H4)(CH2PPh2)2MoO] + , has been obtained by irradiation of [CH3C(CH2- 5 -C5H4)(CH2PPh2)2Mo(CO)2I] in the presence of air[116].

7

 7 Scheme 7

[

  Cp*2Mo2O5] is insoluble in pure water. However, it is rather soluble in MeOH and sufficient amounts remain dissolved in a mixed 20:80 MeOH-H2O solvent (up to ca. 5•10 -4 M) to allow spectroscopic investigations in the entire pH range[80]. The nature of the compound (Scheme 8) depends on the solvent, as shown by electrical conductivity studies[80]. In highly aqueous media, the compound self-ionizes to yield [Cp*MoO2(H2O)] + [Cp*MoO3] -,

8 Figure 1 .

 81 Scheme 8

Cp*MoO 2 (

 2 Scheme 9

Figure 2 .

 2 Figure 2. Energy diagram relating the electronic structure of a [(Cp*M)(-O)6] n+ cluster in ideal D3h symmetry with that of the three individual "Cp*MO4" moieties.In our own laboratory, we have investigated the chemical and electrochemical reduction

Figure 3 .

 3 Figure 3. Structures of the [(Cp*Mo)3(-OH)4(-O)2]X2 compounds, emphasizing the cation-anion H-bonds. Left: X = CF3CO2; right: X = CF3SO3. The Cp* ligands have been removed for clarity. Distances in red are those of the Mo-Mo bonds, those in bold black characters are those of the H-bonded O•••O contacts.

  proposed for [(Cp*Mo)3(-OH)n(-O)6-n]Cl2. An additional interesting observation concerns the simulation of the EPR spectrum. The one reported for [(Cp*Mo)3(-OH)n(-O)6-n]Cl2

Figure 4 . 1 [ 144 ]

 41144 Figure 4. Structures of the [(Cp*W)3(-O)4(-OH)2](CF3SO3)2•2H2O (left) and [(Cp*W)3(-O)6](CF3SO3) (right), emphasizing the cation-anion H-bonds in the former. The Cp* ligands have been removed for clarity. Distances in red are those of the Mo-Mo bonds, those in bold black characters are those of the Hbonded O•••O contacts.

The O exchange process 11 A 12 13 14

 11121314 Scheme 11

16 19

 1619 Scheme16

  certain number of coordination positions (e.g. three for cyclopentadienyl ligands), thereby limiting or eliminating aggregation processes (e.g. formation of high-nuclearity polyoxometallates in acidic media). New catalytic applications in aqueous or biphasic environment may therefore be envisaged. Coming back to a proposition made in the introduction, the envisaged generation of open coordination sites by conversion of oxido ligands to aqua ligands upon metal reduction has already been demonstrated. In fact, even the Cp*Mo VI system where the metal is in the highest oxidation state generates an open coordination site at low pH (rapid equilibrium between [Cp*MoO2(H2O)] + and [Cp*MoO2] + in water) [123]. Upon reduction, compounds [Cp*2M2O5] (M = Mo, W) yield [(Cp*Mo)3O2(OH)4] 2+ or [(Cp*W)3O4(OH)2] 2+ with expulsion of a number of O atoms as

2.7820 Å 2.7854 Å 2.8244 Å 2.71 Å 2.76 Å 2.65 Å 2.76 Å 2.64 Å 2.64 Å 2.59 Å 2.62 Å 2.78 Å 2.78 Å 2.82 Å

Work on this project in my own laboratory has been supported by the CNRS, by CINES and CICT (Project CALMIP) for the DFT calculations, and by the European Commission through the AQUACHEM program (contract HPRN-CT-2002-00176).

Additional funding from the Institut Universitaire de France is also gratefully acknowledged.

The AQUACHEM program has led to a number of fruitful collaborations (Rudi van Eldik in Erlangen, Agustí Lledós in Barcelona, Ovadia Lev and Jenny Gun in Jerusalem, Maurizio Peruzzini in Florence, Armando Pombeiro in Lisbon) that I have greatly appreciated. I also thoroughly enjoy the continuing collaboration with Prof. Funda Demirhan (Celal Bayar University, Manisa), which has been supported by bilateral CNRS-TUBITAK funds and by the French Ministry of Foreign Affairs.

Scheme 17

Complex [Cp*WS3] -also displays nucleophilic reactivity, being alkylated by BzBr and MeI to yield [Cp*WS2(SR)] products (R = Bz, Me). It also adds to acetylenes (RCCH, R = Ph, H) to form ene-dithiolate complexes of W IV , [Cp*WS(SCH=CRS)] - [119].

Replacement of M=O and M=NR bonds with M-X bonds (X = halogen).

This reaction is usually accomplished with HX (gas or Et2O solution) or Me3SiX reagents under anhydrous conditions, because water hydrolyses the Mo-X bonds to Mo=O bonds, as shown in a preceding section. The reaction has been realized on both Mo VI and Mo V complexes. Thus, treatment of [Cp2Mo2O4] or [CpMoOX2] with HX gas affords [CpMoX4] (X = Cl, Br) [9,68]. The resistance of the Cp-Mo in these reactions is remarkable, although the aggressiveness of HX as a proton donor is lower in the gas phase than in aqueous solution.

Analogously

Replacement of M=O with M=NR bonds.

The exchange with imido ligands is conveniently realized by reaction with isocyanates. although this is not very fast under neutral conditions. Therefore, the peroxido product could also be obtained using aqueous 30% H2O2, but timing was essential to minimize the hydrolysis side reaction. The more common tBuOOH reagent, on the other hand, does not readily effect this particular peroxidation reaction. Peroxidized diethyl ether turns out to be the most effective reagent [102], the reactions being complete over 24 h at room temperature.

However, a later study showed that TBHP smoothly converts [Cp*MoO2Cl] to

[Cp*MoO(O2)Cl] in toluene at 45°C (71% isolated yields after 4 h) [86]. Other alkylhydroperoxides such as cumene hydroperoxide and Ph3COOH also effect this transformation, whereas n-hexyl hydroperoxide failed to yield the desired product, leading instead to uncharacterized insoluble white materials with formation of n-hexyl alcohol. An even more recent study shows that complexes [Cp*MCl4] can be converted directly and in high isolated yields (85% for Mo, 80% for W) to [Cp*MO(O2)Cl] by using 30% aqueous H2O2 in diethyl ether [163].

An 18 O labelling study showed that [Cp*MoO(O2)Cl] exchanges with H2 18 O selectively into the oxido position. Furthermore, reaction of 18 O-labelled [Cp*MoO2Cl] with unlabelled peroxide-containing and H2 18 O-saturated diethyl ether yields [Cp*MoO(O2)Cl] with no 18 O label in the peroxido ligand. This clearly demonstrates that the O-O bond of the organoperoxide remains intact during the reaction and in the final product [102]. Thus, formation of the peroxido ligand occurs by ligand exchange rather than by oxygen atom transfer.

As detailed in section 4.1, the peroxidation reaction is particularly easy for alkyl derivatives. Thus, peroxide derivatives are obtained in good yields by addition of H2O2 or TBHP to tricarbonyl adducts [98,108,109].

Cleavage of the Cp-M bond

A rare example of a reaction where the Cp*Mo is cleaved heterolytically (albeit very slowly) was revealed during an attempt to crystallize a salt of the [Cp*MoO2(H2O)] + cation, which was generated by dissolving [Cp*2Mo2O5] in water at low pH. A solution of this cation, left unprotected from air and sunlight for several weeks, led to the deposition of crystals of compound [Cp*2Mo6O17] [164]. The W analogue, obtained by a non-aqueous route, has also been reported [165]. The compound is a member of the isoelectronic family of polyoxometallates, [(Cp*Mo)n(MoO)6-nO13] (n-2)-, of which those with n = 0 [166] and 1 [85,139] are also known.

As already mentioned in a previous section, the unsubstituted analogue, [Cp2Mo2O5], does not resist an aqueous environment for a long time, even at pH 7 [79], but the nature of the resulting products has not been investigated in detail.

While going from Cp to the sterically more encumbered and electronically more donating Cp* greatly increases the stability of all Mo VI compounds in water (with respect to protonolysis of the cyclopentaidneyl-molybdenum bond), moving further on to the even more sterically encumbering but electronically less donating C5Ph4(2,5-C6H3(OMe)2) greatly labilizes the interaction with the metal. Thus, compound [(C5Ph4(2,5-C6H3(OMe)2))-MoO2(OMe)] leads to a mixture of products, some of which could not be characterized, when treated with water, acetic acid, or 4-methoxyphenol. However, all these reactions yielded variable amounts of the free cyclopentadiene, C5Ph4(2,5-C6H3(OMe)2)H, this being isolated in high yields when using a large excess of water in the reaction [87]. Loss of this ring has also been observed during catalytic cyclooctene epoxidation. Given the established reactivity with aliphatic alcohols (vide supra), the reaction between [(C5Ph4(2,5-C6H3(OMe)2))MoO2Br] and TBHP is presumed to afford [(C5Ph4(2,5-C6H3(OMe)2))MoO2(OOtBu)] which serves as an active catalyst. However, NMR monitoring shows that all NMR signals of the coordinated ring are lost and no new peaks grow that could be attributed to another (C5Ph4(2,5-C6H3(OMe)2))Mo VI species or to the free cyclopentadiene, C5Ph4(2,5-C6H3(OMe)2)H. Yet, at the point in which all starting compound is fully decomposed, the catalytic oxidation continues at even faster rate [167]. Therefore, a new and cyclopentadienyl-free species, the nature of which is not yet elucidated, must be an even better catalyst for olefin epoxidation.

This finding serves as a warning that even systems believed to be extremely robust toward protonolysis of the cyclopentadienyl-molybdenum bond, may undergo slight decomposition with loss of the organic ligand, even in trace amounts, to afford a more active molybdenum species.

Condensations to form organometallic polyoxometallates

Compound [Cp*2Mo6O17], serendipitously obtained as described in the previous section, was also prepared directly and selectively by the condensation of [Cp*2Mo2O5] and Na2MoO4 in the appropriate stoichiometric ratio in acidic aqueous solution, see Scheme 19 Na2WO4 and [( 6 -arene)RuCl2]2, although the reaction was conducted in acetonitrile [171].

Compound [(( 6 -arene)Ru)4(WO2)2(-O)6] was also obtained as a by-product. The organic


E Le Flécher

Adrien Durand Petiteville

Frédéric Gouaisbaut

Viviane Cadenat
email: cadenat@laas.fr

Thierry Sentenac
email: sentenac@laas.fr

Stavros Vougioukas

Le Flécher

Nonlinear output feedback for autonomous U-turn maneuvers of a robot in orchard headlands

Keywords: mobile robotics, sensor-based navigation, control theory, agricultural robotics, orchards

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

INTRODUCTION

According to [START_REF] Foley | Solutions for a cultivated planet[END_REF], agriculture will need by 2050 to double its production to feed the increasing population. Mechanization has been identified as one of the best solutions to increase significantly the food production [START_REF] Reid | The impact of mechanization on agriculture[END_REF]. For mowing, spraying, pruning, harvesting in fields or orchards, one of the main challenge consists in navigating. In other words, the robot has to autonomously drive from the entrance of a row to its exit, then navigate in the headlands to reach the entrance of the next row. This process is repeated to cover the area of interest. This problem has been addressed for many years as it is shown in the review proposed in [START_REF] Li | Review of research on agricultural vehicle autonomous guidance[END_REF]. Most of the presented solutions focus on open field navigation and rely on GPS. The presented work is included in the orchards navigation problem where GPS cannot be used because its signal is blocked by the dense canopy. For this reason, orchards navigation systems are based on sensors such as laser range finders, Lidars or cameras. For example, laser range finders and cameras are used to drive through rows [START_REF] Subramanian | Development of machine vision and laser radar based autonomous vehicle guidance systems for citrus grove navigation[END_REF], [START_REF] Sharifi | A novel vision based row guidance approach for navigation of agricultural mobile robots in orchards[END_REF]. The headland navigation problem is addressed in works such as [START_REF] Andersen | Autonomous rule-based robot navigation in orchards[END_REF], [START_REF] Zhang | Mapping orchards for autonomous navigation[END_REF] and [START_REF] Bayar | Localization and control of an autonomous orchard vehicle[END_REF] where sensory data (proprioa https://orcid.org/0000-0002-2683-8675 ceptive and/or exteroceptive) are used to localize the robot in a metric map of the orchard in order to follow a precomputed path. However, keeping an updated metric map seems challenging due to constant variation of the environment. Indeed, over the years, trees grow up, and are pruned; over the seasons, leaves grow then disappear; over the days, fruits grow, bend branches and finally fall. Moreover, metric localization can accumulate errors and lead to the navigation failure. To overcome these drawbacks, we have presented in [START_REF] Durand-Petiteville | Design of a sensor-based controller performing u-turn to navigate in orchards[END_REF] a solution relying on a topological representation of the environment coupled with a set of output feedback controllers. Especially, the headland navigation consists in a U-shape turn around the last tree of the row. It is performed thanks to an output feedback controller tracking a spiral, inspired by the work presented in [START_REF] Boyadzhiev | Spirals and conchospirals in the flight of insects[END_REF].

However, as presented in [START_REF] Bochtis | Minimising the non-working distance travelled by machines operating in a headland field pattern[END_REF] and in [START_REF] Zhang | Mapping orchards for autonomous navigation[END_REF], it is not always possible to perform a simple U-shape turn because of the minimum turning radius of the robot which might be too small to deal with narrow rows. It is then mandatory to consider different shapes for turns. Thus, in this paper, we address the headland navigation by designing controllers allowing to perform two different shapes of turns (see figure 1). The first one, the classical U-shape turn, is used when the robot turning radius is large enough to allow the vehicle directly reach the middle of the next row. When it is not the case, a more complex shape for the turn known as Ω-shape turn is chosen as proposed in [START_REF] Bochtis | Minimising the non-working distance travelled by machines operating in a headland field pattern[END_REF]. This shape is classically used when the robot can only drive forward. The lack of sensors to detect hazards while driving backward usually motivates this choice. The problem considered in this paper is centered around control only (the planning level is not considered here). Thus, this work aims i) at creating the paths of reference to perform U-turn maneuvers (either Ω or U, built from spirals, following the ideas proposed in [START_REF] Boyadzhiev | Spirals and conchospirals in the flight of insects[END_REF]) and ii) at designing nonlinear controllers able to track the predefined shapes. The whole problem is stated in terms of polar coordinates instead of Cartesian ones. This formulation allows to design output feedback controllers which depend on the measurements and not only on the robot absolute position for instance as it has been done in (d'Andrea Novel et al., 1992;d'Andréa Novel et al., 1995). Therefore, unlike the works [START_REF] Asif | Output feedback control for trajectory tracking of wheeled mobile robot[END_REF][START_REF] Yang | Robust tracking control for wheeled mobile robot based on extended state observer[END_REF][START_REF] Shi | Robust finite-time tracking control of nonholonomic mobile robots without velocity measurements[END_REF], there is no need to consider the design of observers to recover the whole state. In this way, metric localization and cumulative errors can be avoided. From a technical point of view, the proposed controllers rely on an input to output linearization based on the expression of the errors dynamics. A nonlinear diffeomorphism based only on the outputs allows to transform the nonlinear model into a linear system. This latter is then controlled using classical linear control laws. Finally, notice that both controllers induce a zero dynamic which is proved to be stable only.

This paper is organized as follows. The next section is devoted to the problem statement. An exogenous model of the spiral shape is first derived using polar information. Then, based on this reference, a nonlinear model of the error dynamics is established. The nonlinear output feedbacks controllers, based on input to state linearization techniques [START_REF] Isidori | Nonlinear control systems[END_REF], allowing to vanish these errors are designed in section 3. Finally, simulation results allowing to highlight the interest and the efficiency of the approach end the ar-ticle.

ROBOT AND U-TURNS MODELING

In this section, the mathematical models used in this paper are introduced. First the model of a differential robot is presented. Next, the spiral model from [START_REF] Boyadzhiev | Spirals and conchospirals in the flight of insects[END_REF] is recalled. Finally, based on the geometry of the spiral, the two shapes of turns are modeled.

System modeling

We consider a differential robot driving in the headland by turning around the last tree of a row. Firstly, a global frame F w = (O w , x w , y w , z w ) represents the position of the tree of interest (see figure 2(a)). Next, the frame F r = (O r , x r , y r , z r ) is attached to the differential robot. The robot states in F w are defined by χ(t) = [d(t) β(t) α(t)] T , where d(t) is the norm of the vector d connecting O w and 0 r , β(t) the angle between x w and d , and α(t) the angle between x r and d. d(t) and β(t) are the polar coordinates of the robot in F w , and α(t) is its orientation. This choice for the state representation is motivated by the fact that both d(t) and α(t) can be directly measured by embedded sensors such as laser range finders, stereo visions systems or Lidars. Thus, it is assumed that the measure is given by:

y(t) = 1 0 0 0 0 1 χ(t) (1)
Finally, the robot control input is defined by [v(t) ω(t)] T , with v(t) the linear velocity along x r and ω(t) the angular velocity around z r . Thus, the kinematic equations are given by:

  ḋ(t) β(t) α(t)   =    -cos(α(t)) 0 -sin(α(t)) d(t) 0 sin(α(t)) d(t) -1    v(t) ω(t) (2)

Spiral modeling

In this work, it is proposed to perform different shapes of turns by tracking paths created thanks to spirals, whose a model is given in [START_REF] Boyadzhiev | Spirals and conchospirals in the flight of insects[END_REF]. Let define a point O p moving on a plane with respect to a fixed point O s (see figure 2(b)). From now on, O s will be considered as the center of the spiral. v is the velocity vector applied to O p and its norm is denoted by v (t). Moreover d is the vector connecting O s to O p whose norm is d (t). Finally α (t) is defined as the oriented angle between v and d . In [START_REF] Boyadzhiev | Spirals and conchospirals in the flight of insects[END_REF] it is shown that if both v (t) and α (t) are constant then O p describes a spiral whose center is O s .

For this reason they are respectively denoted v and α from now on. Moreover the author shows that the dynamics of the distance d is defined as follows:

ḋ = -v cos(α ) (3)
As it can be seen in this equation, the type of performed spiral depends on the sole parameter α Equation ( 3) and its analysis highlight our interest in the spirals. Indeed, adapting the spiral model to our system, i.e., the center of the spiral is the position of the tree of interest with O w = O S , defines a reference path, whose distance dynamics is known, solely based on one point. The reference path is thus defined in the robot sensor space.

. First if 0 < α < π, O p turns counter-clockwise with re- spect to O s otherwise if 0 < α < -π it turns clock- wise. Then if 0 ≤ α < π/2 or 0 ≤ α < -π/2, d (

U-shape turn modeling 2.3.1 U-shape turn in the navigation problem

In the context of an orchard navigation, the robot has to drive from p i , end of the current row, to p f , beginning of the next row (see figure 3(a)). In this specific application, the width of each row is roughly known (e.g., farmers data or Google map) and saved in the robot database. Thus, when the end of the row is detected by a dedicated data processing algorithm, the robot measures d S1 i and extracts d S1 f from its database. It is then possible to compute the spiral S 1 linking those two points, i.e., to compute α for a given v . Moreover, for a given spiral, one can calculate the required turning radius. If this latter is smaller than the robot one, then a U-shape turn is tracked to enter the next row.

U-shape turn error

As previously explained, U-shape turns are performed by tracking a spiral. To do so, it is mandatory to respectively make α(t) and d(t) converge towards α and d (t). Thus, the problem of tracking a U-shape turn can be seen as the problem of finding a controller such that the error (4) converges to zero. 2.4 Ω-shape turn 2.4.1 Ω-shape turn in the navigation problem

e d (t) = d(t) -d (t) e α (t) = α(t) -α (4)
In case the spiral S 1 computed in 2.3.1 requires a too large turning radius, the robot has to perform an Ωshape turn. One recalls that this choice is motivated by the inability of the robot to go backward. An Ωshape turn is made of three parts: i) moving from p i to a spiral S 2 (ϕ 1 ), ii) tracking S 2 (ϕ 2 ), and iii) moving from S 2 to p f (ϕ 3 ). For each part, the robot turning radius has to be large enough to track the reference. Moreover, one has to guarantee the Ω shape, and thus the success of the U-turn. Let define, β ϕ the angular distance to switch from one row to the next one. Moreover, β ϕ 1 , β ϕ 2 and β ϕ 3 are respectively the angular distances of parts ϕ 1 , ϕ 2 and ϕ 3 .

Because

β ϕ 1 + β ϕ 2 + β ϕ 3 = β ϕ , with β ϕ 1 ≥ 0, β ϕ 2 ≥ 0
and β ϕ 3 ≥ 0, it is mandatory that:

β ϕ 1 + β ϕ 3 ≤ β ϕ (5)
Next, let define S min , the spiral centered on O w with the smallest radius d min such that the robot turning radius is large enough to track it. Thus, parts ϕ 1 and ϕ 3 have to make the robot move from/to spiral S 1 to/from a spiral S 2 centered on O w with a radius d S2 ≥ d min while respecting the constraint introduced by equation (5). For this reason, it is proposed to track a distance profile d β (β(t)), guaranteeing that the robot reaches the spiral for a given angular distance.

Distance profile design

To design the distance profile d β (β(t)), we first define normalized angle β(t), with 0 ≤ β(t) ≤ 1, as:

β(t) = β(t) -β(t 0 ) β(t f ) -β(t 0 ) (6)
where t 0 and t f are the initial and final time, and β(t) is based on the encoders measurements. β(t 0 ) and β(t f ) represent therefore the starting angle and the final angle. In addition, dβ ( β(t)) is defined as the normalized distance to track, with 0 ≤ dβ ( β(t)) ≤ 1. In order to obtain a distance profile dealing with the initial and final robot orientations, the path described by the profile is tangent to the spirals. Thus, it is proposed to define dβ ( β(t)) such as:

dβ ( β(t)) = k 0 tan -1 (k 1 ( β(t) + k 2 )) + k 3 (7)
with k 0 , k 1 , k 2 and k 3 , scalar terms used to design the shape of the normalized function. Moreover the choice of those parameters has to guarantee that the distance profile to track does not require a too large turning radius. Finally, using equations ( 6) and ( 7), we obtain:

d β (β(t)) = d(t 0 ) + dβ ( β(t))[d(t f ) -d(t 0 )] (8)
An example of this distance profile is given in figure 3

(b) with k 0 = 0.365, k 1 = 10, k 2 = -0.5, k 3 = 0.5, β(t 0 ) = 0 rad, β(t f ) = π/4 rad, d(t 0 ) = 5 m and d(t f ) = 10 m.

Ω-shape turn error

As previously explained, Ω-shape turns are made of three parts. ϕ 2 consists in a spiral tracking and thus requires to find a controller vanishing the error (4).

Regarding parts ϕ 1 and ϕ 3 , it is mandatory to make d(t) converge towards d β (β(t)). Thus, the problem of making the robot reach a spiral turns into the problem of finding a controller such that the error (9) converges to zero.

e d β (t) = d(t) -d β (β(t)) (9)
U-shape and Ω-shape turns can thus be performed by following a spiral and moving from one spiral to the other. At this stage, as the different turns have been expressed by introducing a set of errors, the next section will be dedicated to the design of controllers allowing to make them vanish.

CONTROLLERS DESIGN

This section proposes to design an output feedback control law which makes the errors (4) and ( 9) converge toward zero asymptotically. As the system is nonlinear, the main idea is to use an exact input to state linearization method proposed by [START_REF] Isidori | Nonlinear control systems[END_REF]. We first present the controller allowing the tracking of a spiral. In a second subsection, an adaptation of this controller is presented to move from one spiral to another one. In this section, it is assumed that v(t) is a constant input set at v . The only remaining input is therefore ω(t).

Spiral tracking

The errors (4) dynamics are defined by:

ėd (t) = v [cos(α ) -cos(α(t))] ėα (t) = -ω(t) + v d(t) sin(α(t)) (10)
Which can be written as follows: t) sin(e α (t) + α ) (11) The main idea of this section is to linearize the error system. To this end, consider the transformation:

ėd (t) = v [cos(α ) -cos(e α (t) + α )] ėα (t) = -ω(t) + v e d (t)+d (
z = z 1 z 2 = e d (t) v [cos(α ) -cos(e α (t) + α )] = T (e) (12)
Notice that T (0) = 0 and in the domain D defined by:

D = (e d , e α ) ∈ ℜ 2 e d ∈ ℜ, e α ∈] -α , -α + π[} (13
) T defines a diffeomorphism. Applying this transformation to the error system leads to: t) sin(e α (t) + α )) (15) Taking:

ż1 = z 2 ż2 = v sin(e α (t) + α ) ėα (t) (14) and ż1 = z 2 ż2 = v sin(e α (t) + α )(-ω(t) + v e d (t)+d (
ω(t) = 1 v sin(e α (t) + α ) ω 2 (t)+ v e d (t) + d (t)
sin(e d (t)+d )

(16) we obtain :

ż1 = z 2 ż2 = ω 2 (t) (17)
where ω 2 (t) is a new control law, which has to be designed. At this stage, as the system ( 17) is linear, a classical linear control law:

ω 2 (t) = -λ 1 z 1 (t) -λ 2 z 2 (t) (18)
with λ 1 , λ 2 > 0, allows to stabilize asymptotically system (17).

Therefore, we propose the following theorem: Theorem 1. Consider two positive scalars, λ 1 , λ 2 , the error system (10) in closed loop with the control law ( 16), ( 17), ( 18), where z = T (e) in ( 12) is locally asymptotically stable. Proof 1. It is sufficient to notice that z converge asymptotically to zero and z = T (e) defines a local diffeomorphism with T (0) = 0. Remark 1. λ 1 and λ 2 are two gains used to tune the speed of convergence of the system in closed loop. Remark 2. The relative degree of the chosen output e d used for the linearization is two while the original state space dimension is three. Therefore, there exists a zero dynamic. Defining β the reference trajectory for β, the β error dynamics is driven by

ėβ (t) = (- sin α d(t) + sin α d )v
Following [START_REF] Isidori | Nonlinear control systems[END_REF], it is straightforward to see that the zero dynamic defined by ėβ (t) = 0 is stable only. As a consequence, in steady state, we may expect a static error between β(t) and its expected value β (t).

Distance profile tracking

From now on, a path leading to the convergence toward a specific spiral is described and thus a function of d β (β(t)) is proposed. Therefore, the error to be minimized can be rewritten as:

e d β (t) = d(t) -[d( β(t 0 )) + dβ ( β(t))d gap ] (19)
The controller designed to ensure the convergence of the error will use the same input to output feedback linearization method than the previous one. Let introduce the following new states

z s = z 1s z 2s = e d β (t) ż1s
and therefore the transformation

T s z s = z 1s z 2s = e d β (t) -v cos(α) + d β (β(t)) sin(α) d = T s (χ(t))
(20) where d β (β(t)) stands for the derivative of d β with respect to β. It can be proved that the transformation T s defines a local diffeomorphism and applying this transformation leads to:

ėd β (t) = -v cos(α(t)) -k 0 k 1 β(t)d gap k 2 1 ( β(t) + k 2 ) 2 + 1 (21)
Calculating the derivative of z 2s along the trajectories of the original system gives:

ëd β (t) =v α(t)sin(α(t)) - k 0 k 1 β(t)d gap (k 2 1 ( β(t) + k 2 ) 2 + 1) + 2k 0 k 3 1 β2 d gap ( β(t) + k 2 ) (k 2 1 ( β(t) + k 2 ) 2 + 1) 2 (22)
where, thanks to the Eq.6, β(t) and β(t) are computed as:

β(t) = v sin(α(t)) d(t)β gap (23) β(t) = - v sin(α(t)) ḋ(t) d 2 (t) β gap + v α(t)cos(α(t)) d(t)β gap (24)
Hence, the z s system can be rewritten as :

ż1s = z 2s , ż2s = f (α(t), β(t), d(t)) + g(α(t), β(t), d(t))ω (25)
where the function f is defined accordingly to ( 22),( 23),( 24).

Choosing the control law for ω such that

ω(t) = 1 g(α, β, d) ( f (α, β, d) + ω 3 (t)) ( 26
)
where ω 3 (t) is a new control allowing to obtain the following system defined by a simple double integrator:

ż1s = z 2s , ż2s = ω 3 (t) (27)
We propose therefore the following control law for

ω 3 (t) ω 3 (t) = -λ 1s ėd β (t) -λ 2s e d β (t), (28)
where λ 1s and λ 2s are two positive scalars which ensure the asymptotic stability of the closed loop system (25) with the control ( 26), ( 28). Therefore we propose the following theorem:

Theorem 2. Consider two positive scalars, λ 1s , λ 2s , the error system (25) in closed loop with the control law (26), ( 28), where z s = T s (χ) in ( 20) is locally asymptotically stable.

Proof 2. Omitted.

Remark 3. The controllers (16), ( 26), shows two singularities, when α(t) = π and α(t) = 0. These singularities are due to the polar coordinates model and the construction of the transformation T . Furthermore, these ones occurs when the robot is facing the last tree or let it on its back. Dealing with these configurations are then easily avoidable.

As previously noted in the last subsection, 3.1, the system in closed loop exibits also a stable zero dynamic.

In this section, we have proposed two output state feedbacks controlling the convergence of the states toward its references. A first one follows a specific spiral based on d (t) and α (t). It is used for the part ϕ 2 of our shapes of turn. The second controller allows performing the convergence toward a spiral with a given angle β through a arctan profile. It will be used during the part ϕ 1 and ϕ 3 of our turns.

SIMULATIONS

In this section, simulations realized with the MATLAB c software are presented. First, we focus on the previously designed controllers, then we couple them in order to simulate Ω-shape turns. For all the simulations: the sampling time is setup as T s = 0.1 s, the spiral center is defined as [0, 0] and the linear velocity is given as v = 0.15 m.s -1 . Moreover, the robot is represented by a set of green and red lines, respectively x r and y r .

Spiral tracking

The first set of simulations shows the performance of the spiral tracking controller given by equation ( 16) setup with λ 1 = 0.1 and λ 2 = 0.5. Here, two different situations are presented. The first column (i.e., figures 4(a), 4(c), 4(e) and 4(g)) presents the behavior of the robot when the initial robot pose is on the spiral to d), 4(f) and 4(h)) presents the behavior of the when its initial pose is not on the spiral. The robot starts at χ(t) = [6, 0, π/2] T and tracks the same spiral. In figure 4(a) it can be seen that the robot tracks accurately the spiral when its initial position belongs to it. In figure 4(b), the robot first converges toward the spiral and then follows it. However, the tracking is not fully accurate and there is an error between the current position and the desired one. As mentioned in section 3.1, it can be explained by a static error on β(t) in steady state. Indeed, as it can be seen in figures 4 (d) and 4(c) displaying the evolution of β(t) and β (t), a static error exists on steady state when the initial robot state does not belong to the spiral. However, the static error on β does not modify the controller performances regarding e α (t) and e d (t) which both converge towards zero (see figures 4(f), 4(e)). Finally, figures 4(h) and 4(g) show the evolution of ω(t) used to track the spiral.

Distance profile tracking

The second set of simulations presents the performances of controller given by equation ( 26), allowing to move toward a specific spiral. The controller is set up with λ S1 = 0.05, λ S2 = 0.1 and d β (β(t)) is defined with k 0 = 0.365, k 1 = 10, k 2 = -0.5, k 3 = 0.5, β(t 0 ) = 0 and β(t f ) = 3π/8. The robot starts at χ(t) = [5, 0, π/2] and has to reach a spiral defined with d (t 0 ) = 10 m and α = π/2 rad. As it can be seen in figure 5(c), the robot reaches the desired spiral at the given angle β(t f ). Moreover, the distance profile is accurately tracked (5(b)) with a smooth command ω(t) (see figure 5(a)).

Ω-shape turn

The third set of simulations presents an example of Ω-turn in the context of a headland navigation. First, a row following task (see [START_REF] Durand-Petiteville | Design of a sensor-based controller performing u-turn to navigate in orchards[END_REF] for more details) is performed, followed by an Ω-turn driving the robot to the next row. Finally, a new row following task is started. For this simulation, the turn is performed with ω MIN-MAX = ±0.08 rad/s -1 and it requires both controllers given by equations ( 16), ( 18), (26), and (28). The parameters λ 1 = 0.01, λ 2 = 0.05 and λ s1 = 0.01, λ S2 = 0.05 are used for their respective controller. A profile is created for ϕ 1 and ϕ 3 with k 0 = 0.365, k 1 = 10, k 2 = -0.5 and k 3 = 0.5. The robot starts at χ(t) = [3, 0, π/2] and needs to reach a spiral defined at d S2 = 8m and α = π/2 rad with a respective angle of reference of β(ϕ 1 ) = 3π/8 rad. Finally, d s1 f = 2.5m with an angle of reference of β(ϕ 3 ) = 3π/8 rad. In addition, to assess the sensitivity of the controllers to errors, a centered Gaussian noise has been added to d(t) and α(t), with a respective amplitude of 0.05m and 1 • . As shown in 6(c), the robot successfully performs a Ω-shape turn by using alternatively both controllers presented above. Indeed the designed distance profiles, as well as the spiral, are accurately tracked by the robot (see figure 6(b)). Moreover, it should be noticed that the static error on steady state during the spiral tracking does not affect the turn. Indeed, the distance profile for ϕ 1 is designed to allow to start the spiral tracking when the robot state belongs to the spiral. Thus, despite this drawback, the controller given by ( 16) provides suitable performances for our specific navigation problem. It can also be noticed that the noise added to the measures does not disturb the behavior of the robot. Finally, an appropriate tuning of parameters defining the distance profiles and spiral allows to perform an Ω-shape turn while dealing with a maximal robot turning radius. In this work, the problem of driving in headlands during an orchard navigation has been addressed. First, U-shape and Ω-shape turns have been modeled using polar coordinates in order to generate the paths to follow when driving in the headland. Then, two nonlinear output state feedbacks controllers have been designed to track the designed paths. Their design was based on input to state linearization techniques.

Simulations have highlighted the suitable performances of the controllers and their usefulness to efficiently drive the robot in orchards headlands. The obtained results are encouraging and must be integrated on our testbed. To do so, it is planned to couple the controllers with a perception system allowing to detect the trees in the surroundings of the robot. Thus, it will be possible to evaluate the efficiency and robustness of the proposed approach.

Moreover, a couple of challenges have to be addressed to obtain a fully autonomous system. The first one consists in automatically generating the spiral and distance profile parameters based on the a priori known and on-line acquired data related to the orchard structure. Moreover, the controller sensitivity to robot state has to be investigated. Thus, it might be required to design a recursive estimation process based on the acquired data to improve the accuracy of the state knowledge. Finally, it seems relevant to guarantee the continuity of the control law when the robot switches from one controller to the other one.

Figure 1 :

 1 Figure 1: Dashed red: U-shape Solid blue: Ω-shape

  Figure 2: Models

  t) decreases with time. In other words, O p is describing an inward spiral around O s . If π/2 < α ≤ π or -π/2 < α ≤ -π then d (t) increases with time which means O p is describing an outward spiral around O s . Finally, if α = π/2 or α = -π/2, d (t) = d (0). O p then describes a circle of radius d (0) around O s .

  Figure 3: U-shape, Ω-shape and distance profile d β (β(t))

Figure 4 :

 4 Figure 4: Simulation of spiral tracking

  Figure 5: Distance profile tracking

Figure 6 :

 6 Figure 6: Ω-shape turn


Thiago A Lima
email: thiago.lima@alu.ufc.br

Sophie Tarbouriech

Bismark C Torrico
email: bismark@dee.ufc.br

Fabrício G Nogueira
email: fnogueira@dee.ufc.br

Energy-based design of dynamic allocation in the presence of saturating actuators *

This paper addresses the design of dynamic allocation functions for systems with saturating actuators. The novelty of the proposal relies on the use of an anti-windup loop to drive the state of the allocator, which in some sense dynamically compensates the difference between the computed control signal and the plant input. Furthermore, the co-design of the allocator and anti-windup loop is presented, allowing to deal with the multiple objective of reducing the allocator error and increasing the closed-loop region of stability. Theoretical conditions are derived in terms of linear matrix inequalities (LMIs) and an optimization scheme consisting in the minimization of the energy of the allocator error is proposed. Two examples borrowed from the literature illustrate the proposed technique and show its effectiveness.

Introduction

Control allocation is widely used in over-actuated systems where the controlled plant is, in general, modeled using torques and forces as inputs, and these forces and torques are generated by the set of actuators (for example microthrusters in space applications) that together produce the desired control effort. As pointed out in [START_REF] Tjønnås | Adaptive control allocation[END_REF], the advantages with the use of the control allocation approach is, in general, modularity and ability to handle contraints.

The control allocation problem is treated in several papers dealing with specific applications, in particular in the aeronautical or spatial contexts: see, for example, the works of [START_REF] Jin | Modified pseudoinverse redistribution methods for redundant controls allocation[END_REF], [START_REF] Oppenheimer | Control allocation for over-actuated systems[END_REF], [START_REF] Boada | Formation flying control for satellites: Anti-windup based approach[END_REF], [START_REF] Durham | Aircraft Control Allocation[END_REF]. Technical solutions are also proposed in the literature from a theoretical point of view. Both the works in [START_REF] Galeani | On input allocation-based regulation for linear over-actuated systems[END_REF] and [START_REF] Serrani | Output regulation for over-actuated linear systems via inverse model allocation[END_REF] consider the output regulation problem of over-actuated systems in the presence of full-information regarding the system states and exogenous inputs. More specifically, [START_REF] Galeani | On input allocation-based regulation for linear over-actuated systems[END_REF] proposes allocation mechanism that takes the form of a hybrid system and accounts for input constraints. In [START_REF] Petersen | Constrained quadratic programming techniques for control allocation[END_REF], optimization-based algorithms, as interior point method, are detailed in order to compute the optimal static allocation. [START_REF] Tor | Control allocation-a survey[END_REF] note that control allocation functions have the primary objective of ensuring that the controller output (noted y c in the paper) is produced jointly by the multiple effectors. Thus, given the influence matrix (noted M in the paper), we want to make sure that the input signal (noted u p in the paper) is equal to the computed control signal (y c ) or that at least the error between them is very small.

The objective pursued in the current paper can be viewed as in the same vein. Indeed, the paper proposes the co-design of a dynamic allocation function and anti-windup loop with the focus of minimizing the allocation error and maximizing the estimation of the region of stability of the closed loop. Although dynamic allocation function has been proposed in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF], there are some fundamental differences with the strategy in 1 this paper that can be shortly listed: i) [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] does not consider the co-design problem of the anti-windup loop and allocation. ii) The parameters of the allocator in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] are manually selected. iii) The formulation in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] is focused in the specific case where the size of the plant input (noted m c in the paper) is equal to the size of allocator output (noted m a in the paper) and the influence matrix is the identity matrix (i.e., M = I). In the current work, we can deal with broader range of systems since we consider the case of m a ≥ m c .

By considering a Lyapunov-based approach, theoretical conditions are derived in terms of linear matrix inequalities (LMIs) in order to solve the co-design of the allocator and anti-windup loop. Furthermore, an optimization scheme consisting in the minimization of the energy of the allocator error is proposed. Its use in two examples borrowed from the literature shows its effectiveness.

The paper is organized as follows. Section 2 is dedicated to present the general view of the control allocation, and to specify the class of the plant, controller and allocation function under consideration. Section 3 presents the main theoretical conditions, together with the associated optimization scheme. In Section 4, two examples borrowed from the literature emphasize the interest of the proposed approach. Finally, in Section 5, concluding remarks and forthcoming issues end the paper.

Notation. For a matrix Y ∈ R n×m , Y ∈ R m×n means its transpose, Y (i) denotes its ith row, while for v ∈ R m , v (i) denotes its ith component. For matrices W = W ∈ R n×n and Z = Z ∈ R n×n , W > Z means that W -Z is positive definite. Likewise, W ≥ Z means that W -Z is positive semi-definite. I and 0 denote identity and null matrices of appropriate dimensions, although their dimensions can be explicitly presented whenever relevant. In this case, 0 n×m represents the n × m null matrix, while I n represents n × n identity matrix. The in A B C denotes symmetric blocks in the expression of a matrix, that is = B . Finally, for matrices W and Z, diag(W, Z) = W 0 0 Z where the null matrices 0 are of appropriate dimensions.

2 Problem formulation

General view

Consider the general view of the control allocation problem shown in Fig. 1. Subsystems C, F , M, and P are the controller, the control allocator, the influence matrix and the plant, respectively. The plant is driven by u p ∈ R mc inputs. The controller computes a set of desired y c ∈ R mc efforts that must be injected in the plant in ideal conditions. The plant input is generated by a set of m a ≥ m c actuators, represented by the signal y f ∈ R ma . In the absence of nonlinearities, u p = My f , and the so-called influence matrix M ∈ R ma×mc maps how each individual effort of the m a actuators combines to generate the inputs acting on the plant. The simplest allocation function is given by the right pseudo-inverse of M, that is, F = M † , with MM † = I.

C F M P • r y c y f u p y p + +
In case the actuators are subject to amplitude limits, the process input, u p , is given by u p = Msat(y f ), where the standard decentralized saturation function is defined as

sat(y f (i) ) = sign(y f (i) ) min{|y f (i) |,u (i) },u (i) > 0, (1)
for i = 1, . . . , m a , where u (i) denotes the amplitude bound in each actuator.

In this case, two main problems arise: i) Guarantees of stability of the closed-loop in the presence of saturation need to be ensured. ii) The application of the pseudo-inverse of M no longer produces null error between the plant input and the controller output. In this context, more complex allocation functions with the ability to handle redundancy and constraints should be applied.

Plant and controller description

Consider the plant P described by the following equations

P ∼ ẋp = A p x p + B p u p , y p = C p x p , (2)
where x p ∈ R np is the plant state vector, u p ∈ R mc is the plant input, y p ∈ R q is the measured output. A p , B p , and C p are all constant and known matrices of appropriate dimensions. Furthermore, the pairs (A p , B p ) and (C p , A p ) are supposed to be controllable and observable.

Let us assume that the plant ( 2) is stabilized by a dynamic output controller C linearly designed via the connection u p = y c , that is without taking into account the saturation. The controller C is defined by the following equations

C ∼ ẋc = A c x c + B c y p + v aw , y c = C c x c + D c y p , (3)
where x c ∈ R nc is the controller state vector and y c ∈ R mc is the controller output. A c , B c , C c , and D c are supposed known. The anti-windup compensation signal v aw is added in order to mitigate the undesired effects of saturation (see, for example, [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Springer[END_REF], [START_REF] Zaccarian | Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation[END_REF]). In this paper, we consider the anti-windup signal

v aw = E c ϕ(y f ), E c ∈ R nc×ma , with the deadzone ϕ(y f ) defined as ϕ(y f ) = sat(y f ) -y f (4)
where the saturation map is defined from (1) and y f is the output of the allocation function.

Remark 1. By construction, the linear connection plant-controller is supposed to be stable. In other words, the controller (3) (with v aw = 0) stabilizes the plant (2) through the linear interconnection u p = y c and therefore the following matrix is Hurwitz

A 0 = A p + B p D c C p B p C c B c C p A c (5)

Dynamic allocation function description

Consider the influence matrix M ∈ R mc×ma , with m a ≥ m c . Let η = {η 1 , η 2 , . . . , η n f }, with n f = m a - rank(M)
, be a basis for the Kernel of M. We then propose the following dynamic allocation function

F ∼ ẋf = A f x f + E f ϕ(y f ), y f = Nx f + M † y c , (6)
where

N = η 1 η 2 . . . η n f ∈ R ma×n f , M † ∈ R ma×mc is such that MM † = I, x f ∈ R n f
is the allocator state vector, and y f ∈ R ma is the allocator output. Matrices A f and E f must be designed to achieve desired behavior of the allocator by taking into account the presence of saturation.

Remark 2. This allocation format is particularly interesting since it has potential to couple with the main requirement of allocation functions, which is to reduce the error between the required plant input signal computed by the controller and the plant actual input. When the system is not saturated, u p = My f = y c since MM † = I and MN = 0. During saturation occurrence, u p = Msat(Nx f + M † y c ) and the state x f is used to improve system behavior and somehow redistribute the control effort between the actuators. The term E f ϕ(y f ) plays an important role since it uses the difference between the saturated actuator output and the required signal.

Let us define the allocator error as e = u p -y c , then e = Msat(y f ) -y c . Using identity (4), the expression e = M (Nx f + ϕ(y f )) is easily obtained. Thus, the mapping y c → e can be defined Note that due to MN = 0, in the absence of saturation one gets e = 0.

y c → e      ẋf = A f x f + E f ϕ(y f ), y f = Nx f + M † y c , e = M (Nx f + ϕ(y f )) (7) F M • y c y f u p e + -
Remark 3. In the rest of the paper, we kept the product MN in the expression of e in order to preserve the general format of the allocator and then to be able to deal with the case studied in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF]. See also Section 3.2.

Closed-loop system and problem formulation

By taking into account the definitions of P , C and F , identity (4) and the connection u p = Msat(y f ), the complete closed-loop system with x = x p x c x f ∈ R n , n = n p + n c + n f , can be written as

     ẋ = Ax + B ϕ ϕ(y f ) y f = Cx e = Hx + Mϕ(y f ), (8)
where

A = A + L f A f L f , B ϕ = B 1 + LE, with A =   A p + B p D c C p B p C c 0 B c C p A c 0 0 0 0   , B 1 =   B p M 0 nc×ma 0 n f ×ma   , E = E c E f L =   0 np×nc L f 0 np×n f I nc 0 nc×n f 0 n f ×nc I n f   , C H = M † D c C p M † C c N 0 0 MN .
The presence of the dead-zone in the closed-loop dynamics (8) implies to characterize a suitable region of the state space in which the stability is ensured (see, for example, [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Springer[END_REF], [START_REF] Zaccarian | Modern Anti-windup Synthesis: Control Augmentation for Actuator Saturation[END_REF]). In general the global asymptotic stability of the origin (that is for any initial condition x(0) ∈ R n ) does not hold except if the open loop has suitable properties of stability [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF]. Hence, the regional stability (that is, only for initial conditions in a neighborhood of the origin) has to be studied. Since exact characterization of the basin of attraction of the origin remains an open problem, a challenging problem consists in providing an estimate of the basin of attraction as accurate as possible. Furthermore, we want to ensure some level of performance to the allocator in terms of the error e, which can be done by imposing conditions that limit the energy of this signal. With respect to (3) and ( 6), the main objective of this paper is to propose the co-design of the dynamic allocation function, that is A f , and E f , along with the controller anti-windup gain E c . Then the problem we intend to solve can be summarized as follows.

Problem 1. Design matrices A f , E f and E c , such that 1. the regional asymptotically stability of the closed-loop system (8) is ensured and the estimate of the region of attraction is maximized.

2. the energy of the allocator error is minimized.

3 Main results

Theoretical conditions

For vectors y f ∈ R ma and θ ∈ R ma , consider the set

S = {y f ; θ; -u (i) ≤ y f (i) -θ (i) ≤ u (i) , i = 1, ..., m a }, (9)
and recall the following Lemma from p.43 in [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Springer[END_REF].

Lemma 1. If y f and θ belong to set S, then the deadzone nonlinearity ϕ(y f ) satisfies the following inequality, which is true for any diagonal positive definite matrix

T ∈ R ma×ma ϕ (y f )T[ϕ(y f ) + θ] ≤ 0. (10)
The following theorem provides a solution to Problem 1.

Theorem 1. Assume there exist positive definite matrices

P = P ∈ R n×n , R = R ∈ R mc×mc , positive definite diagonal matrix S = S ∈ R ma×ma , matrices G ∈ R ma×n , X 11 ∈ R np×np , X 12 ∈ R np×nc , X 21 ∈ R nc×np , X 22 ∈ R nc×nc , X 31 ∈ R n f ×np , X 32 ∈ R n f ×nc , X 33 ∈ R n f ×n f , K f ∈ R n f ×n f , K e ∈ R (nc+n f )×ma such that Ψ =     -X -X Ψ 12 Ψ 13 0 Ψ 22 Ψ 23 -XH -2S -SM -R     < 0, (11)
P G (i) u 2 (i) ≥ 0, (12)
hold with Ψ 12 = P + J + AX -X, Ψ 13 = B 1 S + LK e , Ψ 22 = J + J + AX + XA ,

Ψ 23 = Ψ 13 + G -XC , X =   X 11 X 12 0 X 21 X 22 0 X 31 X 32 X 33   , J =   0 0 0 0 0 0 0 0 K f   . Then, matrices E = E c E f = K e S -1 , A f = K f X -1
33 are solution to Problem 1. In other words:

1. the closed-loop system (8) is asymptotically stable in the ellipsoid ε(P, 1) = {x ∈ R n ; x Px ≤ 1}, with P = XPX and X = X -1 ;

2. the energy of the error signal is limited and given by ∞ 0 e(τ ) Re(τ )dτ < 1, with R = R -1 .

Proof. Consider a quadratic Lyapunov function V(x) = x Px, with symmetric P > 0 ∈ R n×n . Note first that the satisfaction of inequality [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF] means that matrix X is non-singular. The satisfaction of relation [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Springer[END_REF] ensures the inclusion of the ellipsoid ε(P, 1) in the set given by S

= {x ∈ R n ; -u (i) ≤ G (i) x ≤ u (i) },
where G = GX , P = XPX and X = X -1 .

Next we want to verify that V(x) < 0. By choosing θ = Cx -Gx, Lemma 1 applies and one gets -2ϕ (y f )T[ϕ(y f ) + θ] ≥ 0, for any x ∈ S. Hence, if we verify that

L = V(x) -2ϕ (y f )T[ϕ(y f ) + θ] + e Re < 0, (13)
with R > 0, T > 0, then V(x) < 0, and stability of ( 8) is assured. Consequently, V(x) + e Re < 0 is also which can be integrated leading to

∞ 0 e(τ ) Re(τ )dτ < V(x(0)) ≤ 1, (14)
which shows the bound on the energy of the signal e(t). By defining the augmented vector ζ = ẋ x ϕ(y f ) , we can rewrite inequality L < 0 as

L = ζ Qζ < 0, with Q =    0 P 0 H RH G T -C T + H RM -2T + M RM    . (15)
We also have that the relation Bζ = 0 holds for B = -I A B ϕ . From the Finsler Lemma, we have that the following facts are equivalent (see, for example, [START_REF] Maurício | Stability tests for constrained linear systems[END_REF]):

• ζ Qζ < 0, ∀ζ such that Bζ = 0, ζ = 0.

• ∃W such that Q + WB + B W < 0.

By considering W = X X 0 , we obtain the new condition Ψ = Q + WB+B W < 0. By applying a Schur complement to Ψ, followed by pre-and post-multiplying by diag(X -1 , X -1 , T -1 , I) and its transpose, respectively, and making changes of variable R = R -1 , X = X -1 , P = XPX , S = T -1 , G = GX , K f = A f X 33 , K e = ES we obtain condition Ψ < 0. Hence, it follows that if relations [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF] and ( 12) are satisfied then one gets L < 0, or equivalently V(x) < 0, for any x ∈ ε(P, 1). Then the two items of Theorem 1 are proven and the proof is completed.

Remark 4. In case the plant state matrix A p is Hurwitz stable, global stability of the closed loop can be achieved and the design of A f , E f , E c can also be realized by solving LMI [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF] with G = 0. Indeed, this case is straightforward by noticing that making G = 0 imposes that G = 0 and therefore corresponds to making θ = y f , thus turning set S into the whole state-space.

Case when M enters the plant

In some papers, as the one in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF], the influence matrix M enters the plant model. In this case, m a = m c , the system has more inputs than states (m c > n p ) and the input-redundancy nature of the plant is explicit. All the results in this paper can straightforwardly be applied in this case by making M = I and choosing N as a base for the null space of B p , that is, B p N = 0. Although in this case MN = 0, convergence of the allocator error to zero takes place due to the fact that the allocator states x f converges to zero at steady-state, thus the term Nx f also tends to zero and the allocator recovers the property u p = y c after some time.

Optimization issues

From [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF], it becomes clear that maximization of the trace of R leads to minimization, in some sense, of the allocator error e(t). Therefore, while solving the LMIs in Theorem 1 (or in Remark 4), we can accomplish better results for the allocator by minimizing the trace of R. In case of Theorem 1, the maximization of the ellipsoid ε(P, 1) is also of interest. Therefore, a multi-objective convex optimization procedure applies. Consider a positive definite matrix P 0 and the following matrix inequality

P 0 I X + X -P ≥ 0. (16)
Then, minimization of the trace of P indirectly leads to minimization of the trace of P and, therefore, to maximization of the ellipsoid ε(P, 1). Consider weighting parameters ρ 1 , ρ 2 . Then the following convex optimization procedure takes place in case of Theorem 1

min ρ 1 λ 1 + ρ 2 λ 2
subject to ( 11), ( 12), ( 16)

, P 0 ≤ λ 1 I,R ≤ λ 2 I (17)
In case global asymptotic is sought (Remark 4), the following convex optimization procedure applies min λ subject to [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF] with G = 0 and R ≤ λ

Since optimization problems (17) and ( 18) are convex, they can be easily solved with the help from standard LMI solvers and parsers (for example the YALMIP toolbox from [START_REF] Löfberg | Yalmip : A toolbox for modeling and optimization in matlab[END_REF]).

4 Simulation results

Example 1

Consider the satellite formation flying control problem from [START_REF] Boada | Formation flying control for satellites: Anti-windup based approach[END_REF], where the controlled output y p represents the relative position between two satellites in a vertical axis. Given two satellites, the objective is to cancel the lateral position error between them in the zaxis. The process can be represented by the following model

A p B p C p D p =   0 1 0 0 0 0 m -1 1 -m -1 2 1 0 0 0   ,
where m -1 1 and m -1 2 are the masses of the two satellites. The plant input is given by u p = u p1 u p2 , where u p1 and u p2 are forces that act individually in each satellite. Each satellite possesses 4 thrusters that jointly produce the force applied in each of them. The influence matrix is given by M = diag(M 1 , M 2 ), with M 1 = M 2 = 1 -1 -1 1 . We assume that each thruster can produce a a force between 0 mN and 100 mN , therefore the saturation limits are not symmetric. In order to apply the developed conditions, the same symmetrizing technique of [START_REF] Boada | Formation flying control for satellites: Anti-windup based approach[END_REF] takes place, consisting of substituting the asymmetric saturation by a symmetric one with limits u i = 50 mN, i = 1 . . . 8, followed by addition of the kernel symmetrizing vector ξ = u. After choosing m 1 = m 2 = 1000 kg, a stabilizing LQG controller is designed using identity matrices for all the weights. The resulting controller is given by

A c B c C c D c =     -1.7321 1
    , We then compute M † =0.25diag(M 1 , M 2 ), N = diag(N 1 , N 2 ), with N 1 =N 2 = 1 1 -1 I 3
. We use optimization procedure (17) with weights ρ 1 = 1, ρ 2 = 0.15, so that regional asymptotic stability can be guaranteed by means of Theorem 1. The obtained anti-windup and allocator matrices for this example are given by . We simulate the system response for an initial condition of x p (0) = -0.25 0.001 , with x c (0) = 0 and x f (0) = 0. Figure 3 shows the obtained results. It can be observed that the allocation error is indeed reduced by application of the proposed technique.

E c E f =             -0.
            , A f =         -1.

Example 2

Consider the exponentially stable plant from [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF]. The plant is defined by the following data: There is no loss of generality in considering this example since D p = 0. This process is input redundant, as noted in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF]. The saturation limits are given by u = 1 0.01 0.02 . To control the system and guarantee asymptotic tracking of constant references in the absence of saturation, [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] inserts an integrator and designs a stabilizing LQG controller which only uses the first two input channels. The resulting controller is given by

A c B c C c D c =           -1.
0 0 0 0 0 0           .
For this example, m a = m c and M = I. We select then N as the Kernel of B p , resulting in N = -0.4726 -1.3143 1 . In this case, we utilize optimization procedure (18) which allows to establish global asymptotic stability results (Remark 4). We obtain, thus

E c E f =       0.
      , A f = -72.8135.
The parameters of the allocator and the anti-windup of [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] can be found therein. Since we are dealing with the stabilization case, we consider null reference and simulate the system response for an initial condition x p (0) = 0.4 0.4 , with x c (0) = 0 and x f (0) = 0. Figure 4 shows the output response and the allocator errors for both strategies. The computed control signal and the plant input for the proposed strategy and the one from [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF] are shown in Figures 5 and6, respectively. The fundamental difference in the results relies in the fact that for the proposed strategy, the allocator error is kept very small during the whole simulation and converges to zero after the system is out of saturation. It should be reminded, however, that minimizing the allocator error was not a priority in [START_REF] Zaccarian | Dynamic allocation for input redundant control systems[END_REF].

Conclusion

In this paper, we proposed the co-design of dynamic allocation function and anti-windup loop in the presence of saturation of actuators. The allocation function has a general format, which allows to include multiple objective criteria into the design conditions. The developed equations can be used in both the cases when the controller is designed by taking into account the process model from forces and torques and the case when the process model explicitly exhibits the over-actuation phenomenon, that is, when the influence matrix M enters the process model. Simulation case studies were used to illustrate both cases. The results open the door for several future work as in particular to consider other nonlinearities affecting the actuator and the fact to deal with event-triggered control.

Figure 1 :

 1 Figure 1: General view of control allocation problem.

Figure 2 :

 2 Figure 2: Diagram illustrating mapping from y c to allocator error e.

Figure 3 :

 3 Figure 3: Example 1: Output and allocation errors.

A

Figure 4 :

 4 Figure 4: Example 2: Output and allocation errors for both the proposed strategy and [14].

Figure 5 :Figure 6 :

 56 Figure 5: Example 2: Controller output and plant input signals for the proposed strategy.

Acknowledgment

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (Capes) -Finance Code 001, and by ANR project HANDY 18-CE40-0010.

* Manuscript accepted for publication at the 24th International This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal HANDY 18-CE40-0010.