Changhong Wang

Zhaochong Zhang
email: zczhang@cugb.edu.cn

Andrea Giuliani

Sylvie Demouchy

Catherine Thoraval

Lukáš Krmíček

Hongze Bo

Wanfeng Zhang

Xiaoping Xia

Hydrogen Concentrations and He Isotopes in Olivine From Ultramafic Lamprophyres Provide New Constraints on a Wet Tarim Plume and Earth's Deep Water Cycle

Olivines in the Wajilitag aillikites show high water contents (75-168 ppm H 2 O).

• Hydrogen loss in olivines during decompression was limited to the outermost rims, and yields dehydration durations of 15-417 min.

• H 2 O in these rocks derived from a plume source slightly contaminated by recycled hydrous crustal material.

Introduction

Water is the dominant volatile component in the Earth and plays a crucial role in controlling surface climate, the emergence and evolution of life as well as rheology and partial melting processes in the Earth's interior (e.g., [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF][START_REF] Hirschmann | Water, melting, and the deep Earth H 2 O cycle[END_REF][START_REF] Hirschmann | Comparative deep Earth volatile cycles: The case for C recycling from exosphere/mantle fractionation of major (H 2 O, C, N) volatiles and from H 2 O/Ce, CO 2 /Ba, and CO 2 /Nb exosphere ratios[END_REF][START_REF] Kurokawa | Subduction and atmospheric escape of Earth's seawater constrained by hydrogen isotopes[END_REF][START_REF] Marty | The origins and concentrations of water, carbon, nitrogen and noble gases on Earth[END_REF]. It is well established that water is continuously transferred into the mantle through subduction of hydrous minerals, but largely returns to the ocean-atmosphere system through arc volcanism (e.g., [START_REF] Bodnar | Whole Earth geohydrologic cycle, from the clouds to the core: the distribution of water in the dynamic Earth system[END_REF]Cerpa et al., 2022). However, the proportion of water in the slab that can ultimately be transferred to the deep mantle and perhaps contribute to subsequent wet plume formation and flood basalt magmatism is poorly constrained (e.g., [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF][START_REF] Hwang | A role for subducted super-hydrated kaolinite in Earth's deep water cycle[END_REF]. One hypothesis entails that a large fraction of subducted water can be carried into deep mantle [START_REF] Hu | Dehydrogenation of goethite in Earth's deep lower mantle[END_REF][START_REF] Hwang | A role for subducted super-hydrated kaolinite in Earth's deep water cycle[END_REF], whereas other authors favor a scenario whereby water is efficiently released to mantle wedge, crust and ocean-atmosphere system during subduction, leading to low time-integrated H 2 O contents in the deeply subducted slab (Cerpa et al., 2022;[START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF]. Combining He isotope with H 2 O measurements has been proven valuable for tracking the source of water in volcanic rocks including inputs of crustal volatiles into the mantle [START_REF] Marty | The origins and concentrations of water, carbon, nitrogen and noble gases on Earth[END_REF]. Helium isotopes are indeed powerful tracers of primordial contributions to mantle-derived magmas, but can also pinpoint the occurrence of (U-bearing) subducted material in the mantle (e.g., [START_REF] Hanyu | The uniform and low 3 He/ 4 He ratios of HIMU basalts as evidence for their origin as recycled materials[END_REF][START_REF] Jackson | Ancient helium and tungsten isotopic signatures preserved in mantle domains least modified by crustal recycling[END_REF][START_REF] Kurz | Helium isotopic systematics of oceanic islands and mantle heterogeneity[END_REF] especially when combined with other geochemical tracers such as Sr, Nd, Pb and Os isotopes.

Alkaline ultramafic magmas such as ultramafic lamprophyres (UMLs), kimberlites and lamproites are volatile-rich (e.g., H 2 O, CO 2 ) and can provide valuable information on the nature of fusible components in the mantle (>100 km) beneath stable continents [START_REF] Giuliani | Southwestern Africa on the burner: Pleistocene carbonatite volcanism linked to deep mantle upwelling in Angola[END_REF][START_REF] Giuliani | Kimberlites: From Deep Earth to Diamond Mines[END_REF][START_REF] Rock | The nature and origin of ultramafic lamprophyres: alnöites and allied rocks[END_REF]Tappe et al., 2006). Olivine is the first phase to crystallize and, therefore, provides a unique tool for deciphering the primitive composition of these magmas and the nature of their mantle source [START_REF] Giuliani | Kimberlite genesis from a common carbonate-rich primary melt modified by lithospheric mantle assimilation[END_REF][START_REF] Nosova | Olivine in ultramafic lamprophyres: chemistry, crystallisation, and melt sources of Siberian Pre-and post-trap aillikites[END_REF]Sokol et al., 2013;[START_REF] Veter | Trace elements in olivine of ultramafic lamprophyres controlled by phlogopite-rich mineral assemblages in the mantle source[END_REF]. Despite being a nominally anhydrous mineral (NAM), olivine can incorporate trace amounts of hydrogen (H, bonded as hydroxyl) in crystal defects (ionic vacancies and interstitials; e.g., [START_REF] Berry | Fingerprinting the water site in mantle olivine[END_REF][START_REF] Demouchy | Hydrogen, trace, and ultra-trace element distribution in natural olivines[END_REF][START_REF] Mosenfelder | Hydrogen incorporation in olivine from 2-12 GPa[END_REF][START_REF] Newcombe | Water-inolivine magma ascent chronometry: Every crystal is a clock[END_REF][START_REF] Tollan | The role of trace elements in controlling H incorporation in San Carlos olivine[END_REF], and can act as a potential important reservoir for H in the mantle [START_REF] Peslier | A review of water contents of nominally anhydrous natural minerals in the mantles of Earth, Mars and the Moon[END_REF][START_REF] Ohtani | Hydration and Dehydration in Earth's Interior[END_REF].

Alkaline ultramafic magmas generally ascend from the deep mantle very rapidly (10-30 m/s) as indicated by the presence of diamonds [START_REF] Kelley | Rapid kimberlite ascent and the significance of Ar-Ar ages in xenolith phlogopites[END_REF]Sparks et al., 2006) and thus can preserve the original water contents in the core regions of olivine phenocrysts [START_REF] Matveev | FTIR spectroscopy of OH in olivine: A new tool in kimberlite exploration[END_REF]. For example, kimberlitic olivine shows the highest water contents in natural olivine with up to 600 ppm H 2 O (Bell et al., 2004;[START_REF] Hilchie | The origin of high hydrogen content in kimberlitic olivine: evidence from hydroxyl zonation in olivine from kimberlites and mantle xenoliths[END_REF][START_REF] Matsyuk | Hydroxyl in olivines from mantle xenoliths in kimberlites of the Siberian platform[END_REF][START_REF] Matveev | FTIR spectroscopy of OH in olivine: A new tool in kimberlite exploration[END_REF].

The occurrence of fresh, olivine-rich aillikites (or carbonate-rich ultramafic lamprophyres) in the northwest Tarim large igneous province (LIP) (Fig. 1a) provides an ideal opportunity to examine Earth's deep water cycle because these magmas are probably genetically linked to the Tarim plume (Wang et al., 2021a;[START_REF] Zhang | Perovskite and baddeleyite from kimberlitic intrusions in the Tarim large igneous province signal the onset of an end-Carboniferous mantle plume[END_REF]. Based on detailed bulk-rock major-and trace-element and Sr-Nd-Os isotope, olivine major-and trace-element and oxygen isotope, and bulk-carbonate C-O isotope analyses, it was previously suggested that these rocks originated from a moderately depleted sub-lithospheric (plume) source which was marginally contaminated by deeply subducted oceanic crustal components based on radiogenic Os isotopes (Wang et al., 2021a(Wang et al., , 2022)). In this work, we combine new H 2 O content and He isotope data for olivine with new and existing olivine major-and trace-element composition and bulk-rock Sr-Nd-Os isotope results (Wang et al., 2021a(Wang et al., , b, 2022)), including newly acquired bulk-rock Pb isotope data, to constrain the source of water in these rocks.

The water contents and distribution in the olivines were determined by secondary ion mass spectroscopy (SIMS) and Fourier transform infrared spectrometry (FTIR).

Noble-gas analyses were undertaken by in-vacuo crushing of bulk olivine separates.

The goals of this study are (1) to quantify the H 2 O contents of primitive aillikite magmas and their mantle source; and (2) thereby to estimate the source and role of water in the generation of the Tarim LIP.

Geological Background and Petrography

The Tarim craton in northwestern China occupies an area of 600,000 km 2 (Fig. 1a) and is composed of Archean and Neoproterozoic crystalline basement overlain by Phanerozoic sedimentary strata [START_REF] Tian | The Tarim picrate-basalt-rhyolite suite, a Permian flood basalt from northwest China with contrasting rhyolites produced by fractional crystallization and anatexis[END_REF]Zhang et al., 2013a). Permian magmatic activity identified in this region produced more than 260,000 km 2 continental flood basalts (Fig. 1a; [START_REF] Tian | The Tarim picrate-basalt-rhyolite suite, a Permian flood basalt from northwest China with contrasting rhyolites produced by fractional crystallization and anatexis[END_REF]. In addition to the erupted flood basalts and rhyolites as well as minor nephelinites, the Tarim LIP also contains a diverse assemblage of mafic-ultramafic intrusive rocks including aillikites, nepheline syenites, lamprophyres, carbonatites and mafic dyke swarms (Fig. 1b; e.g., [START_REF] Cheng | Subducted slabplume interaction traced by magnesium isotopes in the northern margin of the Tarim Large Igneous Province[END_REF][START_REF] Tian | The Tarim picrate-basalt-rhyolite suite, a Permian flood basalt from northwest China with contrasting rhyolites produced by fractional crystallization and anatexis[END_REF]. The Permian magmatism can be divided into two major pulses according to geochronological studies: (1) flood basalts and felsic volcanic rocks at ~290 Ma [START_REF] Jiang | Petrogenesis and magma source of the ultramafic rocks at Wajilitag region, western Tarim Plate in Xinjiang[END_REF][START_REF] Tian | The Tarim picrate-basalt-rhyolite suite, a Permian flood basalt from northwest China with contrasting rhyolites produced by fractional crystallization and anatexis[END_REF][START_REF] Zhou | OIB-like, heterogeneous mantle sources of Permian basaltic magmatism in the western Tarim Basin, NW China: implications for a possible Permian large igneous province[END_REF]; and (2) mafic-ultramafic intrusions and dyke swarms at ~280-270 Ma (e.g., [START_REF] Zhang | Diverse Permian magmatism in the Tarim Block, NW China: genetically linked to the Permian Tarim mantle plume?[END_REF][START_REF] Zhang | Perovskite and baddeleyite from kimberlitic intrusions in the Tarim large igneous province signal the onset of an end-Carboniferous mantle plume[END_REF].

Wajilitag is situated in the northwestern portion of the Tarim LIP (Fig. 1b).

Permian magmatic rocks in this region consist of diverse coeval intrusions and dyke swarms of variable composition, i.e., aillikites, gabbros, pyroxenites, aegirine nepheline syenites and alkaline lamprophyres-tephrites. Based on geological observations and geochronological data, the formation sequence of these magmas is inferred to be as follow: mafic-ultramafic intrusions (gabbros and pyroxenites; ~284-281 Ma) → aillikites (284-268 Ma) → carbonatites and nepheline syenites (~273 Ma) → alkaline lamprophyres-tephrites and nephelinites (~268 Ma) (Wang et al., 2021a and references therein). The Wajilitag aillikites are the first diamond-bearing rocks identified in NW China (Fig. 1b; [START_REF] Jiang | Petrogenesis and magma source of the ultramafic rocks at Wajilitag region, western Tarim Plate in Xinjiang[END_REF][START_REF] Zhang | Perovskite and baddeleyite from kimberlitic intrusions in the Tarim large igneous province signal the onset of an end-Carboniferous mantle plume[END_REF]. The NWW-SEE trending dykes intruded upper Devonian strata, and were, in turn, crosscut by tephrite and alkaline lamprophyre dykes. The aillikite pipes exhibit elliptical to rounded shape with exposures at surface up to 180 m × 100 m, but their extension at depths is not known. The dimensions of the dykes range between 0.5-2 m in width and 5-7 m in length, but their actual length should be longer due to ubiquitous desert cover.

These aillikites display a distinctive inequigranular porphyritic texture with abundant phenocrysts (30-40%) and nodules (polygranular aggregates, 5-10%) set in a fine-grained matrix (Fig. 2a-c). The ferromagnesian silicate phenocrysts include olivine, clinopyroxene and hydrous minerals (i.e., amphibole and phlogopite). Olivine can be divided into two textural groups: fine-grained microcrysts (<5 mm) and coarse-grained macrocrysts (>5-25 mm). The former are common in all aillikite pipes and dykes (e.g., Dyke 1, 2 and 3), while the latter only occur in the most evolved Dyke 3 aillikites (Fig. 2d-f vs Fig. 2f). As shown in Fig. 2i, primary melt inclusions generally present in abundance in the olivine microcrysts. The matrix consists of phlogopite, apatite, perovskite, titanomagnetite and calcite. Quantitative mineral mode abundances for a representative aillikite sample are present in Fig. S1 in Supporting Information S1. Herein, we mainly focus on olivines separated from three fresh aillikite dykes previously named Dyke 1, 2 and 3 (Fig. 1b). Detailed petrographic descriptions of these dykes can be found in Wang et al. (2021a).

Analytical Methods

Major and Trace Element Analyses

Petrographically fresh aillikite samples collected from three Wajilitag aillikite dykes (Dyke 1, 2 and 3; Fig. 1b) were initially crushed to millimeter sized fragments using a jaw crusher and then fragments of the mafic-ultramafic nodules and crustal xenoliths in the aillikites were carefully removed using a stereoscopic microscope.

Then, the freshest olivine crystals were handpicked under a binocular microscope either mounted in epoxy resin for H 2 O, major-, trace-element determinations or treated separately for bulk-mineral He isotope analyses.

The major and minor element abundances of olivine were analyzed using an EMPA1600 electron probe micro analyzer at the Chinese University of Geosciences, Beijing. Analytical conditions were 20 nA probe current, 15 kV accelerating voltage, 5 μm beam diameter, and peak counting times of 20 s. The results were calibrated by employing synthetic oxides and natural minerals, and matrix corrections were undertaken using a ZAF program. The analytical results are present in Table S1.

The minor and trace elements in olivine selected for major element and water content analyses were measured by LA-ICP-MS employing an Element XR HR-ICP-MS instrument (Thermo-scientific) coupled to a 193 nm ArF excimer laser-ablation system at the National Research Centre for Geoanalysis, Beijing and the Institute of Geology and Geophysics, Chinese Academy of Sciences. Ablation was undertaken in a He atmosphere and Ar was used as the carrier gas (1.5 l/min). Every analysis was performed using a laser spot size of 50-100 μm at 6 Hz with an energy of ~4 J/cm 2 for 40 s preceded by 20 s of gas blank measurement. 29 Si was used as the internal standard. Three glass standards (GSE-1G, NIST SRM 610, NIST SRM 612; Jochum et al., 2011) and one olivine standard (XEN; Batanova et al., 2015;Wu et al., 2020) were analyzed as external references after every eight unknowns. Details of analytical procedures can be found in [START_REF] Wu | Comparison of ultrafine powder pellet and flux-free fusion glass for bulk analysis of granitoids by laser ablation-inductively coupled plasma-mass spectrometry[END_REF]. The minor and trace element analytical results are given in Table S2. The analytical precision and standard deviation are listed in Table S3.

Secondary Ion Mass Spectrometry (SIMS)

Olivine water contents were determined by a CAMECA IMS 1280-HR multicollector ion microprobe at the Guangzhou Institute of Geochemistry, Chinese Academy of Sciences, following the procedures described by [START_REF] Zhang | Optimization of SIMS analytical parameters for water content measurement of olivine[END_REF].

Prior to the analysis, the machine (including the sample chamber without the sample mounts) was baked for 24 hours. The vacuum in the sample chamber was enhanced by keeping an alloy sample mount (52% Sn + 48% Bi) in the high-vacuum storage chamber overnight and then further vacuumed to 5 × 10 -9 mbar. Moreover, an automatic liquid nitrogen cooling technique was applied to cool the sample chamber in order to improve the vacuum performance. Analyses were conducted using a primary Cs + beam (4-5 nA) accelerated at 10 kV. Each spot was pre-sputtered for 200 seconds using a 30 × 30 μm raster beam and then the raster size was reduced to 20 × 20 μm using a narrow field aperture to minimize the water background signals. A normal incidence electron gun was applied to compensate for sample charging, and a nuclear magnetic resonance (NMR) controller was employed to stabilize the magnetic field during the measurement campaign. Other analytical conditions include a 400contrast aperture, ~60 entrance slit, and 50 eV energy slit with a 5 eV energy window.

Although we focused on olivine H concentration quantification, the oxygen isotopic values were also analyzed simultaneously. The 16 O -, 18 O -, and 16 O 1 H -were detected by two Faraday cup detectors and an electron multiplier of monocollector, respectively. Collector slits of 500 mm were employed to collect 16 O -and 18 O -, which yielded a mass resolving power (MRP) of ~2500. A collector slit of ~173 μm was applied for 16 O 1 H -to minimize the isobaric interference of 17 O -, which corresponds to a ~7000 MRP. The olivine water contents were calculated from measured 16 O 1 H -/ 16 O - ratios based on the calibration curve constructed by FTIR H 2 O contents and SIMS 16 O 1 H -/ 16 O -ratios for four olivine reference materials (San Carlos, KLB-1, ICH-30, Mongok; following [START_REF] Zhang | Optimization of SIMS analytical parameters for water content measurement of olivine[END_REF]. The calibration curve ([H 2 O] = 5.5 × 10 6 x -9.12; R 2 = 0.99, Fig. 3) was used here to invert the measured x = 16 O 1 H -/ 16 O -ratios into H 2 O contents. The analytical accuracy and precision of water content measurement were generally better than 10 % in this study.

Fourier-Transform Infrared Spectroscopy (FTIR)

The transmission FTIR spectroscopy analyses were carried out to determine the speciation of water and structural position of hydroxyl groups in olivine at the Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing.

Unpolarized and polarized infrared spectra were obtained using a Bruker Vertex 70V coupled with a mercury cadmium telluride (MCT) detector cooled with liquid nitrogen and a Bruker HYPERION 2000 IR microscope. Unpolarized mid-infrared radiation between 650-4000 cm -1 was produced by a Ge-KBr beam splitter and a Globar light source. The polarized IR radiation was generated using a KRS-5 polarizer (thallium bromideiodide) placed between the sample and the mid-IR source.

The stage was flushed with nitrogen gas during the analysis to limit impact from ambient atmospheric moisture and a background determination was carried out prior to each analysis. Optically clean areas of olivine in doubly polished section (100-150 μm in thickness) were measured with a square aperture of sizes between 50 and 100 μm. For each measurement more than 128 scans were accumulated at 4 cm -1 resolution. Cross-sections with the di-electric vector (E) of the incident infrared light parallel to the principal optical axes (E//α, E//β, or E//γ) of olivine were acquired for olivine grains in three rock samples. The crystallographic orientations of the olivine were obtained from electron backscatter diffraction (EBSD, see Text 1 in Supporting Information S1) analyses prior to FTIR measurements. The integrated absorbance (A) for the olivine, corresponding to the integrated areas beneath the O-H bands, was calculated using the Origin pro8 software. It should be noted that as water contents in the Wajilitag olivine have been quantified using SIMS analysis, the unpolarized and polarized FTIR spectroscopy mainly focused on examining hydroxyl speciation and relative differences of hydrogen concentration (or integrated hydroxyl absorbance) in olivine, respectively.

Helium Isotope Measurements

Helium isotope analyses were performed at the Institute of Mineral Resources, Chinese Academy of Geological Sciences. Before analysis, mineral grains were cleaned ultrasonically in deionized water and washed in acetone for 30 min, and finally dried. Mineral separates (>500 mg each) were loaded into a crusher, baked online at 150 ℃ for 48 h, and then pumped for 7-14 days until blanks were low and stable. The gases released by crushing were purified using a titanium sponge pump at 780 ℃ and Zr-Al pump at 300 ℃ for 20 min to eliminate active gases including water and CO 2 . The remaining gases were exposed to a second cold Zr-Al pump for 10 min.

Comparatively pure He was introduced into the analytical system and further purified using a liquid nitrogen cooled titanium sublimation pump to remove other trace gases such as H 2 . The noble gases isotope compositions were analyzed using a Helix SFT noble gas mass spectrometer. 4 He was analyzed by a Faraday cup and 3 He by an electric multiplier. The mass resolution of >700 can separate 3 He from the HD + H 3 doublet peak. Air standard was analyzed during each analytical session. The hot blank levels of He at 1600 ℃ were 4 He = 2 × 10 -11 cm 3 STP/g (standard temperature and pressure). The contribution of the blank to the analyzed samples was negligible. The analytical results and standard deviation are listed in Table S4.

Bulk-sample Pb Isotopes

The Pb isotope compositions of nine representative aillikite samples were measured by a GV Isoprobe-T thermal ionization mass spectrometer monitored by standard NBS 981 at the Beijing Research Institute of Uranium Geology. About 100-200 mg of rock powder (<200 meshes) was completely dissolved in PFA beakers using a mixture of HF + HNO 3 + HClO 4 for 24 h. Pb was separated and purified using anion exchange resin AG1-X8 with diluted HBr as an eluant. The purified Pb was loaded onto rhenium filaments using a phosphoric acid-silica gel emitter. Total procedure blanks were <100 pg for Pb. During the measurement campaign, repeated analyses of the NBS 981 standard yielded values of 206 Pb/ 204 Pb = 16.937 ± 0.002, 207 Pb/ 204 Pb = 15.457 ± 0.002, and 208 Pb/ 204 Pb = 36.611 ± 0.004, in agreement with the reference values (e.g., [START_REF] Belshaw | A new variable dispersion double-focusing plasma mass spectrometer with performance illustrated for Pb isotopes[END_REF]. The Pb isotope analytic errors were reported as ± 2σ.

Results

Olivine Major and Trace Element Concentrations

The major and minor element compositions of olivine grains which have been selected for water content analyses are present in Figs. 4-6 and reported in Table S1.

The forsterite contents (Fo = 100 × Mg/(Mg + Fe) a.p.f.u.) of the olivine in the aillikites range from 78 to 88. Similar to previous studies (Wang et al., 2021a, b), the olivine can be distinguished between high-Ni (>0.10 wt.% NiO) and low-Ni (<0.10 wt.% NiO) groups (Fig. 4) based on major and minor element compositions. The high-Ni olivine group corresponds to fine-grained microcrysts (<5 mm) identified petrographically in Dyke 1, 2 and 3 (Fig. 2d-f), while the low-Ni olivine group includes large macrocrysts (5-25 mm; Fig. 2f) that only occur in Dyke 3. The high-Ni olivines have higher Fo 78-88 and NiO (0.10-0.47 wt.%) contents but lower MnO (0.10-0.47 wt.%) contents than the low-Ni ones (Fo = 78-81; NiO <0.10 wt.%; MnO = 0.40-0.74 wt.%). Their CaO concentrations overlap (0.06-0.18 wt.% and 0.09-0.14 wt.%, respectively). These compositions are entirely consistent with those reported by Wang et al. (2021a, b;Fig. 4). X-ray element maps show that the SiO 2 , FeO and MgO contents of the olivine microcrysts are almost homogenous with only slight variation along the edges (Fig. S2 in Supporting Information S1).

The trace element compositions of olivine grains are quantified by LA-ICP-MS (Table S2). Concentrations of Ni, Cr, Ti and Al are higher in the high-Ni olivine group compared to the low-Ni one, i.e., Ni = 1149-4164 vs. 23.4-259 ppm, Cr = 16.3-106 vs. 3.6-11.9 ppm, Ti = 26.7-117 vs. 44.7-82.9 ppm and Al = <22.9 vs. <2 ppm. In contrast, Mn, Zn and Li contents are enriched in the latter: Mn = 1319-3133 ppm vs. 3835-5588 ppm; Zn = 109-231 vs. 277-349 ppm; Li = 1.8-4.8 vs. 3.6-7.3 ppm. Other elements show little difference between the two olivine groups, including Ca (579-1101 vs. 559-871 ppm), P (88.9-178 vs. 91.4-218 ppm), and Sc (2.4-4.6 vs. 1.4-9.1 ppm). These trace element compositions are consistent with those reported by Wang et al. (2021a, b;Fig. S3 in Supporting Information S1). A LA-ICP-MS transect across a representative high-Ni olivine microcryst shows that Ni contents decrease and Mn and Ca contents increase from core to rim (Fig. 6).

Content and Distribution of Hydrogen in Olivine

Water contents in the Wajilitag olivine grains measured by SIMS vary from 75 to 168 ppm H 2 O (Table S1; Fig. 5). To avoid potential inclusions of hydrous phases such as serpentine or clinohumite, we used the small beam (<30 μm) and shallow depth of penetration (<2 μm) of SIMS analysis, as well as monitored the H/Si and O/Si count to notice potential contamination by silicate rich (and hydrous) components. For the high-Ni olivine microcrysts, the H 2 O contents increase gradually with decreasing Fo values, and increasing Mn, Ca and Ti contents (Fig. 5). Based on their water contents, the microcrysts can be further divided into high-H 2 O (144-168 ppm H 2 O) and low-H 2 O (75-146 ppm H 2 O) subgroups, with the latter showing higher Fo, Ni, Cr and lower Ca and Ti concentrations, i.e., more primitive composition. The lower-H 2 O, higher-Fo microcrysts predominantly occur in the less evolved Dyke 1 and 2 aillikites, while the higher-H 2 O, lower-Fo microcrysts are mainly present in evolved Dyke 3 (Table S1). The H 2 O contents of the low-Ni macrocrysts are 103-161 ppm H 2 O, and do not show any correlation with Fo or minor elements, except for a positive correlation with Ti concentrations (Fig. 5).

To assess hydroxyl speciation in complement to SIMS measurements, unpolarized FTIR spectra were obtained from optically clear areas of crystallographically unoriented olivine crystals in thick sections from the aillikite pipe and dykes (Fig. 7 and Table S5). Absorption bands due to O-H vibrations in the olivine consist of prominent overlapping peaks at 3572 and 3563 cm -1 , a peak at 3525 cm -1 with a broad asymmetric shoulder at 3542 cm -1 , and a peak at 3484 cm -1 .

Additional four low intensity peaks are observed at 3637, 3620, 3612, 3597 cm -1 .

These spectra resemble hydroxyl spectra of kimberlitic olivines from the Siberian [START_REF] Matsyuk | Hydroxyl in olivines from mantle xenoliths in kimberlites of the Siberian platform[END_REF] and Kaapvaal [START_REF] Peslier | Fast kimberlite ascent rates estimated from hydrogen diffusion profiles in xenolithic mantle olivines from southern Africa[END_REF] cratons. Although the beam was focused to <100 μm, the FTIR collects data through the entire sample thickness; hence, it was unavoidable to measure minor inclusions. The peak at 3693 cm -1 is related to serpentine inclusions (e.g., [START_REF] Matsyuk | Hydroxyl in olivines from mantle xenoliths in kimberlites of the Siberian platform[END_REF]. The infrared absorption spectra of low-Ni olivines are similar to those of high-Ni ones (Fig. 7). No hydrous fluid inclusions related peaks are present in the spectra as no characteristic broad absorption bands centered around 3400 cm -1 were observed (Fig. 7).

Hydrogen distribution within individual crystals was also determined for three high-Ni and one low-Ni olivine grains using polarized FTIR analyses on crystallographically oriented olivine grains (Figs. 6a and8). Cross sections of these olivines reveal bell-shaped profiles with higher and homogeneous absorbance at the center (long flat plateau) compared to the grain edges (Fig. 8).

Helium Isotopes in Olivine

Helium contents and isotopic compositions in the mineral separates from the Wajilitag aillikites are summarized in Table S4. Helium isotopic ratios are reported as R/Ra ratios, i.e., 3 He/ 4 He ratio of measured sample is normalized to the atmospheric ratio (Ra =1.39 × 10 -6 ). As the olivine macrocrysts and microcrysts in Dyke 3 aillikites could not be confidently separated using a binocular microscope, only olivine separates from Dyke 1 and 2 were analyzed because they only contain high-Ni microcrysts. The measured 4 He abundances of the high-Ni olivine separates are 6.27-38.8 × 10 -8 cm 3 STP/g and the 3 He/ 4 He ratios are 5.31-5.84 Ra.

Bulk-Rock Pb Isotopic Compositions

The Pb isotopic compositions of the Wajilitag aillikite pipe and dykes are present in Fig. 9 and Table S6

Discussion

In the following sections, we first use major-and trace-element compositions of olivine combined with FTIR analyses to investigate the mechanism that may control the incorporation of water into the Wajilitag olivines. We then discuss the factors that contribute to the high and variable water contents in these olivines, and assess H 2 O contents of primitive aillikite magmas and their mantle source based on H concentrations in high-Fo olivines. Finally, by combining new olivine water contents and bulk-mineral He isotope analyses with new and existing Pb-Sr-Nd-Os isotope constraints, we attempt to examine the origin and role of water in the generation of the Tarim LIP.

Hydrogen Incorporation in Olivine

As mentioned above, olivines in the Wajilitag aillikites can be divided into high-Ni microcrysts and low-Ni macrocrysts. The well-defined linear correlations between Fo, minor and trace elements (e.g., increasing Mn with decreasing Fo and Ni) (Fig. 4) in the olivine microcrysts suggest that they are phenocrysts crystallized during magmatic differentiation of the host aillikite magmas (Wang et al., 2021a, b). In comparison, the origin of the macrocrysts, whose compositions are displaced from the evolution trend of the microcrysts (Fig. 4), is more complex. These grains probably represent antecrysts derived from precursor highly evolved aillikite melts based on their large size, homogeneous composition, very low Fo 78-81 , Ni and Cr and high Mn and Zn contents (Wang et al., 2021a).

Water contents in the analyzed high-Ni olivine microcrysts exhibit a relatively wide variation of 75-168 ppm H 2 O (Table S1; Fig. 5). For these olivines, water contents are positively correlated with Ti concentrations, but increase gradually with decreasing forsterite contents from Fo 78 to Fo 88 (Fig. 5a). The positive correlation (R 2 = 0.71 (square of the correlation coefficient); Fig. 5b) between H 2 O and Ti contents likely suggests a common incorporation mechanism in these olivines. As shown in Fig. 7, the high-Ni olivine grains show prominent peaks at 3572 and 3525 cm -1 in the FTIR spectra, reflecting the significance of Ti-related point defects in the olivine structure (e.g., [START_REF] Berry | Fingerprinting the water site in mantle olivine[END_REF][START_REF] Kovacs | Site-specific infrared O-H absorption coefficients for water substitution into olivine[END_REF]. Alternatively, these peaks could also relate to the presence of Ti-clinohumite exsolutions lamellae (e.g., [START_REF] Matsyuk | Hydroxyl in olivines from mantle xenoliths in kimberlites of the Siberian platform[END_REF]. However, the homogeneous BSE response of these olivine grains (Figs. 2g-h) demonstrates that the exsolved phases, if present, must be of nanometer size. It is notable that these olivines have considerably higher water contents relative to hypothetical pure Ti-clinohumite-like point defects or lamellae [START_REF] De Hoog | Titanium-and water-rich metamorphic olivine in high-pressure serpentinites from the Voltri Massif (Ligurian Alps, Italy): evidence for deep subduction of high-field strength and fluid-mobile elements[END_REF], thus the composition of the point defects or exsolution phases must deviate from the ideal stoichiometry MgTi[]O 2 (OH) 2 . Additional peaks at 3637, 3620, 3612, 3597, 3563, 3542 and 3484 cm -1 (Fig. 7) are linked to Si-related substitution mechanisms, i.e., OH is incorporated as Mg 2 [V Si '''']OH 4 , where [V Si ''''] represents a Si vacancy (e.g., [START_REF] Berry | Fingerprinting the water site in mantle olivine[END_REF][START_REF] Kovacs | Site-specific infrared O-H absorption coefficients for water substitution into olivine[END_REF]. The lack of correlation (R 2 = 0.13) between H 2 O and SiO 2 seems to be in contradiction with the FTIR observations. However, since Si is a major element in olivine, trace quantities of H substitutions in the tetrahedral site are unlikely to be detected in SiO 2 variations. As shown in Fig. 7d, the two spectra from the low-Ni olivine macrocrysts are similar to the high-Ni microcrysts, suggesting that H-related substitutions associated with Ti-related point defects and Si vacancies may also account for the OH incorporation in the low-Ni olivines. We thus propose that hydrogen in the Wajilitag olivines is related to coupled substitution of H with various vacancies in olivine, not only Ti or not solely Si vacancies, even if dominant. Consequently, the main H diffusion mechanism will be rate-controlled by vacancy diffusivity and not by the faster proton-polaron exchange or interstitials jumps (see [START_REF] Thoraval | Numerical models of ionic diffusion in one and three dimensions: application to dehydration of mantle olivine[END_REF].

Origin of H 2 O Variations in Olivine

Water contents in olivine from the Wajilitag aillikites are higher (75 to 168 ppm (2) assimilation of hydrous crustal materials; (3) addition of H by ionic diffusion from the host volatile-rich aillikite melts after olivine crystallization (Demouchy & Mackwell, 2006); (4) incomplete H loss from olivine upon magma ascent (Demouchy et al., 2006;[START_REF] Ferriss | Rates of dehydration of olivines from San Carlos and Kilauea Iki[END_REF]; and (5) H acquired by magmatic olivine during crystallization (Sokol et al., 2013).

Aillikites may contain abundant serpentine precipitated from post-emplacement hydrothermal fluids (Tappe et al., 2006) and the Wajilitag aillikites are no exception.

Thus, H could be added into olivine during interaction with post-emplacement hydrothermal fluids. However, the extremely low diffusivity of H in olivine at typical serpentinization temperatures for these or similar rocks (commonly ≤400 ℃; Stripp et al., 2006) and at the shallow pressures (<3 km depth) of magma intrusion implies that more than 10,000 years would be required for a millimeter sized olivine crystal to attain high water contents (e.g., >30 ppm H 2 O; Demouchy & Mackwell, 2006).

Considering that the timescale for alkaline ultramafic rocks cooling and hydrothermal activity is centuries to a few millennia for large pipes [START_REF] Sparks | Dynamical constraints on kimberlite volcanism[END_REF], hydrothermal alteration is thus an unlikely reason for the high H concentrations in the Wajilitag olivines. In addition, at these low-temperature conditions, interaction of olivine with hydrous fluids would be more likely to serpentinize olivine rather than increasing water contents by diffusion.

Assimilation of hydrous crustal materials can lead to high water contents in magma and olivine crystallized from the system. However, the presence of diamonds in these aillikites [START_REF] Zhang | Perovskite and baddeleyite from kimberlitic intrusions in the Tarim large igneous province signal the onset of an end-Carboniferous mantle plume[END_REF] indicates that these magmas ascended rapidly toward the Earth's surface and did not stagnate in crustal magma chambers where assimilation would have occurred. Moreover, the lack of correlations between bulkrock SiO 2 contents, trace element ratios (Nb/U and Ce/Pb), 87 Sr/ 86 Sr (i) and 187 Os/ 188 Os (i) (Fig. S4) also rule out a prominent role for crustal assimilation (see Wang et al., 2021bWang et al., , 2022)). Thus, it is less likely that the magma became more hydrous through crustal contamination.

Alkaline ultramafic rocks such as aillikites may contain several weight percent H 2 O retained in various hydrous magmatic phases such as amphibole and phlogopite (Figs. 2c and S1 in Supporting Information S1), suggesting that their parental magmas are enriched in water (e.g., [START_REF] Rock | The nature and origin of ultramafic lamprophyres: alnöites and allied rocks[END_REF]Tappe et al., 2006). Moreover, as H is an incompatible element during igneous differentiation [START_REF] Demouchy | Hydrogen, trace, and ultra-trace element distribution in natural olivines[END_REF], the water content in evolving mafic-ultramafic magmas will continuously increase following fractional crystallization of early-stage nominally anhydrous mineral (NAM) such as olivine and pyroxene. Coupled with the high diffusivity of H in olivine at magmatic temperatures, addition of H into olivine via ionic diffusion from the evolved H 2 O-rich aillikite melts is thus possible (Demouchy & Mackwell, 2006;[START_REF] Peslier | Fast kimberlite ascent rates estimated from hydrogen diffusion profiles in xenolithic mantle olivines from southern Africa[END_REF]. However, this process would generate reversely zoned diffusion profiles with lower H in the cores than at the edges. In contrast, as shown in Figs. 6a and8, OH absorbance profiles across the olivine grains display distinctive core-to-rim depletion of H rather than enrichment. These results rather suggest that the Wajilitag olivines may have been affected by H loss during syn-eruptive degassing.

Alkaline ultramafic magmas typically ascend very rapidly through the lithosphere [START_REF] Sparks | Dynamical constraints on kimberlite volcanism[END_REF], and hence olivine phenocrysts and megacrysts in these rocks can preserve pristine water contents (i.e., minimally affected by diffusion) at least in the cores (Bell et al., 2004;[START_REF] Matveev | FTIR spectroscopy of OH in olivine: A new tool in kimberlite exploration[END_REF][START_REF] Peslier | Fast kimberlite ascent rates estimated from hydrogen diffusion profiles in xenolithic mantle olivines from southern Africa[END_REF].

As shown in Figs. 6a and8, the Wajilitag olivine crystals have homogenous H distribution in the core regions forming flat plateaus (see also [START_REF] Thoraval | Numerical models of ionic diffusion in one and three dimensions: application to dehydration of mantle olivine[END_REF]. In addition, the water contents of high-Ni olivine microcrysts increase gradually with decreasing Fo values (Fig. 5a) and all the examined olivine grains have almost homogeneous Fo values within individual grains (e.g., Figs. 6b andS2) but exhibit wide variability in the grain population . Moreover, the lower-H 2 O, higher-Fo 81-88 olivine microcrysts predominantly occur in the less evolved Dyke 1 and 2 aillikites, while the higher-H 2 O, lower-Fo 78-81 microcrysts are mainly present in evolved Dyke 3 (Table S1). These lines of evidence suggest that (1) the variation of water contents in the Wajilitag olivines is likely mainly controlled by fractional crystallization (recorded by Fo decrease, Fig. 5a); and (2) the H values observed in the cores of most grains represent pristine water contents gained by olivine during crystallization from differentiating and gradually water-richer aillikite magmas.

Estimation of Dehydration Duration

As illustrated in Figs. 6a and8, the H values at the edges of the Wajilitag olivines are generally lower than those in the cores, possibly indicating partial loss of H during transport of the magmatic olivine to the surface. We modelled the H profiles of these olivine crystals using the diffusion equations of Demouchy & Mackwell (2006) to estimate the duration of H loss. Hydrogen diffusion is governed by the following parameters: (1) temperature and thus H diffusion coefficients (D) in olivine at the given temperature; (2) the diffusion mechanism, either the proton-polaron exchange or proton-vacancy diffusion; and (3) the crystallographic direction of H diffusion in olivine crystal (Demouchy & Mackwell, 2006). The equilibrium temperatures of the Wajilitag olivines were computed from olivine-spinel thermometers and yielded 950-1150 ℃ (average 1040 ℃; Wang et al., 2021b). The crystallographic orientations for olivine profiles were established through EBSD analysis. The hydrogen diffusion coefficient in the olivine can be calculated using the following equations established for olivine of Fo 90 composition (Demouchy & Mackwell, 2006): D α and γ = 10 -4.5 exp(-204,000/RT) D β = 10 -1.4 exp(-258,000/RT) where D α, β, γ are the H diffusion coefficients in olivine along the α, β and γ optical axes (γ = [100], α = [010] and β = [001]), R is the gas constant (8.31447 JK -1 mol -1 ) and T is the temperature (K).

The 1-dimensional diffusion equation of Demouchy & Mackwell (2006) was used to model the entire H diffusion profiles (e.g., from edge to edge) across the analyzed olivine grains (Fig. 8):

𝐶 (𝑥) = 𝐶 (erfc( x 2√𝐷𝑡 ) + erfc( (X -x) 2√𝐷𝑡
))

where C OH (x) represents the hydrogen concentrations (or as in this study, the integrated hydroxyl absorbance) at a distance x (m) from the edge, C 0 represents the initial integrated hydroxyl absorbance (e.g., the core absorbance), D is the diffusion coefficient (m 2 s -1 ), X is the width of the diffusion profile from edge to edge, t is the diffusion time (s). Using the H diffusion coefficients (e.g., D α, β or γ ) as calculated above and the olivine equilibration temperature of 1040 ℃, resulting diffusion profiles produced by the fitting curves yield a dehydration duration of between 15 and 417 min (Fig. 8). As aillikite melts do not stagnate in crustal magmatic chambers and ascend rapidly to the surface (e.g., [START_REF] Sparks | Dynamical constraints on kimberlite volcanism[END_REF], thus the temperature difference between olivine crystallization and H diffusion during magma degassing is expected to be small (e.g., <100 ℃). Using experientially determined diffusion coefficients of H (e.g., Dα, β or γ) for a shallow magma temperature of 940 ℃, the calculated diffusion profiles yield durations of between 40-660 min for the four olivine grains in Fig. 8. These dehydration durations significantly overlap with the range (15-417 min; Fig. 8) calculated from a higher temperature of 1040 ℃, indicating a limited effect of a slightly lower temperature. This is in agreement with previous study by [START_REF] Thoraval | Numerical models of ionic diffusion in one and three dimensions: application to dehydration of mantle olivine[END_REF].

We also modelled diffusion in 3-dimensions following the method of [START_REF] Thoraval | Numerical models of ionic diffusion in one and three dimensions: application to dehydration of mantle olivine[END_REF] to calculate the duration of dehydration for the observed concentration profiles across the four olivine grains (Ol-1, Ol-2, Ol-3, and Ol-4; Fig.

S5 in Supporting Information S1). The fastest direction of diffusion for H is [001] and dehydration is predominantly driven by this crystallographic direction. As profiles were acquired on thin section, there is uncertainty on the initial geometry of each olivine grain (only 2 directions known and not 3). Nevertheless, we decided to approximate the grain shape with a rectangle shape with [001] twice the length of ) calculated above for a temperature of 1040 ℃. Figure S5 displays the results of the modelling for each crystallographic axis and for each olivine. Similar to 1D diffusion models, the 3D model yields very short durations (16 min) of dehydration for Ol-1 and Ol-2 and longer durations (135-592 min) for Ol-3 and Ol-4 (see Fig. S5 in Supporting Information S1).

Water Content in the Parental Aillikitic Magmas and their Mantle Source

As the water contents of the Wajilitag olivines are affected by magma differentiation and, to a very localized extent, volatile degassing, only the cores of olivine crystallized at the earliest stage can be used to infer the H 2 O contents of the primitive aillikite melt [START_REF] Peslier | Fast kimberlite ascent rates estimated from hydrogen diffusion profiles in xenolithic mantle olivines from southern Africa[END_REF]. The highest forsterite values (Fo 87.7 ) of the analyzed olivines (Table S1) are similar to those of the most forsteritic olivine phenocrysts crystallized from primitive mantle-derived melts [START_REF] Herzberg | Identification of source lithology in the Hawaiian and Canary Islands: Implications for origins[END_REF].

Furthermore, the Mg# value (74.1) of the equilibrium melts for these high-Mg olivine grains calculated using the Fe-Mg partition coefficient (𝐾𝑑 ) of 0.40 for silicate-carbonate melts [START_REF] Giuliani | Insights into kimberlite petrogenesis and mantle metasomatism from a review of the compositional zoning of olivine in kimberlites worldwide[END_REF] and references therein) is marginally higher than the aillikite sample with highest Mg# (73.1, Table S7; Wang et al., 2021a).

Therefore, it is suggested that these high-Fo olivines crystallized from primitive aillikite melts and can be used to infer the initial melt H 2 O contents. The experimentally determined partition coefficient for H between olivine and carbonated silicate melt (𝐷 ) is the key parameter for this calculation and ranges from 0.006-0.009 to 0.002-0.004 with molar CO 2 /(CO 2 +SiO 2 ) in the melt varying from 0 to 0.4 [START_REF] Aubaud | Hydrogen partition coefficients between nominally anhydrous minerals and basaltic melts[END_REF]Bell et al., 2004;Sokol, et al., 2013). Using a H 2 O content of 93 ppm H 2 O for olivine (e.g., average value for the four highest-Fo olivine compositions, Fig. 5a), and partition coefficients between olivine and primitive aillikite melt (CO 2 /(CO 2 +SiO 2 ) = 0.3-0.4; Tappe et al., 2006;Wang et al., 2021a) of 0.002-0.006, the water contents of the melt are estimated as 1.6 to 4.7 wt.%. The water content of the primitive aillikite melt is therefore higher than the range shown by ocean island basalts (OIB, 0.3-1.0 wt.%) and mid-ocean ridge basalts (MORB, 0.1-0.3 wt.%), but overlaps those of arc magmas (2.0-8.0 wt.%) (Xia et al., 2016 and references therein). The water content resembles primary H 2 O content values proposed for natural kimberlite melts (2-9 wt.%) [START_REF] Moussallam | H 2 O-CO 2 solubility in low SiO 2melts and the unique mode of kimberlite degassing and emplacement[END_REF].

Next, we estimate the H 2 O content of the mantle source to infer if aillikites derive from mantle domains enriched in H 2 O. The garnet-bearing peridotite versus carbonated silicate melt partition coefficient (𝐷 ) for H 2 O depends on mineral abundances in the peridotite source and their respective mineral D values (Bell et al., 2004;Sokol et al., 2013). This bulk 𝐷 for peridotite-melt is calculated by combining individual mineral D values with their modal proportions in typical garnet-bearing peridotite source (mineral modes taken from Sokol et al. (2013): 72% olivine, 21% orthopyroxene, 6% garnet, 3% clinopyroxene, and 1% spinel).

Assuming that 𝐷 values = 0.002-0.006, 𝐷 = 2 [START_REF] Ardia | H 2 O storage capacity of olivine at 5-8 Gpa and consequences for dehydration partial melting of the upper mantle[END_REF][START_REF] Aubaud | Hydrogen partition coefficients between nominally anhydrous minerals and basaltic melts[END_REF][START_REF] Hirschmann | Dehydration melting of nominally anhydrous mantle: The primacy of partitioning[END_REF] and 𝐷 = 5 [START_REF] Mookherjee | Aluminate speciation in H 2 O at high pressures and temperatures[END_REF], 𝐷 between garnet-bearing peridotite source and carbonate-silicate melt (CO 2 /(CO 2 +SiO 2 ) = 0.3-0.4) ranges from 0.005 to 0.01. The water content of the peridotite source for the Wajilitag aillikites is estimated using models for both batch and fractional melting, and assuming a partial melting degree of 0.7-1.5% (Tappe et al., 2006; detailed calculations are present in Text 2 in Supporting Information S1).

The results show that the calculated water contents in the mantle source of these aillikites are 150-1200 ppm H 2 O (average 520 ppm H 2 O), which is significantly higher than those of the depleted upper mantle (MORB source) (e.g., <200 ppm H 2 O; [START_REF] Hirschmann | Comparative deep Earth volatile cycles: The case for C recycling from exosphere/mantle fractionation of major (H 2 O, C, N) volatiles and from H 2 O/Ce, CO 2 /Ba, and CO 2 /Nb exosphere ratios[END_REF][START_REF] Shimizu | Identifying volatile mantle trend with the water-fluorine-cerium systematics of basaltic glass[END_REF], but falls in the range of source water contents for OIBs (300-1000 ppm H 2 O; [START_REF] Bizimis | Water in Hawaiian garnet pyroxenites: implications for water heterogeneity in the mantle[END_REF]. A previous study [START_REF] Xia | High water content in primitive continental flood basalts[END_REF] based on the analyses of clinopyroxene phenocrysts in flood basalts also proposed that the Tarim mantle plume was enriched in water (up to 3000-5000 ppm H 2 O). If these elevated water contents are typical of the Tarim plume, they help explain the occurrence of voluminous magmatism in the Western Tarim LIP due to the dumping effect of water on the peridotite solidus (e.g., [START_REF] Kawamoto | Melting temperature and partial melt chemistry of H 2 O-saturated mantle peridotite to 11 gigapascals[END_REF][START_REF] Liu | Compositional variation of picrites in the Emeishan large igneous province modulated by water in the mantle plume[END_REF].

Implications for Earth's Deep Water Cycle

It has been previously shown that the majority of igneous units in the Tarim LIP including basalts, aillikites, alkaline lamprophyres, nephelinites and mafic-ultramafic intrusions are characterized by Fe-and Ti-rich signatures and enriched (OIB-like)

primitive mantle nominalized trace element patterns [START_REF] Cheng | Subducted slabplume interaction traced by magnesium isotopes in the northern margin of the Tarim Large Igneous Province[END_REF][START_REF] Zhou | OIB-like, heterogeneous mantle sources of Permian basaltic magmatism in the western Tarim Basin, NW China: implications for a possible Permian large igneous province[END_REF]. Combination of these geochemical features with constraints from Sr-Nd-Os-C-O-Mg isotopes has been used to invoke a mantle plume source polluted by subduction-related components for these magmas and related carbonatites (e.g., [START_REF] Cheng | Subducted slabplume interaction traced by magnesium isotopes in the northern margin of the Tarim Large Igneous Province[END_REF]Wang et al., 2022). This interpretation is further reinforced by previous regional geological studies, which suggest that the Southern Tianshan oceanic plate was subducted beneath the Tarim craton in Paleozoic period (e.g., ca.

200 Ma before the formation of the Tarim LIP; [START_REF] Charvet | Paleozoic tectonic evolution of the Tianshan Belt, NW China[END_REF][START_REF] Ge | The Paleozoic northern margin of the Tarim Craton: passive or active?[END_REF][START_REF] Wang | Geochemical constraints on the nature of mantle source for Cenozoic continental basalts in east-central China[END_REF]. Our new results for the Wajilitag aillikites that were emplaced shortly after the eruption of the Tarim flood basalts provide further insights into the nature of the Permian Tarim plume.

The high-Ni olivine microcrysts in Dyke 1 and 2 originated from primitive aillikite melts (Wang et al., 2021a). Therefore, the noble gases of these olivines can be used to constrain the mantle source compositions of the aillikites. The 3 He/ 4 He isotope ratios (5.31-5.84 Ra; Table S4) of these olivine grains are much higher than those of continental crust and atmosphere [START_REF] Allegre | Rare gas systematics: formation of the atmosphere, evolution and structure of Earth's mantle[END_REF][START_REF] Ballentine | Production, release and transport of noble gases in the continental crust[END_REF], but slightly lower than those of MORBs (8 ± 1 Ra; [START_REF] Graham | Noble gases in MORB and OIB: observational constraints for the characterization of mantle source reservoirs[END_REF] and significantly lower than those of hot and buoyant plumes sourced from the lowermost mantle (~8-50 Ra; [START_REF] Jackson | Primordial helium entrained by the hottest mantle plumes[END_REF]. These He isotope compositions overlap the range of xenoliths from the sub-continental lithospheric mantle (SCLM, 5-6 Ra) [START_REF] Day | Absence of a high time-integrated 3 He/(U + Th) source in the mantle beneath continents[END_REF][START_REF] Day | The helium flux from the continents and ubiquity of low-3 He/ 4 He recycled crust and lithosphere[END_REF][START_REF] Gautheron | He, Ne and Ar composition of the European lithospheric mantle[END_REF], which may indicate that these aillikites derived from a SCLM source. However, the depleted Sr-Nd isotopes ( 87 Sr/ 86 Sr (i) = 0.70340-0.70390; εNd (i) = +1.3 to +4.9; Wang et al., 2022) of primary magmatic apatite and perovskite in the aillikites are more consistent with an asthenospheric (plume) mantle source rather than metasomatised domains in the lithospheric mantle, which would rather feature geochemically enriched Sr-Nd isotope compositions.

Therefore, the lower-than-MORB He isotopes are better attributed to the influence of deeply recycled hydrous crustal material in the source, which contained some U and therefore in-grew 4 He during mantle residence (e.g., [START_REF] Hanyu | The uniform and low 3 He/ 4 He ratios of HIMU basalts as evidence for their origin as recycled materials[END_REF]. This interpretation is consistent with moderately radiogenic Os isotope compositions of these rocks (Table S7; Wang et al., 2022) and new bulk-sample Pb isotope data, which are intermediate between PREMA (i.e., the possible deep-mantle component common to aillikites and kimberlites; [START_REF] Giuliani | Remnants of early Earth differentiation in the deepest mantle-derived lavas[END_REF] and EM (i.e., the recycled hydrous crustal component) (Fig. 9). This multi-isotopic approach highlights the complex make-up of the Tarim plume and the presence of deeply subducted (or delaminated) hydrous crustal components, which contributed to the formation of voluminous volcanic outpours.

The subducted (or delaminated) hydrous crustal components in the Tarim plume are likely derived from the deeply subducted Southern Tianshan oceanic slab [START_REF] Cheng | Subducted slabplume interaction traced by magnesium isotopes in the northern margin of the Tarim Large Igneous Province[END_REF] [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF]. If so, the deeply subducted oceanic slab might have lost more than 93% of its original water content since subduction.

These dehydration estimates are consistent with other approaches (i.e., based on H isotope and H 2 O/Ce variation of oceanic crust before and after subduction; [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF][START_REF] Shaw | Hydrogen isotopes in Mariana arc melt inclusions: Implications for subduction dehydration and the deep-Earth water cycle[END_REF].

Conclusions

This study presents H 2 O, major-, trace-element and bulk-mineral He isotope The positive correlation between H 2 O and Ti in these olivines, combined with infrared hydroxyl bands at 3572 and 3525 cm -1 , suggests that H is mainly incorporated as Ticlinohumite-like point defects. FTIR profile analyses using polarized infrared radiation reveal water loss during decompression, but limited to the rims of olivines.

Conversely, long flat plateaus with homogeneous water concentration in the cores provide pristine magmatic water contents of liquidus olivine. Using experimental diffusion data for H, the duration of H loss is estimate as 15-417 min for the olivines with lower water contents in the rim zones. Based on the water contents of the cores of primitive high-Fo olivines, the minimum water contents of the primitive aillikite melts and their mantle source are estimated as 1.6-4.7 wt.% and 150-1200 ppm H 2 O, respectively. These water contents in the mantle source are higher than those typical of the depleted upper mantle (i.e., MORB source) hence requiring addition of a waterbearing component. Additionally, the 3 He/ 4 He ratios (5.31-5.84 Ra) of primitive high-Ni olivines are slightly lower than the range of MORBs (i.e., depleted upper mantle).

In combination with depleted Sr-Nd isotopes of primary apatite and perovskite, and moderately radiogenic Pb and Os isotopes of bulk samples (Wang et al., 2022), these new results suggest that water in the aillikites was mainly sourced from a plume contaminated by deeply recycled (U-bearing) hydrated crustal material. This study reinforces the view that the Tarim plume was wet [START_REF] Xia | High water content in primitive continental flood basalts[END_REF], but clarifies that much of this water was stored in deeply subducted material that polluted the plume source in the lowermost mantle. PREMA mantle are also plotted for comparison (Stracke, 2012 and references therein). NHRL: Northern Hemisphere Reference Line [START_REF] Hart | Heterogeneous mantle domains: signatures, genesis and mixing chronologies[END_REF].

  . The Pb isotopic compositions of the examined aillikite samples yield present-day values for 206 Pb/ 204 Pb = 19.21-20.52, 207 Pb/ 204 Pb = 15.65-15.74 and 208 Pb/ 204 Pb = 39.31-41.00. The measured U/Pb and Th/Pb ratios range between 0.24-0.71 and 0.87-2.51, respectively. The age-corrected (270 Ma) initial Pb isotope ratios define a relatively small range, including 206 Pb/ 204 Pb (i) = 18.30-18.90, 207 Pb/ 204 Pb (i) = 15.60-15.66 and 208 Pb/ 204 Pb (i) = 38.54-39.50. These compositions feature relatively high in 207 Pb/ 206 Pb and 208 Pb/ 206 Pb and are broadly intermediate between typical Enriched Mantle (EM) and Prevalent Mantle (PREMA) compositions (Fig. 9).

  H 2 O; Fig.5) than those from alkali basalts (4-70 ppm H 2 O,[START_REF] Demouchy | Distribution and transport of hydrogen in the lithospheric mantle: A review[END_REF] and South Africa cratonic peridotites (up to 120 ppm H 2 O, see[START_REF] Peslier | A review of water contents of nominally anhydrous natural minerals in the mantles of Earth, Mars and the Moon[END_REF], while the highest water contents are as high as those reported from the Siberian cratonic peridotites (200 ppm H 2 O,Doucet et al., 2014, within 20 % error bar). Five possible mechanisms may account for these observations, including (1) post-emplacement hydrothermal processes (e.g., serpentinization;[START_REF] Stripp | Post-emplacement serpentinization and related hydrothermal metamorphism in a kimberlite from Venetia, South Africa[END_REF];

[

  100] and[010] for Ol-1, and [010] twice the length of [001] and[100] for Ol-3, with the initial water concentration set as the concentration in the central plateau. Modeled diffusion profiles are calculated using the H diffusion coefficient (e.g., D [100],[ 010] or[001]

  compositions of olivine from the Wajilitag aillikites in the Tarim large igneous province, NW China. Olivines in the aillikites are divided into two groups: (i) high-Ni microcrysts (<5 mm) and (ii) low-Ni macrocrysts (5-25 mm). The former are phenocrysts crystallized directly from the aillikite melts, while the latter probably represent antecrysts produced from evolved aillikite melts during early pulses of magmatic activity. The water contents are high for both the microcrysts and macrocrysts and range between 75-168 ppm H 2 O and 103-161 ppm H 2 O, respectively.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. (a) Representative field photograph of the Wajilitag aillikites. (b)

Figure 3 .

 3 Figure 3. Water content calibration curve for olivine analyses by SIMS. Independent

Figure 4 .

 4 Figure 4. Fo versus NiO (a), CaO (b), MnO (c) and Ti concentrations for olivine

Figure 5 .

 5 Figure 5. Water contents determined by SIMS as a function of selected major and

Figure 6 .

 6 Figure 6. Compositional traverse for a representative olivine microcryst shown in

Figure 7 .

 7 Figure 7. Unpolarized Fourier transform infrared absorption spectra of randomly

Figure 8 .

 8 Figure 8. Profiles of H absorbance values (normalized to thickness of 1 cm) and

Figure 9 .

 9 Figure 9. Age-corrected (270 Ma) initial 206 Pb/ 204 Pb (i) vs. 87 Sr/ 86 Sr (i) (a), εNd (i) (b), 207 Pb/ 204 Pb (i) (c) and 208 Pb/ 204 Pb (i) (d) diagram for the Wajilitag aillikites. The Pb, Sr and Nd isotope compositions of MORB, HIMU (high 238 U/ 204 Pb ratio), enriched and

  Figure 1.

  Figure 2.

  Figure 3.

  Figure 4.

  Figure 5.

  Figure 6.

  Figure 7.

  . To constrain how much water was retained in the subducted crustal component, we here employ the approximate H 2 O contents (i.e., 520 ppm) of the mantle plume source of the Wajilitag aillikites. Based on Os isotope mass balance calculationsWang et al. (2022) suggested that less than one third of recycled oceanic crustal material can occur in the Tarim plume. If we assume 30% of water in the mantle plume source was originated from deeply subducted oceanic material and the

	remaining 70% from a depleted mantle source containing 200 ppm H 2 O, the recycled
	oceanic crust should contain ca. 1300 ppm H 2 O. It is suggested that globally oceanic
	crust contains 2-3 wt.% H 2 O

Acknowledgements

We are indebted to Weiliang Kong, Bingxiang Liu, and Mingde Lang for their invaluable support during field work. The authors would like to acknowledge Jinwu Yin, Jinhua Hao, Linghao Zhao, Xiaoguang Li, Bo Zhang, Feng Liu and Xiaojia Wang for their technical support. Qunke Xia, Xiaoyan Gu and Shitou Wu are thanked for their constructive suggestions and discussion. This paper benefitted from the constructive reviews of Anna Nosova and an anonymous referee, and the editorial handling of Mark Dekkers and Susanne Straub. This work was supported by the National Natural Science Foundation of China (42030302, 42102054) and China Postdoctoral Science Foundation (H29291). LK was supported by the EXPRO 2019 project of the Czech Science Foundation (No. 19-29124X).

Data Availability Statement

The authors comply with the AGU's data policy, and the data sets of this paper are available at https://doi.org/10.6084/m9.figshare.21067804.v5.


Mari Chaikovskaia
email: mari.chaikovskaia@etu.uca.fr

Jean-Philippe Gayon
email: j-philippe.gayon@uca.fr

Zine Elabidine Chebab
email: ze.chebab@mecabotix.com

Jean-Christophe Fauroux

Sizing of a fleet of cooperative robots for the transport of homogeneous loads

We consider the problem of determining the number of robots necessary to transport a set of homogeneous loads in a given time interval from a zone A to a zone B, at minimum cost. The cost is function of the number of robots and of the distance travelled by robots. The operations are divided into several phases: loading, loaded travel, unloading, empty travel and battery charging. We first consider the case of noncooperative robots for which we derive a closed-form expression for the optimal number of robots. We then consider the case of cooperative robots where loads can be carried either by a single robot (mono-robot) or by several robots that cooperate (poly-robot). The fleet sizing problem can be formulated as a mathematical programming. We distinguish several scenarios, depending on the respective carrying capacity of mono-robots and poly-robots. We also address the infinite horizon problem which models a fleet of vehicles operating permanently and leads to simpler results.

I. INTRODUCTION

A warehouse of a manufacturing company today is characterized by dynamic production processes governed by the demands of a rapidly changing global economy, such as the increasing number of product variants, customization of products and responsiveness to changing market conditions. In order to be competitive, companies are forced to seek innovative robotic solutions to operate their warehouses. Thus, certain giants of the online trade organize competitions to develop autonomous robots for the pick-and-place tasks [START_REF] Morrison | Cartman: The low-cost cartesian manipulator that won the amazon robotics challenge[END_REF]. A particular development over the last decade has taken place for AGVs (Automated Guided Vehicles) and AMRs (Autonomous Mobile Robots) [START_REF] Oyekanlu | A review of recent advances in automated guided vehicle technologies: Integration challenges and research areas for 5gbased smart manufacturing applications[END_REF], especially in logistics warehouses and industrial production [START_REF] Andersson | Agv & amr robotics[END_REF]. In our work, we use the term AGV in both cases, considering the robots individually and also in cooperation.

The benefits of cooperation are clearly demonstrated by the animal world. Collaboration can be of different types, for example, the execution of a task by several subordinates under the direction of a leader, or the execution of the same task with the same level of responsibility [START_REF] Dugatkin | Cooperation among animals: an evolutionary perspective[END_REF]. The cooperative approach can be applied not only in nature, but Fig. 1: Different modes of cooperation [START_REF] Chebab | Conception et commande collaborative de manipulateurs mobiles modulaires[END_REF] in industry [START_REF] Le | Coopération dans les systèmes multi-robots: contribution au maintien de la connectivité et à l'allocation dynamique de rôles[END_REF]. Cooperative robots capable of working in parallel on the same task open wide perspectives [START_REF] Noreils | Cooperation between mobile robots and industrial applications: Some perspectives[END_REF]. For example, a large load can be transported by several small robots connected to the load (co-manipulation mode) or one robot can transport the load while connecting to another robot to increase stability (connection mode) [START_REF] Chebab | Conception et commande collaborative de manipulateurs mobiles modulaires[END_REF] as illustrated in the Figure 1.

We use the concepts and terminology developed in [START_REF] Chebab | A method for structural synthesis of cooperative mobile manipulators[END_REF]:

• Mono-robot (m-bot) = An elementary robot, which is designed to work on its own or with others; • Poly-robot (p-bot) = A set of p m-bots, which cooperate on the same task. Different questions arise when operating a fleet of robots, such as the design of the warehouse architecture [START_REF] Van Geest | Design of a reference architecture for developing smart warehouses in industry 4.0[END_REF], trajectory planning with obstacle and collision avoidance [START_REF] Cardarelli | Cooperative cloud robotics architecture for the coordination of multiagv systems in industrial warehouses[END_REF], [START_REF] Lee | Smart robotic mobile fulfillment system with dynamic conflict-free strategies consid-ering cyber-physical integration[END_REF], service policy [START_REF] He | Differentiated service policy in smart warehouse automation[END_REF] and battery charging [START_REF] Zou | Evaluating battery charging and swapping strategies in a robotic mobile fulfillment system[END_REF].

In this article, we are interested in the sizing of a fleet of cooperative robots for the transport of standardized loads that are all identical, which we will refer to as homogenous loads.

AGVs are generally expensive and determining the right type and number of vehicles is crucial. The pioneering work, [START_REF] Egbelu | The use of non-simulation approaches in estimating vehicle requirements in an automated guided based transport system[END_REF] highlights several factors influencing the required number of vehicles, such as system layout, location of load transfer points, trips frequency, vehicle-dispatching strategy, system reliability and speed of travel.

Different works are interested in the sizing of robot fleets. They can be divided schematically into two categories: the stochastic models [START_REF] Arifin | Determination of vehicle requirements in automated guided vehicle systems: a statistical approach[END_REF], [START_REF] Koo | Estimation of part waiting time and fleet sizing in agv systems[END_REF], [START_REF] Choobineh | Fleet sizing of automated guided vehicles: a linear programming approach based on closed queuing networks[END_REF] and the deterministic models [START_REF] Rjeb | Fleet-sizing of robots in a logistics warehouse -Transport operation between reception area and storage area[END_REF], [START_REF] Egbelu | The use of non-simulation approaches in estimating vehicle requirements in an automated guided based transport system[END_REF]. As in our work we are developing a deterministic mathematical model, we will focus on the last two works in more detail. The work [START_REF] Egbelu | The use of non-simulation approaches in estimating vehicle requirements in an automated guided based transport system[END_REF] proposes four analytical approaches to estimate the number of robots, giving examples for each of them. The author proposes to consider add dispatching rules and then simulate his models with varying incoming material flow. These models are optimistic according to the author. [START_REF] Rjeb | Fleet-sizing of robots in a logistics warehouse -Transport operation between reception area and storage area[END_REF] refines the results of [START_REF] Egbelu | The use of non-simulation approaches in estimating vehicle requirements in an automated guided based transport system[END_REF] in the case of homogeneous loads by providing an analytical formula for the optimal number of robots. In the case of heterogeneous loads, the authors formulate the problem as a bin packing problem.

Different objective functions are considered in the literature. In particular, several works are interested in minimizing the cost or in maximizing the profit. [START_REF] Beaujon | A model for fleet sizing and vehicle allocation[END_REF] maximizes the total profit (difference between revenues and total transportation costs, including penalty costs for unmet demand). [START_REF] Etezadi | Vehicle fleet composition[END_REF] minimizes the cost of a fleet of purchased or leased vehicles. [START_REF] Sinriech | An economic model for determining agv fleet size[END_REF] minimizes the cost by applying penalties if performance is not achieved in terms of quality of service.

Contributions

To our knowledge, this article is the first to focus on the sizing of a fleet of cooperative robots. To simplify, we first consider that the loads are homogeneous. The section II considers the sizing of a fleet of non-cooperative robots and extends the results of [START_REF] Rjeb | Fleet-sizing of robots in a logistics warehouse -Transport operation between reception area and storage area[END_REF] by adding the concept of transport capacity. The section III considers the problem of sizing a fleet of cooperative robots. Finally, the section IV offers a conclusion and research perspectives, in particular concerning the problem with heterogeneous loads.

II. SIZING OF A FLEET OF NON-COOPERATIVE ROBOTS

In this first part, we determine the number of robots necessary to transport a set of loads over a time interval [0, T ] where T is the horizon.

A. Assumptions and notations

We use the following notations: Robot capacity is denoted c. The capacity of a robot is related to the size and mass of the loads, here we assume that capacity represents the number of loads that a robot can carry simultaneously.

We call τ the cycle time that represents the sum of the loaded travel time (d/2v l where v l is the travelling speed of a loaded robot), the empty travel time (d/2v e where v e is Fig. 2: Transport of identical loads by a fleet of homogeneous robots the displacement speed of an empty robot), the loading time (t l )and the unloading time (t u ):

τ = d 2v l + d 2v e + t l + t u (1)
Over the time interval [0, T ], the robot is immobilized during t b (battery recharge, maintenance, failure, etc.). The remaining available time is then (Tt b ).

The cost per unit of time of a fleet of N > 0 robots traveling a total distance D over the [0, T ] is

f (N) = αN + β D T + γ (2)
where α represents the fixed cost of a robot per unit of time (cost related to maintenance, purchase or rental), β the cost per meter traveled by one robot and γ the cost per unit of time independent of the number of robots (hardware and software infrastructure). Note that the total distance traveled D is directly related to the number of loads n to be transported. Indeed, it takes n c round trips to transport the n loads, where x is the least integer greater than or equal to x. We then have D = d n c and the cost function is

f (N) = αN + β d T n c + γ. ( 3
)
The objective is to determine the number of required robots, N * , to transport the set of n loads over time interval [0, T ] at minimum cost, considering A as the starting point and return point of the robots. For this simple problem, it is equivalent to determining the minimum number of robots allowing all loads to be transported over the time interval. To have feasible solutions, we assume that Tt b ≥ τ.

We also make the following assumptions:

• The robot storage place is located at point A. There is no waiting to load in A or to unload in B (the loads are available immediately to be loaded and the robots do not hinder each other).

• The problem of traffic jams for robots is not taken into account. We could nevertheless take into account these different elements by introducing an efficiency coefficient, as proposed in [START_REF] Egbelu | The use of non-simulation approaches in estimating vehicle requirements in an automated guided based transport system[END_REF].

B. Optimal number of robots (finite horizon)

One The number of robots being an integer, the minimum number of robots to transport n loads during the time interval [0, T ] is then

N * = n c (T -t b )/τ (4)
The minimum cost is then

f * = f (N * ) = αN * + β d T n c + γ (5) = α n c (T -t b )/τ + β d T n c + γ (6

C. Optimal number of robots (infinite horizon)

We are also interested in the limit case where the time horizon T tends to infinity. This allows, on the one hand, to avoid side effects (if horizon T is not a multiple of cycle time τ) and, on the other hand, to model a fleet of vehicles operating permanently.

We denote by µ = c/τ the maximum flow rate of loads per robot (maximum number of loads that a robot can carry per unit of time), by λ = n/T the demand flow of loads to transport and by δ = t b /T the immobilization rate.

If we make T tend to infinity, keeping λ and δ constant, we get:

N * = λ µ(1 -δ ) (7)
f * = α λ µ(1 -δ ) + β d λ c + γ (8)
A detailed proof of this result is provided in the Appendix.

III. SIZING OF A FLEET OF COOPERATIVE ROBOTS

In this section, we assume that robots can cooperate to transport loads. We remind that a m-bot is an elementary robot which can work on its own or with others while a p-bot is a set of p m-bots that cooperate together.

A. Assumptions and notations

We use the following notations: We will make the following additional assumptions:

• τ m ≤ τ p as a p-bot may waste time in cooperation.

• There is no additional cost associated to a p-bot. The costs of a p-bot are simply those induced by the m-bots constituting it. • There is no possible reconfiguration . A p-bot always remains a p-bot and a m-bot that is alone always remains alone.

B. Optimal fleet (finite horizon)

The fleet sizing problem can then be modeled by the following mathematical program which aims at minimizing the cost function [START_REF] Le | Coopération dans les systèmes multi-robots: contribution au maintien de la connectivité et à l'allocation dynamique de rôles[END_REF] :

min α(N m + pN p ) + β d T n m c m + p n p c p + γ (9)
subject to:

N m c m T -t b τ m + N p c p T -t b τ p ≥ n (10) n m ≤ N m c m T -t b τ m (11) n p ≤ N p c p T -t b τ p ( 12
)
n = n m + n p (13) n m ∈ IN, n p ∈ IN, N m ∈ IN, N p ∈ IN ( 14
)
Constraints interpretation :

• Constraint ( 9): the fleet of robots must have enough capacity to carry the n loads over the interval [0, T ]; • Constraint ( 11): the number of loads carried by m-bots must be less than or equal to the maximum number of loads that m-bots can transport on interval [0, T ]; • Constraint ( 12): the number of loads carried by p-bots must be less than or equal to the maximum number of loads that p-bots can transport on interval [0, T ]; • Constraint ( 13): the sum of the number of loads carried by m-bots and p-bots must be equal to the total number of loads to be transported.

When β = 0, the problem comes down to determining the minimum number of robots allowing all loads to be transported. When α = 0, the problem comes down to achieving the smallest number of round trips to transport all the loads (a round trip from a p-bot counts as p round trips).

We will distinguish three cases linked to the respective capacities of m-bot and p-bot.

a) c m = 0 : In this first case, we assume that a m-bot can't carry a load on its own (c m = 0). This scenario may appear for a load of great mass, great volume or even great length. For example, as shown in Figure 4, the robot cannot transport a load much larger than itself for stability reasons. The problem then consists in determining the number of p-bots needed to carry all the loads and the results of Section II can be re-used. So we have:

N * p = n c p (T -t b )/τ p , N * m = 0.
b) c m ≥ c m : In this second scenario, a single m-bot has a greater capacity than a m-bot in p-bot configuration. Let us give a first example where this scenario occurs. If the capacity constraint is related to the transported mass and an additional pallet is needed in p-bot mode, then we lose mass capacity in p-bot mode. Another example would be the case where the load is transported by manipulator arms installed on the mobile platform in p-bot mode. Figure 5 shows the case when the robot loses its mass capacity due to the pallet, and thus the m-bots transport more than the p-bots. We can then use again the results from the previous section. So we have

N * p = 0, N * m = n c m (T -t b )/τ m . c) 0 < c m < c m :
In this 3rd case, a single m-bot has a lower capacity than a m-bot in p-bot configuration. This scenario may arise for the transport of long objects (for example tubes) or even objects of large volumes but of low density. Figure 6 shows an example where a p-bot can have a capacity greater than a m-bot, when we can not stack loads on top of each other. Figure 7 shows the Gantt diagram of the optimal solution when both configurations are allowed.

α(N m + pN p ) + β d λ m c m + p λ p c p + γ (15)
s.t. N m µ m (1 -δ ) + N p µ p (1 -δ ) ≥ λ (16)
λ m ≤ N m µ m (1 -δ ) (17)
λ p ≤ N p µ p (1 -δ ) (18)
λ = λ m + λ p (19)
λ m ∈ IR, λ p ∈ IR, N m ∈ IN, N p ∈ IN (20)
In two cases, we can re-use the results of Section II-C. a) c m = 0 :

N * m = 0, N * p = λ µ p (1 -δ ) b) c m ≥ c m : N * m = λ µ m (1 -δ ) , N * p = 0 IV. CONCLUSION
This paper describes a deterministic mathematical framework for the sizing of a fleet of identical robots, named mbots, with have the possibility to cooperate. A set of p m-bots that cooperate on a given task constitute a p-bot.

When cooperation is not allowed, we obtain a closed-form expression for the optimal number of m-bots. When cooperation is allowed and reconfiguration forbidden, we formulate the fleet sizing problem by a mathematical program with linear constraints and non-linear objective function.

Our mathematical model allows us to determine the most profitable number of robots that should cooperate. If the capacity of p m-bots is smaller than the capacity of a single p-bot, then using exclusively p-bots or a mix of m-bots can lead to a significant cost decrease. Otherwise, it is optimal to use exclusively m-bots.

The next step of our work will be to consider the case where a p-bot can be reconfigured as a m-bot, leading to higher utilization of robots. We will also consider the case of heterogeneous loads in mass and dimensions, where a mbot can be used for a small load and a p-bot for a large load.

APPENDIX

Using the fact that x-1 < x ≤ x, we can limit the optimal number of robots obtained in ( 4

•

  d: round trip distance from A to B • τ: cycle time • v l : travelling speed of a loaded robot • v e : travelling speed of a empty robot • t l : loading time • t u : unloading time • D: total distance traveled by a robot • α: fixed cost per unit of time of a robot • β : cost per meter traveled by a robot • γ: fixed cost per unit of time, independent of the number of robots • N: number of robots for loads transportation • n: number of loads transported • T : planning horizon In this paper, we consider a fleet of N identical mobile robots which must transport a set of n identical loads unload in B and return empty to A, as shown in Figure 2.

  robot can make at most Tt b τ round trips over the time interval [0, T ] where x denotes the greatest integer less than or equal to x. Thus N robots with capacity c can carry at most Nc Tt b τ loads over the time interval. To transport the n loads, it is therefore necessary that Nc Tt b τ ≥ n and therefore that N n c (Tt b )/τ .

)

  Consider the following example: n = 5, c = 2, τ = 0.4, t b = 0, T = 1α = 10, β d = 2, γ = 1.Then N * = 2, f * = 27 and a possible scheduling is represented as a Gantt diagram in Figure 3.

Fig. 3 :

 3 Fig. 3: Gantt diagram for optimal transport of 5 loads

•

  p: number of m-bots constituting one p-bot • c m : m-bot capacity • c p : p-bot capacity • c m = c p /p: virtual m-bot capacity when it evolves as part of a p-bot • τ m : m-bot cycle time • τ p : p-bot cycle time (including possible cooperation time) • α m : fixed cost per unit of time of a m-bot • β m : cost per meter traveled by a m-bot • γ m : fixed cost per unit of time, independent of the number of m-bots • N m : number of m-bots working alone • N p : number of p-bots • n m : number of loads transported by m-bots working alone • n p : number of loads transported by p-bots The other assumptions remain unchanged from Section II: • T : planning horizon • n: number of loads to be transported from A to B n = n m + n p • N: number of robots N = N m + pN p

(a) 1 2 Fig. 4 :

 124 Fig. 4: Illustration for case c m = 0

  Fig. 5: Illustration for case c m ≥ c m

Fig. 6 :

 6 Fig. 6: Illustration for case 0 < c m < c m

Fig. 7 :

 7 Fig. 7: Gantt diagram of the optimal solution of 4 loads

  the notations λ , µ, δ , this frame is re-writtenλ µ(1δ ) ≤ N * < λ µ 1δ -c T (23)If we tend T to infinity, keeping λ and δ constant, we get --------→T →+∞ α λ µ(1δ ) + β d λ c + γ(27)

Table I

 I presents the optimal solution according to the type of authorized robots.

TABLE I :

 I Optimal solutions according to the types of authorized robots

		N m	N p	n m	n p	total cost
	m-bots only	2	0	4	0	42
	p-bot only	0	1	0	4	42
	Mix of m-bots and of p-bots	1	1	1	3	33

Jean-Christophe Fauroux is CEO of MecaBotiX and associated researcher Université Clermont Auvergne, CNRS, Institut Pascal, Clermont Auvergne INP, France JC.Fauroux@MecaBotiX.com (a) co-manipulation mode (b) connection mode

This research was financed by the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25) and by the Auvergne-Rhône-Alpes region under the program "Pack Ambition Recherche" 1 Mari Chaikovskaia is member of LIMOS, INP Clermont Auvergne,