Sujing Wang
email: sjwang4@ustc.edu.cn

Liyu Chen

Mohammad Wahiduzzaman

Antoine Tissot

Lin Zhou

Ilich A Ibarra

Aída Gutiérrez-Alejandre

Ji Sun Lee

Jong-San Chang

Zheng Liu

Jérôme Marrot

William Shepard

Guillaume Maurin

Qiang Xu
email: q.xu@aist.go.jp

Christian Serre
email: christian.serre@ens.fr

A Mesoporous Zirconium-Isophthalate Multifunctional Platform

Mesoporous materials suffer from limitations including poor crystallinity and hydrolytic stability, lack of chemical diversity, insufficient pore accessibility, complex synthesis and toxicity issues. Here the association of non-toxic Zr-oxo clusters and feedstock isophthalic acid (IPA) via a Homometallic-Multicluster-Dot strategy results in a robust crystalline mesoporous MOF, denoted as MIP-206, that overcomes the aforementioned limitations. MIP-206, built up from an unprecedented combination of Zr 6 and Zr 12 oxo-cluster inorganic building units into a single structure, exhibits accessible meso-channels of ca. 2.6 nm and displays excellent chemical stability under different hydrolytic and harsh conditions. Owing to the abundant variety of functionalized IPA linkers, the chemical environment of MIP-206 can be easily tuned without hampering pore accessibility due to its large pore windows. As a result, MIP-206 loaded with palladium nanoparticles acts as an efficient and durable catalyst for the dehydrogenation of formic acid under mild conditions, outperforming benchmark mesoporous materials. This paves the way towards the utilization of MIP-206 as a robust mesoporous platform for a wide range of potential applications.

INTRODUCTION

Mesoporous solids, a category of materials possessing structural voids of 2-50 nm in which large guest molecules can be accommodated, are of interest to sustainable development with potentially broad applications in fields related to energy, environment and health. Owing to their high availability and tunable pore size, traditional mesoporous materials, such as silica, metal oxides and activated carbons, are prevalent in current usage, particularly in heterogeneous catalysis [START_REF] Liang | Heterogeneous Catalysis in Zeolites, Mesoporous Silica, and Metal-Organic Frameworks[END_REF] , energy conversion/storage 2 , analytical science [START_REF] Wang | New insights into the structure-performance relationships of mesoporous materials in analytical science[END_REF] and medical areas [START_REF] Nguyen | Mesoporous Silica as a Versatile Platform for Cancer Immunotherapy[END_REF] , although they suffer from well-recognized limitations such as lack of crystallinity, chemical diversity, structural uniformity, stability and/or reproducibility. These limitations have inhibited pursuits to improve their performance, extend their use to other applicable fields and accumulate fundamental understanding 5 of these materials. Considerable efforts have been devoted in the past two decades to develop substitutes, promote new strategies and develop alternative candidates [START_REF] Lebeau | Introduction for 20 years of research on ordered mesoporous materials[END_REF] . Among them, mesoporous metal-organic frameworks (MOFs) are a promising family of materials that exhibits an ordered porosity and successfully addresses some limitations of conventional benchmarks.

MOFs can be composed of an almost unlimited selection of inorganic building blocks (metal ions or clusters) and organic linkers in periodic structures which are merged efficiently, exhibiting high crystallinity with atomic precision, good reproducibility of preparation over control, and, most importantly, remarkable structural and chemical tunability 7; 8 . Two strategies for preparing mesoporous MOFs are well-documented. Extending the organic linker to construct mesoporous MOFs following the isoreticular chemistry strategy is the most straightforward route and illustrates well the ability to manipulate MOF structures to achieve diverse topologies and functionalities [START_REF] Furukawa | Ultrahigh Porosity in Metal-Organic Frameworks[END_REF][START_REF] Eddaoudi | Systematic Design of Pore Size and Functionality in Isoreticular MOFs and Their Application in Methane Storage[END_REF][START_REF] Moreau | Unravelling exceptional acetylene and carbon dioxide adsorption within a tetra-amide functionalized metal-organic framework[END_REF] . The practical applicability is, however, severely limited by the costly and unscalable linkers, significantly reduced stability and, in some cases, structural interpenetration. On the contrary, the super-molecular building block (SBB) construction approach, which involves the use of small and rigid linkers (similarly to the construction of crystalline mesoporous zeolites from small inorganic building units 12; 13 ) to minimize structural interpenetration issues, leads to accessible mesoporous voids for utilization [START_REF] Férey | A Chromium Terephthalate-Based Solid with Unusually Large Pore Volumes and Surface Area[END_REF][START_REF] Férey | A Hybrid Solid with Giant Pores Prepared by a Combination of Targeted Chemistry, Simulation, and Powder Diffraction[END_REF][START_REF] Wang | Colossal cages in zeolitic imidazolate frameworks as selective carbon dioxide reservoirs[END_REF][START_REF] Liu | Mesoporous Cages in Chemically Robust MOFs Created by a Large Number of Vertices with Reduced Connectivity[END_REF][START_REF] Drache | The modulator driven polymorphism of Zr(IV) based metal-organic frameworks[END_REF] . For instance, terephthalic acid, a polymer industry feedstock, when assembled with homometallic monocluster superoctahedra dots composed of Cr(III) trimers of octahedra, leads to the most chemically stable mesoporous MOF (MIL-101) [START_REF] Férey | A Chromium Terephthalate-Based Solid with Unusually Large Pore Volumes and Surface Area[END_REF] .

However, its practical use is limited by insufficient accessibility to the mesopores due to their microscale pore windows, toxic chromium species and harsh, poorly scalable synthesis conditions. Similarly, mesoporous MOFs based on heterometallic multicluster dots, such as the Zr/Cu carboxylate/azolate MOF-818 (Figure S1) [START_REF] Liu | Mesoporous Cages in Chemically Robust MOFs Created by a Large Number of Vertices with Reduced Connectivity[END_REF] , enhance the pore accessibility through the inclusion of a larger number of clusters within each pore window (12 versus 6 in benchmark mesoporous MOFs such as MIL-101 or NU-1000) [START_REF] Wang | Scalable synthesis and post-modification of a mesoporous metal-organic framework called NU-1000[END_REF] , but suffer from limitations such as the formation of competitive byproducts, difficulty with the introduction of organic functional groups and limited chemical stability, thus making it still challenging to consider these mesoporous MOFs for most practical applications.

To date, there is no satisfying strategy to construct robust mesoporous MOFs with adequate pore accessibility and chemical tunability based on feedstock linkers and nontoxic metals. To address this challenge, we present here a Homometallic-Multicluster-Dot (HMD) strategy to efficiently blend the connection diversity of two different homometallic inorganic clusters into one structural dot for the construction of robust mesoporous MOFs. As a proof of concept, we selected the nontoxic Zr 6 and Zr 12 oxo-clusters to build the inorganic dot and isophthalic acid (IPA), a feedstock produced at billion-kilogram scale annually in industry, as the linker to synthesize a robust mesoporous Zr-IPA MOF, denoted as MIP-206 (MIP stands for the Materials of the Institute of Porous Materials of Paris). It displays excellent chemical and thermal stability as well as large accessible mesopores of ca.

2.6 nm in diameter. One can introduce diverse functional groups at the 5-position of the IPA moiety that faces the mesopores in the structure, allowing fine-tuning of the chemical environment of the pores by simply interchanging various cost-effective substituted IPAs as linkers. MIP-206 can thus act as an ideal mesoporous matrix to support palladium nanoparticles (Pd NPs) for the heterogeneous catalytic release of hydrogen from formic acid under mild conditions. The tailorable nature of the MIP-206 structure enables Pd(0)@MIP-206s catalysts to outperform current benchmark mesoporous catalysts, while exhibiting excellent stability and durability under experimental conditions. Hence, our work paves the way towards the design of robust mesoporous MOFs by applying the HMD strategy, which not only promotes the expanding structure diversity of MOFs, but also holds great promises in achieving multifunctional mesoporous MOF materials.

RESULTS AND DISCUSSION

Zr-MOFs assembled from Zr 6 or Zr 12 oxo-clusters separately with small linkers have been well-investigated in the past years despite the fact that only microporous structures could be achieved due to the short distances separating each cluster. While the Zr 6 oxo-cluster shows a flexible connectivity tolerance ranging from 4 to 12, the Zr 12 oxo-cluster displays an even wider connectivity range as a result of its intrinsic thirty-six terminal coordination sites. Thus, the combination of Zr 6 and Zr 12 oxo-clusters was anticipated to display considerable flexibility in forming homometallic multiclusters dots with large sizes suitable to construct new mesoporous MOFs. Following this expectation, MIP-206 was synthesized by simply heating a mixture of ZrCl 4 , isophthalic acid and formic acid under solvothermal conditions, which is also suitable for the preparation of MIP-206s with substituted IPA linkers with diverse functional groups as well as multivariate MIP-206s based on mixed linkers. It is noteworthy that the syntheses of MIP-206 series are facilely scalable with, for instance, MIP-206 or MIP-206-OH at a 10 g scale production from a one-pot reaction.

The crystal structure of MIP-206 was determined by a computational reverse topological construction approach combined with synchrotron X-ray diffraction single crystal data and high-resolution powder X-ray diffraction (PXRD) data. The structural model was fine-tuned until the calculated PXRD patterns of the DFT-derived minimum energy structure of MIP-206 was found to be consistent with the experimental structure (Figure S2). MIP-206 with a simulated structural formula of Each Zr 12 oxo-cluster is centered between three adjacent Zr 6 oxo-clusters, through linkages formed by six pairs of bridging IPA molecules, forming a triangular HMD (Figure 1C). Each Zr 6 oxo-cluster is at one triangle vertex position, sharing a pair of IPA linkers which connect the nearest Zr 6 oxo-clusters to the neighboring HMDs, forming to a mesoporous window encircled by three Zr 12 oxo-clusters and six Zr 6 oxo-clusters (Figure 1D). This explains why a large window of ca. 2.6 nm of free aperture is formed, of a similar size as that of MOF-818 [START_REF] Liu | Mesoporous Cages in Chemically Robust MOFs Created by a Large Number of Vertices with Reduced Connectivity[END_REF] but significantly larger than those of topical mesoporous MOFs whose hexagonal windows involve only six oxo-clusters, such as MIL-101 [START_REF] Férey | A Chromium Terephthalate-Based Solid with Unusually Large Pore Volumes and Surface Area[END_REF] or DUT-126 [START_REF] Drache | The modulator driven polymorphism of Zr(IV) based metal-organic frameworks[END_REF] . Noticeably, all IPA molecules share the same connection mode (Figure S3) and adopt a configuration facing the 5-position to the meso-void inner surface. This is of a great advantage for tailorable window size, shape and chemical environment by substitution with functionalized IPA linkers. A two-dimensional (2D) layer is generated via the interconnection of adjacent HMDs. In addition to the mesoporous window, there is a small triangular cavity with a diameter of ca. 7 Å between the contiguous Zr 6 oxo-clusters and a slit-like void between the IPA linker pairs (Figure 1E). However, these small interspaces are largely inaccessible when the adjacent 2D layers are stacked together in an ABAB fashion, in which the Zr 12 oxo-clusters from the B layer occupy the space between the two small triangle voids from the two A layers above and below while the Zr 6 oxo-clusters occupy the gap between the bottom and top slits (Figure S4). As a result, a porous architecture with accessible meso-channels running along the c-axis is generated via the tight packing of each layer, with arrays of functional groups attached at the channel surface (Figure 1F). Unlike benchmark mesoporous materials which display limited chemical diversity, MIP-206 is able to facilitate excellent tunability of the pore chemical environment through customization of the IPA linker, in particular as a considerable number of cost-effective functionalized IPA compounds are commercially available at large scales and more diverse, complex moieties can be generated through well-documented organic synthesis. To this end, a series of substituted IPA linkers were applied to the fabrication of isostructural MIP-206 analogues. As shown in Figure 2A, functional groups such as electron donating/withdrawing groups, halogens and heteroatoms, that are well-known functional groups enabling typically a careful tuning of the chemical environment of the pores of MOFs, did not cause a noticeable disruption to the synthesis, with all variations leading to highly crystalline isostructural materials despite slight variations of peak widths in the PXRD patterns due to different particle sizes (Figure 2B).

As a result of the tailorable chemical environment, the corresponding pores' properties could be easily tuned. For instance, as a model characterization, the hydrophilicity/hydrophobicity of the mesochannels in MIP-206s can be easily altered as evidenced by water adsorption measurements (Figure 2C). While the non-functionalized MIP-206 displays an α value of 0.45, α decreased to 0.30 in MIP-206-OH, reflecting the influence of the hydrophilic hydroxyl group. In contrast, the more hydrophobic bromo substituent in MIP-206-Br shifted the α value to 0.53, indicating the largely reduced hydrophilicity of the structural cavity. Similarly, the adsorption step and capacity of MIP-206s in alcohol adsorption isotherms could be adjusted as well (Figure S5), which is of interest for a wide range of applications including heat reallocation, separation and drug delivery 20; 21 . However, some IPA-type compounds could not successfully assemble into the corresponding MIP-206 isostructures, such as 5-NH 2 -IPA, 5-SO 3 H-IPA, pyrazole-3,5-dicarboxylic acid (PDA) and furan-2,5-dicarboxylic acid (FDA). To further extend the chemical diversity of the pores in this case, we considered the construction of multivariate MIP-206 counterparts (MTV-MIP-206s) following the solid-solution mixed linker strategy [START_REF] Helal | Multivariate metal-organic frameworks[END_REF][START_REF] Deng | Multiple Functional Groups of Varying Ratios in Metal-Organic Frameworks[END_REF][START_REF] Wang | Toward a Rational Design of Titanium Metal-Organic Frameworks[END_REF] . A mixture of the dominant compatible IPA linker with the minor, less compatible IPA-type compound functioned similarly with the major linker alone appearing in the MOF structure (Figure 2D). Phases other than MIP-206 obtained when using the secondary linkers separately were not observed in the mixed linker systems (Figure S6). It indicates the determining role of the major linker in the MOF structure formation process to avoid the generation of unexpected by-products. For instance, 5-SO 3 H-IPA led to an unknown crystalline phase under the standard reaction conditions when used as a sole linker. In contrast, a mixture of IPA/5-SO 3 H-IPA (4/1 molar ratio) yielded the mixed linker MIP-206/5-SO 3 H-IPA compound with 60 mol% IPA and 40 mol% 5-SO 3 H-IPA in the final structure (Figure S7 and Table S1). Different secondary linkers could also yield the MTV-MIP-206 phases with varied ratios of the two linkers involved, as evidenced by the 1 H NMR data collected from decomposed samples in KOH/D 2 O solution. The linker content could be adjusted simply by altering the starting material ratios. For example, the 5-NH 2 -IPA content in the MIP-206/5-NH 2 -IPA product could be tuned from 9 mol% to 28 mol% when the amount of 5-NH 2 -IPA added as starting material increased from 10 mol% to 25 mol% (Figure S7 and Table S1). This tailorable multivariate inclusion is very helpful to functionalize the material according to the application requirement.

Nitrogen porosimetry data of MIP-206 samples were collected at 77 K to characterize their porosity (Figure S8). MIP-206 displays a three-divisional adsorption isotherm with a notable step occurring in the 0.1-0.2 P/P 0 range, which is consistent with its mesoporous character. A nitrogen uptake of 290 cm 3 g -1 was recorded at the relative pressure point of P/P 0 =0.9 associated with an accessible pore volume of 0.45 cm 3 g -1 , conforming well with the value calculated from the crystal structure (Tables S2 andS3). The other samples exhibit isotherms of similar shapes to that of MIP-206 despite distinct nitrogen uptake profiles. It is important to note that the steps in the adsorption isotherms occur in a very narrow range of relative pressures (Figure S8, inset), which supports that the variations of the material have similar pore diameters, no matter the type of functional groups included. Though functional groups such as -OH and -OMe lead to a slightly larger size of the linker and more steric hindrance, the corresponding MOF samples show higher nitrogen uptake than that of the non-functionalized MIP-206, which could be ascribed to a larger number of structural defects and/or less residual impurities trapped in the functionalized structures other than the pure MIP-206 (Figures S9-S11, Tables S2-S6).

Pore Size Distribution (PSD) analysis of MIP-206 (Figure S8) suggests the presence of uniform meso-voids with a diameter of ca. 2.6 nm, which corresponds well to the estimation from the crystal structure as well as the direct visualization of the hexagonal pore organization on the sample surface in high-resolution transmission electron microscopy (HR-TEM) images (Figure S12). As a result of the similar transition steps in the adsorption isotherms, samples tested for nitrogen sorption all possess similar mesopore diameters, within the range of 2.5-2.7 nm, which further highlights the negligible influence of bulky substitutes on the dimensions of the MOF porosity (Table S2).

Stability of the MOF is a very important parameter to be considered in assessing its suitability for practical use. Thermal stability of MIP-206 was first tested by thermogravimetric analysis (TGA) under an oxygen atmosphere and a temperature-dependent PXRD measurement carried out in open air. The long-range order within the 2D layer of the structural framework along the crystal b-axis is maintained up to ca. 450 °C, which corresponds to the maximum temperature point before the sharp drop in the TGA curve and the last crystalline PXRD pattern in the thermodiffraction profile (Figures S9 andS13). Hence MIP-206 is on the whole as stable as benchmark microporous Zr-MOFs such as UiO-66, and is significantly more thermally stable in the presence of oxygen than most of the carbon-based benchmarks and even comparable to some advanced molecular sieves [START_REF] Shah | Hydrogen Sulfide Capture: From Absorption in Polar Liquids to Oxide, Zeolite, and Metal-Organic Framework Adsorbents and Membranes[END_REF] .

Chemical stability of a material is required for industrial applications. In particular, hydrolytic resistance is highly prized as water is present in many processes. To assess its hydrolytic stability, a sample of MIP-206 was first soaked in liquid water either at room temperature (R.T.) or at boiling point. Neither cold water or hot water led to a degradation of the MOF structural integrity even after long time periods (Figures S14 andS15, and Table S2). In this respect, MIP-206 clearly outperforms some conventional mesoporous benchmarks, such as MCM-4 [START_REF] Perego | Porous materials in catalysis: challenges for mesoporous materials[END_REF] . MIP-206 also displays an excellent inertness in acidic conditions, such as concentrated HCl and even super acidic aqua regia, similar to the best microporous Zr-MOFs reported to date such as UiO-66 and MIP-200 [START_REF] Wang | A robust large-pore zirconium carboxylate metal-organic framework for energy-efficient water-sorption-driven refrigeration[END_REF] , highlighting a key advantage of this MOF that the vast majority of inorganic mesoporous materials do not share. In comparison with reported Zr-MOFs (microporous or mesoporous with micro-windows) constructed from similar small linkers with angled coordination sites 27; 28 , the superior stability of MIP-206 under acidic conditions might be related to the presence of the Zr 12 oxo-clusters SBUs.

Such an exceptional acidic resistance has indeed previously been documented for group 4 metal-carboxylate MOFs with high nuclearity SBUs, such as Hf 12 [START_REF] Lan | Nanoscale Metal-Organic Layers for Radiotherapy-Radiodynamic Therapy[END_REF] and Ti 12 30 oxo-clusters. In addition, its crystalline integrity can tolerate basic conditions until pH=12, which significantly extends the chemical stability of MIP-206 in aqueous solutions. To that end, it was reasonable to assume that MIP-206 could handle the harsh conditions related to hydrogen release from the catalytic decomposition of formic acid, which will be presented in the following sections. It should be noted that MIP-206 also displayed a good tolerance toward corrosive gas pollutants such as hydrogen sulfide (H 2 S) and sulfur dioxide (SO 2 ) (Figures S15 andS16), which suggests that MIP-206 could also be a promising sorbent platform for potential toxic chemicals removal from contaminated environments such as polluted air and water. To demonstrate the potential of MIP-206 as a mesoporous support for heterogeneous catalysis, hydrogen generation from formic acid (FA) was selected as the model reaction. FA has been recognized as one of the most promising and safest hydrogen storage carriers for clean energy sources mainly due to its biomass origin, high hydrogen content, and easy transportation in its liquid state 31; 32 . Catalytic release of hydrogen from FA has been investigated and developed extensively 31; 33 .

Heterogeneous catalysts have attracted particular attentions due to ease of separation and efficient recycling. As a class of promising heterogeneous catalysts, metal nanoparticles on porous supports (including oxides, silica, modified carbons and microporous MOFs) have performed adequately in the dehydrogenation of FA 34; 35 .

Though encouraging progress was achieved, notable flaws have been gradually noticed, including limitations in substrate diffusion and active species leaching and aggregation, resulting in inadequate catalytic activity and stability. Therefore, the design of supported metal catalysts featuring high activity, long-term stability, good reusability and facile process is highly sought after to advance the application of FA in hydrogen storage and generation. We expected that the unique features of MIP-206s could be suitable to match these requirements.

Due to their high activity toward this catalytic reaction [START_REF] Naviani-Garcia | New Approaches Toward the Hydrogen Production From Formic Acid Dehydrogenation Over Pd-Based Heterogeneous Catalysts[END_REF] , palladium nanoparticles (Pd NPs) templated by MIP-206 were tested initially to evaluate their corresponding performance. Encouraging activity of the catalyst was observed (Figure S17), which prompted us to further investigate its properties. MIP-206-OH was selected for all the following related investigations owing to its following features: 1) In comparison with the other functionalized MIP-206 family members, the hydroxyl-rich pore environment of MIP-206-OH is more comparable to those of mesoporous benchmarks, such as the silanol-rich silica, and hydroxyl or carboxylate-containing activated carbon. 2) MIP-206-OH was determined to be the optimal variation after taking into account MOF crystallinity, sample activation, pore accessibility and linker cost. After the systematic optimization of catalyst preparation and reaction conditions (Figures S18-S21), MIP-206-OH loaded with 3.1 wt% of Pd NPs (Pd(0)@MIP-206-OH) exhibited the best catalytic performance at 60 °C under the optimized condition.

High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images of the catalyst show a homogeneous distribution of small Pd NPs with a uniform size supported by the MIP-206-OH particles; the MOF template presents a similar morphology with no notable change compared to the pristine material (Figure 3A andB). Pd(0)@MIP-206-OH exhibits a complete release of H 2 in less than ten minutes under the optimized reaction condition, which is among the most active heterogeneous catalysts reported to date. [START_REF] Li | Metal-Nanoparticle-Catalyzed Hydrogen Generation from Formic Acid[END_REF] As shown in Figure 3C, the state-of-the-art mesoporous MOF support (MIL-101-Cr) suffers from an incomplete catalytic decomposition of FA due to the deactivation of Pd [START_REF] Gu | Synergistic Catalysis of Metal-Organic Framework-Immobilized Au-Pd Nanoparticles in Dehydrogenation of Formic Acid for Chemical Hydrogen Storage[END_REF] . Control experiments using the benchmark mesoporous Zr-MOFs, including NU-1000 [START_REF] Wang | Scalable synthesis and post-modification of a mesoporous metal-organic framework called NU-1000[END_REF] and PCN-224, [START_REF] Xu | Seed-Mediated Synthesis of Metal-Organic Frameworks[END_REF] resulted in significantly lower activity and incomplete conversion of reactants due to the instability of the MOF structures and considerable Pd leaching under the catalytic conditions. The stable microporous UiO-66(Zr) displayed a poor performance, probably due to its micropores with limited accessibility. Benchmark mesoporous silica such as MCM-41 and SBA-15 demonstrated poor templating ability to form active Pd NPs. A commercially available mesocarbon material was also tested and exhibited moderate performance, mainly due to the considerable Pd leaching and eventual deactivation during catalysis. To further enhance the catalytic efficiency, we employed the aforementioned solid-solution-mixed-linkers strategy: the MTV-MIP-206-OH with partial PDA substitution was used as the Pd NPs support, as the pyrazole moiety is known to interact specifically with Pd species [START_REF] Li | Metal-Nanoparticle-Catalyzed Hydrogen Generation from Formic Acid[END_REF] , leading to the formation and stabilization of homogeneously distributed and smaller Pd NPs with even higher catalytic activity (Figure 4A andB).

Cycling ability is of utmost importance in heterogeneous catalysis. The durability of Pd(0)@MIP-206-OH was therefore evaluated by carrying out the catalysis for five consecutive runs, as shown in Figure 3D. It is worth noting that the catalytic performance is constant over the cycles with no obvious decrease, which indicates good reusability of the catalyst. Characterizations of the sample after cycling revealed that the characters of the catalyst, such as the Pd NP dispersion, the Pd loading content and the template integrity, was maintained (Figures S14, S15 and S18, and Tables S2 andS7). Therefore, both cycling catalytic performance tests and material characterizations fully support the good reusability of Pd(0)@MIP-206-OH in this reaction.

Furthermore, the influence of the MOF mesopore character on the catalytic performance was discussed. Firstly, catalytic activity of the composite increased with the decrease in size of the supported Pd NPs (Figure 4). In summary, we have demonstrated a Homometallic-Multucluster-Dot strategy to construct robust mesoporous MOFs materials from small linkers and non-toxic metals.

MIP-206, a proof-of-concept example of this strategy that constructed from binary Zr 6 and Zr 12 oxo-cluster dot and feedstock isophthalic acid linker, possesses mesopores of ca. 2.6 nm in free diameter with a good stability and tunability of pore chemical diversity. Our findings suggest that the MIP-206s not only function as ideal robust mesoporous supports for heterogeneous catalysts, but also hold considerable promises as a platform for a wide range of potential applications, such as heat reallocation, separation, removal of toxic chemicals and sensing.

EXPERIMENTAL PROCEDURES

Dehydrogenation of formic acid (FA)

A mixture of as-synthesized catalyst and H 2 O (0.6 mL) was placed in a two-necked round bottom flask (30 mL), which was placed in a water bath at 60 °C. A gas burette filled with water was connected to the reaction flask to measure the volume of released gas (room temperature kept at 25 °C). The reaction started when 0.33 mL of the mixed aqueous solution containing FA (46.7 mg) and sodium formate (SF) (68 mg) was injected into the mixture. The volume of the released gas was monitored by recording the displacement of water in the gas burette. In this reaction system, the SF was used as a catalyst promoter. The durability test of Pd(0)@MIP-206-OH was performed at 60 °C. After each run, the catalyst was separated from the mixture and washed 3 times with H 2 O. Then the catalyst was dispersed in 0.6 mL of H 2 O and reused for next cycle. For each cycle, 0.33 mL of FA/SF solution (46.7 mg/68 mg) was injected to the reaction system. The experiment was repeated for 5 times.

[

  Zr 12 (μ 3 -O) 8 (μ 3 -OH) 8 (μ 2 -OH) 6 (formate) 6 ][Zr 6 (μ 3 -O) 4 (μ 3 -OH) 4 (formate) 4 ] 3 (IPA) 18 crystallizes in a hexagonal P-62C space group with unit cell parameters of a = b = 33.7520(5) Å, c = 21.4610(3) Å, V = 21172.86(5) Å 3 . There are two types of Zr oxo-cluster secondary building units (SBUs) in the structure, namely 8-connected Zr 6 and 12-connected Zr 12 oxo-clusters (Figure 1A and B).

Figure 1 .

 1 Figure 1. Crystal structure detail of MIP-206 (A) A 8-connected Zr 6 oxo-cluster SBU. (B) A

Figure 2 .

 2 Figure 2. Functional MIP-206s with tunable chemical environment and related performance in

Figure 3 .

 3 Figure 3. Hydrogen release from formic acid catalyzed by MIP-206-OH supported Pd NPs. (A)

  For instance, fine Pd NPs loaded homogeneously on MTV-MIP-206-OH/PDA have a size distribution centered around 2 nm, and thus display an improved catalytic activity in comparison with that of the Pd NPs in a size of ca. 5 nm templated by MIP-206-OH when a similar pore accessibility of these two solid supports was observed. Similarly, lower activity of the Pd(0)@MIP-206 was noticed compared with Pd(0)@MTV-MIP-206-OH/FDA for the same reason (FiguresS22 and S23). Secondly, the increased pore accessibility had a positive effect on the catalytic activity, possibly due to the facilitation of reactant diffusion in larger pores. Pd NPs based on MIP-206-OH and MTV-MIP-206-OH/FDA have comparable size distributions but showed distinct catalytic performances, likely related to the differences in pore accessibility. Therefore, tunable catalytic activity of supported Pd NPs towards hydrogen generation from FA could be achieved via selecting a certain template candidate from the MIP-206s family with a suitable mesopore accessibility and a specific functional group attached.

Figure 4 .

 4 Figure 4. Catalytic activity comparison between Pd NPs supported on different MIP-206s. (A)

  Synthesis of MIP-206MIP-206 was synthesized solvothermally by the reaction of ZrCl 4 and IPA in formic acid. IPA (1.1 g, 6.6 mmol) was weighed into a 23 mL Teflon reactor, formic acid (5 mL) was added followed by stirring at room temperature for 5 minutes until a homogeneous suspension was formed. ZrCl 4 (2 g, 8.6 mmol) was added to the suspension followed by 10 minutes stirring at room temperature to disperse the reactants uniformly. Afterwards, the reaction was sealed in an autoclave and heated to 180 °C in 2 hours and was kept at 180 °C for 24 hours. After cooling down to room temperature, the expected product of MIP-206 (1.98 g) was collected by filtration, washed with acetone and air dried. A scale-up synthesis could be achieved easily under the same reaction condition using a 125 mL Teflon reactor with a mixture of IPA (5.0 g, 30.0 mmol), ZrCl 4 (10.4 g, 44.4 mmol) and formic acid (30 mL) as reactants. 10.10 g of crude product of MIP-206 could be isolated. It is noteworthy that only high concentration of reactants could generate the MOF product. Otherwise clear solutions with no solid product were observed. The reaction duration was notably shortened and the reaction yield was significantly improved along the increase of reaction temperature in the range of 120-180 °C.Synthesis of MIP-206-OH5-OH-IPA (0.72 g, 4.0 mmol) was weighed into a 23 mL Teflon reactor, formic acid (5 mL) was added followed by stirring at room temperature for 5 minutes until a homogeneous suspension was formed. ZrOCl 2 •8H 2 O (1.93 g, 6 mmol) was added to the suspension followed by 10 minutes stirring at room temperature to disperse the reactants uniformly. Afterwards, the reaction was sealed in an autoclave and heated to 180 °C in 2 hours and was kept at 180 °C for 20 hours. After cooling down to room temperature, the expected product of MIP-206-OH (2.01 g) was collected by filtration, washed with acetone and air dried. A scale-up synthesis could be achieved easily under the same reaction condition using a 125 mL Teflon reactor with a mixture of 5-OH-IPA (3.60 g), ZrOCl 2 •8H 2 O (9.65 g) and formic acid (25 mL) as reactants. 10.30 g of crude product of MIP-206-OH could be isolated.Synthesis of MTV-MIP-206/PDAIPA (0.88 g) and PDA (0.23 g, 20 mol%) were weighed into a 23 mL Teflon reactor, formic acid (5 mL) was added followed by stirring at room temperature for 5 minutes until a homogeneous suspension was formed. ZrCl 4 (2 g) was added to the suspension followed by 10 minutes stirring at room temperature to disperse the reactants uniformly. Afterwards, the reaction was sealed in an autoclave and heated to 180 °C in 2 hours and was kept at 180 °C for 24 hours. After cooling down to room temperature, the expected product of MTV-MIP-206/PDA (1.83 g) was collected by filtration, washed with acetone and air dried.Synthesis ofPd(0)@MIP-206s 10.8 mg of Pd(OAc) 2 was dissolved in 5 mL of acetone in a beaker under vigorous stirring at room temperature. 50 mg of MIP-206s was added and mixed for 1 hour for sufficiently absorbing the Pd precursor into the MOF pores. Subsequently, the solvent and residual metal precursor were removed by centrifugation. The collected solid was washed with anhydrous acetone twice. The sample was finally dried under vacuum at room temperature to give the Pd(II)@MIP-206s. Then the resulting solid was dispersed in 2.3 mL of H 2 O. The required amount of NaBH 4 was dissolved in 1 mL of H 2 O and added into the above mixture. After 5 min, the solid was collected by centrifugation, rinsed with H 2 O for three times. The catalyst was dispersed in 0.6 mL of H 2 O for dehydrogenation of formic acid (FA).

ACKNOWLEDGMENTS

Authors from France acknowledge the projects of ANR Meacopa and M4CO2.

Korean authors are grateful to the Global Frontier Center for Hybrid Interface Materials of Korea (GFHIM) (Grant No. NRF-2013M3A6B1078879). We thank Dr.

Charlotte Martineau-Corcos (Institut Lavoisier de Versailles) for solid-state NMR data collection and analysis, Dr. Takeyuki Uchida for microscopic measurements, Hazelle Wang for helpful discussion and revision.

AUTHOR CONTRIBUTIONS

Conceptualization, S.W., C.S. and Q.X.; Investigation, S.W., L.C., M.W., A.T., L.Z., I.I., A.G., J.L., J.C., Z.L., J.M., W.S., G.M., Q.X. and C.S.; Writing -Original Draft, S.W.; Writing -Review & Editing, S.W., L.C., M.W., A.T., I.I., J.C., G.M., Q.X. and C.S. Supervision, Q.X., and C.S.

DECLARATION OF INTERESTS

The authors declare no competing interests.


T Hamel
email: thamel@i3s.unice.fr

M-D Hua
email: hua@i3s.unice.fr.

C Samson

Deterministic observer design for vision-aided inertial navigation

Tarek Hamel, Minh-Duc Hua and Claude Samson

Abstract-The purpose of this tutorial is to report results developed over the last three years for state estimation problems arising with unmanned mobile robots equipped with a monocular camera and a 3-axis gyrometer, complemented with either a velocity sensor, or a 3-axis accelerometer, or an optical flow sensor. Definition and characterization of uniform observability for linear time-varying systems, followed by an observer design framework exploiting first-order approximations of a class of nonlinear systems, are first recalled. The resulting Riccati observers are locally uniformly exponentially stable when associated uniform observability conditions are satisfied. This framework is subsequently applied, with detailed explanations, to a set of practical problems, namely i) classical PnP camera pose estimation using known source points and bearing measurements, ii) the adaptation of this problem to unknown source points by using epipolar constraints, iii) camera pose and velocity estimation using bearing and IMU measurements, and iv) camera velocity and depth estimation using optical flow and IMU measurements from the observation of a planar target. The observer solutions proposed for these last two problems are validated with experimental data.

I. INTRODUCTION

State estimation (position, velocity, orientation, etc.) is a central problem in the fields of robotics and autonomous systems. It consists in designing sensor fusion algorithms that reconcile various measurements through the use of a dynamical model to produce state estimates. When the state estimates are used in a control law, these algorithms are required to operate in real time, with high bandwidth and low latency. This puts severe constraints on the sensor fusion algorithms, from both theoretical and practical points of view. A variety of methods and sensory modalities have been proposed in the literature for state estimation of autonomous systems.

The most commonly used proprioceptive sensors are inertial sensors (e.g., accelerometer, gyrometer) and Doppler velocity sensors that have the advantage of producing measurements at a high rate with relatively low latency. However, they only provide relative motion measurements that are furthermore often impaired by significant drifts. Proprioceptive sensors are often complemented with Global Positioning Systems (GPS) to provide a global position measurement at a relatively high rate [START_REF] Farrell | Aided Navigation: GPS with High Rate Sensors[END_REF], [START_REF] Hua | Attitude estimation for accelerated vehicles using gps/ins measurements[END_REF], but with low precision and, more annoyingly, with limited applicability when the GPS signal is severely degraded in urban canyon environments, to the point of being inoperable in indoor environments. These sensors are also often complemented with onboard exteroceptive sensors, such as laser range finders, acoustic sensors and stereo cameras that, beyond the estimation of the system state, provide information about the surrounding environment. Early approaches for state estimation were based on Extended Kalman Filters (EKF) [START_REF] Anderson | Optimal Filtering[END_REF], unscented Kalman filters (UKF) [START_REF] Julier | New extension of the Kalman filter to nonlinear systems[END_REF] and particle filters [START_REF] Samson | An Introduction to Sequential Monte Carlo Methods[END_REF]. However, these early solutions have limitations, mostly in terms of robustness. The nonlinearity of autonomous systems dynamics is much related to the inherent nonlinearity of their state space, well exemplified by the Lie groups of rotation SO(2) and SO [START_REF] Anderson | Optimal Filtering[END_REF]. Exploiting the structure of Lie groups for state estimation is a fruitful approach that has motivated, during the last fifteen years, an increasing interest in the design of nonlinear observers [START_REF] Aghannan | An intrinsic observer for a class of lagrangian systems[END_REF], [START_REF] Bonnabel | Non-linear symmetrypreserving observers on lie groups[END_REF], [START_REF] Mahony | Observers for kinematic systems with symmetry[END_REF] and invariant extended Kalman filters (IEKF) [START_REF] Barrau | The invariant extended kalman filter as a stable observer[END_REF] for autonomous systems. All observers in this class take into account the specificity of sensory devices that translate the action of a Lie group on a measurement space. However, this entails a certain number of complications at the design level that need to be overcome. In particular, the estimation of the pose of a moving monocular camera on the basis of proprioceptive sensors measurements, complemented with bearing measurements of a set of source points identified in the camera image, the coordinates of which in the inertial frame are either known (the case of the classical Perspective-n-Point (PnP) problem), or unknown (the essential matrix estimation problem), yields challenging problems. A variety of methods dealing with these problems have been proposed. Let us cite, for instance, algebraic algorithms [START_REF] Dhome | The inverse perspective problem from a single view for polyhedra location[END_REF], [START_REF] Nistér | An efficient solution to the five-point relative pose problem[END_REF] and iterative algorithms based on gradient search [START_REF] Haralick | Pose estimation from corresponding point data[END_REF] for the PnP problem, and nonlinear optimization algorithms for essential matrix estimation [START_REF] Ma | Optimization criteria and geometric algorithms for motion and structure estimation[END_REF]. There is also a rich literature addressing these problems with EKF algorithms, typically in the context of Simultaneous Localization And Mapping (SLAM) and Visual Odometry [START_REF] Jones | Visual-inertial navigation, mapping and localization: A scalable real-time causal approach[END_REF].

In this paper we show that the deterministic Riccati observer framework developed in [START_REF] Hamel | Riccati observers for the nonstationary pnp problem[END_REF], which exploit firstorder approximations of a class of nonlinear systems, and may formally be viewed as a generalisation of the so-called multiplicative extended Kalman filter (MEKF) [START_REF] Markley | Attitude error representations for kalman filtering[END_REF], [START_REF] Martin | Generalized multiplicative extended Kalman filter for aided attitude and heading reference system[END_REF], applies to a number of applications involving proprioceptive sensors and monocular vision.

The remainder of the paper is organized as follows. In Section II, after specifying the notation used throughout the paper, basic definitions related to uniform observability of a time-varying linear system -a property whose satisfaction is central to proving the exponential stability of the proposed observers-and a generic dynamic system verifying a set of structural properties, complemented with a generic Riccati observer, are recalled. Section III shows how the proposed observer framework applies to the estimation of the camera pose when using bearing measurements of a set of source points whose coordinates in the inertial frame are either known (the classical PnP problem) or unknown (use of epipolar constraints). In Section IV, an adaption of the observer to the generalized PnP problem for which the linear velocity is unknown is proposed, and its performance is validated with experimental data. In Section V, another dynamic problem involving the estimation of the camera's attitude, of its linear velocity and of the distance between the camera and a planar target (depth), on the basis of optical flow and IMU measurements, is addressed. The observer's performance is also validated with experimental data. Concluding remarks are presented in Section VI.

II. PRELIMINARY MATERIAL A. Notation

We denote by R n the n-dimensional Euclidean space, by {e 1 , . . . , e n } the canonical basis of R n and by |x| the Euclidean norm of the vector x ∈ R n . The set B n r := {x ∈ R n : |x| ≤ r} is the closed ball in R n with radius r. The null matrix and the identity matrix of dimension n×n are denoted by 0 n and I n , respectively. The null matrix of dimension m × n is denoted by 0 m×n .

The set S n := {x ∈ R n+1 : |x| = 1} is the n-dimensional sphere embedded in R n+1 with radius equal to one. For any p ∈ S n , the projection onto the tangent space of the unit n-dimensional sphere at the point p is denoted by Π p := (I n+1 -pp ). The Special Orthogonal group of order n is denoted by SO(n) := {R ∈ R n×n : det(R) = 1, R R = RR = I n }. For any x, y ∈ R 3 , x × is the skew symmetric matrix associated with the cross product, i.e. x × y = x × y.

With f denoting a vector-valued function depending on the two variables x and y, and on the time variable t, we write f = O(|x| k1 |y| k2 ) with k 1 ≥ 0 and k 2 ≥ 0 if ∀t : |f (x, y, t)|/(|x| k1 |y| k2 ) ≤ γ < ∞ in the neighbourhood of (x = 0, y = 0). If f depends only on x and t then we write

f (x, t) = O(|x| k ) if ∀t : |f (x, t)|/|x| k ≤ γ < ∞ in the neighbourhood of x = 0.

B. Uniform observability of a linear time-varying system

Consider a Linear Time-Varying (LTV) system of the form:

ẋ = Ā(t)x + B(t)u y = C(t)x with x ∈ R n , u ∈ R l , y ∈ R m ,
W Ā, C Q (t, t + δ) := 1 δ t+δ t Φ (s, t) C (s)Q(s) C(s)Φ(s, t)ds
where Φ(s, t) is the transition matrix associated with Ā, i.e.

d ds Φ(s, t) = Ā(s)Φ(s, t), ∀s ≥ t, Φ(t, t) = I n .
If Ā(t) and C(t) are bounded and if there exist δ > 0 and > 0 such that W Ā,

C

In (t, t + δ) > I n , ∀t ≥ 0, then we say that the pair ( Ā(t), C(t)) is uniformly observable.

C. Riccati Observer for a Class of nonlinear Systems

In this section we define a class of nonlinear systems of interest, along with a concise description of the proposed observer design methodology. This material is taken from [START_REF] Hamel | Riccati observers for the nonstationary pnp problem[END_REF].

Every member of this class of systems lives in B n1 r × R n2 , and its state is denoted as

x = [x 1 , x 2 ] , with x 1 ∈ B n1 r and x 2 ∈ R n2 .
The output of this system, denoted as y ∈ R m , is assumed to be a function of x, x2 , and t, with x2 denoting an estimate of x 2 . This system evolves according to the following equations:

         ẋ = A(x 1 , t)x + u 1 u 2 (t) + O(|x 1 | 2 ) + O(|x 1 ||u 1 |) y(x, x2 , t) = C 1 (x 1 , x2 , t)x 1 + C 2 (x 1 , x2 , t)x 2 +O(|x 1 | 2 ) + O(|x 1 ||x 2 -x2 |) + O(|x 2 -x2 | 2 ) (1)
where

• A(x 1 , t
) is a continuous matrix-valued function uniformly bounded with respect to (w.r.t.) t and uniformly continuous w.r.t. x 1 . Furthermore, it is of the form

A(x 1 , t) = A 11 (t) 0 n1×n2 A 21 (x 1 , t) A 22 (t) • C(x, x2 , t) := [C 1 (x, x2 , t) , C 2 (x, x2 , t) ] ∈ R m×(n1+n2
) is a continuous matrix-valued function uniformly bounded w.r.t. t and uniformly continuous w.r.t. x and x2 . It is also assumed that u 2 (t) is a bounded input and that the solutions to this system belong to a compact set D independently of the choice of the input u 1 , whose role is to make x 1 converge to zero. By setting

u 1 = -K 1 (y -C 2 x2 ) , (2)
an estimate of x is (x 1 , x 2 ) , with x1 ≡ 0 and x2 calculated according to:

ẋ2 = A 22 x2 + u 2 + K 2 (y -C 2 x2 ) Ṗ = AP + P A -P C Q(t)CP + V (t) (3)
with x2 (0) ∈ R n2 , P (0) ∈ R (n1+n2)×(n1+n2) a symmetric positive definite matrix, Q and V bounded continuous symmetric positive semi-definite matrix-valued functions, and

K := [K 1 , K 2 ] given by K = k(t)P C Q (4)
with 0.5 ≤ k(t) ≤ k max < ∞ and P the solution, obtained by numerical integration, to the continuous Riccati equation (CRE) specified in (3). Let x := ((x 1 -x1 ) , (x 2 -x2 ) ) denote the estimation error. Then, as proven in [8, Corollary 3.2], the equilibrium x = 0 is locally uniformly exponentially stable when Q(t) and V (t) are both larger than some positive matrix and the pair (A(0, t), C(0, x 2 (t), t)) is uniformly observable.

III. OBSERVER DESIGN FOR KINEMATIC SYSTEMS INVOLVING MONOCULAR VISION

In this section we focus on kinematic systems evolving in SO(3) × R 3 , with measurements performed in the camera frame, and we show how the equations of this type of system can fit the framework described in the previous section in order to derive observers of interest in practice. The problem at hand consists in providing an estimate of the camera pose given measurements of the camera linear and angular velocities (both assumed to be bounded), and by using either the inertial coordinates of a set of n ≥ 3 static source points identified in the image, or epipolar constraints when these coordinates are not known.

Let {I} denote an inertial (fixed) frame of reference and let {B} denote a frame of reference attached to the camera. Let R ∈ SO(3) denote the orientation of the camera frame {B} w.r.t. the inertial frame {I}. Let ξ ∈ R 3 denote the position of the camera in the inertial frame {I} and define ξ := R ξ ∈ R 3 , its expression in {B} w.r.t. {I} expressed in {B}. Then the equations of motion of the camera satisfy

Ṙ = RΩ × ξ = -Ω × ξ + V (5)
where V ∈ R 3 and Ω ∈ R 3 denote the linear and the angular velocity of the camera expressed in the frame {B}, respectively. Assume that V and Ω are measured by means of a linear velocity sensor (e.g., a Doppler sensor) and an Inertial Measurement Unit (IMU), respectively. The inertial coordinates of the source points are denoted by Pi , and we assume that their calibrated projective coordinates p p i onto the camera plane expressed w.r.t. the frame {B} are measured (see Figure 1). Denoting by P i the coordinates of the i th source point w.r.t. the frame {B} expressed in {B}, and by z i its third component, one verifies that p p i = P i /z i . Rather than perspective outputs, typically used in computer vision algorithms, we prefer using the direction (bearing) vectors:

p i := p p i |p p i | = R ( Pi -ξ) | Pi -ξ| = R Pi -ξ |R Pi -ξ| ∈ S 2 (6)
obtained by projecting the source points onto a virtual spherical image. These outputs can always be calculated from the image data. They differ from perspective outputs only by a scaling factor, and using them contributes to the simplicity of the observer design methodology exposed in the present paper.

A. Observer equations and model adaptation

The estimate of R is denoted by R ∈ SO(3), and the estimate of ξ (resp. ξ) is denoted by ξ ∈ R 3 (resp. ξ ∈ R 3 ). The proposed observer has the following form:

Ṙ = RΩ × -Rσ R× ξ = -Ω × ξ + V -σ ξ (7)
with initial conditions ( R(0), ξ(0)) ∈ SO(3) × R 3 , and σ R , σ ξ ∈ R 3 the innovation terms to be determined thereafter. In practice, it is advantageous to calculate the unit quaternion ( λ0 ∈ R, λv ∈ R 3 ) associated with R, and then translate this quaternion into a rotation matrix. The corresponding equation, which then replaces the first equation in [START_REF] Farrell | Aided Navigation: GPS with High Rate Sensors[END_REF], is:

d dt ( λ0 , λv ) = 0.5 ( λ0 , λv ) • (0, Ω -σ R ) + k (1 -λ2 0 -| λv | 2 ) ( λ0 , λv ) (8)
with • denoting the quaternion product, and k a positive gain ensuring the numerical stability of the quaternion's norm (equal to one). Define the attitude error matrix

R := R R (9)
One could also use R := R R as an alternate definition of this error. From ( 5) and ( 7) one verifies that the attitude error kinematics are given by

Ṙ = -Ω × R + RΩ × + σ R× R (10)
The following step involves developing a first-order approximation of [START_REF] Hua | Attitude estimation for accelerated vehicles using gps/ins measurements[END_REF]. From the Rodrigues' formula, the firstorder approximation of R is given by

R = I 3 + λ× + O(| λ| 2 ) ( 11
)
with λ ∈ B 3 2 equal to twice the vector part of the quaternion associated with the attitude error matrix R, and whose convergence to zero implies the convergence of R to R. One then deduces from [START_REF] Hua | Attitude estimation for accelerated vehicles using gps/ins measurements[END_REF], [START_REF] Hua | Attitude, linear velocity and depth estimation of a camera observing a planar target using continuous homography and inertial data[END_REF], and the identity

a × b × - b × a × = (a × b) × , ∀a, b ∈ R 3 , that λ = -Ω × λ + σ R + O(| λ| 2 ) + O(| λ||σ R |) (12)
From ( 12) and the second equation of ( 5), and by setting x = [x 1 , x 2 ] := [ λ , ξ ] , u 1 := σ R and u 2 (t) := V , one obtains the system equation in [START_REF] Samson | An Introduction to Sequential Monte Carlo Methods[END_REF] with

A(x 1 , t) = A(0, t) := -Ω × 0 3 0 3 -Ω ×
Note that this latter matrix valued-function is independent of x 1 for this particular case. Using R R , instead of R R for the attitude error, and using the corresponding error quaternion, would have led to the same result but with a different matrix A and a different input u 1 .

Once there, it remains to determine the output y and the matrix-valued function C, depending on the problem at hand, i.e. either the PnP problem with known source points coordinates in the inertial frame, or the case of unknown source points' coordinates and the use of epipolar constraints.

B. System output associated with the PnP problem

In this case the inertial coordinates Pi of the source points are known. From [START_REF] Dhome | The inverse perspective problem from a single view for polyhedra location[END_REF], and by definition of the projector Π pi , one has Π pi p i = 0, ∀i = {1, . . . , n}. It follows that

0 = -| Pi -ξ|Π pi p i = Π pi ξ -Π pi R Pi = Π pi ξ -Π pi R R Pi = Π pi ξ -Π pi (I 3 -λ× ) R Pi + O(| λ| 2 )) = Π pi ξ -Π pi R Pi -Π pi ( R Pi ) × λ + O(| λ| 2 )
By setting y i := Π pi R Pi one obtains

y i = Π pi ξ -Π pi ( R Pi ) × λ + O(| λ| 2 ) (13)
for i = 1, . . . , n. Now, by defining the system output as y := [y 1 , . . . , y n ] , one easily verifies that the output equation in (1) is satisfied with

C 1 =    -Π p1 ( R P1 ) × . . . -Π pn ( R Pn ) ×    , C 2 =    Π p1 . . . Π pn   
From there, one deduces from (3) and ( 7) the innovation terms

σ R = u 1 = -K 1 (y -C 2 ξ) and σ ξ = -K 2 (y -C 2 ξ).
Uniform observability of (A(0, t), C(0, x 2 (t), t)) then guarantees the (local) exponential stability of the equilibrium ( R, ξ -ξ) = (I 3 , 0). This observability issue is not treated here due to space limitation, but one may refer to [START_REF] Hamel | Riccati observers for the nonstationary pnp problem[END_REF] where it is addressed in relation to the number and possible singular locations of source points, combined with body motion and the measurement of the camera linear velocity, either in the body-fixed frame or in the inertial frame.

C. System output when using Epipolar Constraints in the case of Unknown Source Points

In the case where the camera pose w.r.t. to the inertial frame is unknown, alike the inertial coordinates of the source points, it is not possible to recover the camera pose from the sole bearing measurements of the source points. However, if the inertial frame is taken as the initial frame of the camera (i.e. at time t=0), then the camera pose (R = I 3 , ξ = 0) is initially known, by definition of this particular frame. This entails that, ∀i = {1, . . . , n}, the (constant) unit vector pi :

= Pi | Pi|
, which is initially equal to p i , is measured. It is then possible to estimate future camera poses w.r.t. this frame by using source points bearing measurements (that do not depend on the chosen inertial frame). To this aim, one can exploit the fact that P i = R ( Piξ) = R Pi -ξ, from which the i th epipolar constraint:

p i Rξ × p i = 0, (i = 1, . . . , n) (14)
is easily deduced. This constraint is typically presented in the form p i Ep i = 0, where E := Rt × (with t := ξ/|ξ|) is the essential matrix involved in the constraint [START_REF] Julier | New extension of the Kalman filter to nonlinear systems[END_REF]. From ( 14), and using [START_REF] Hua | Attitude, linear velocity and depth estimation of a camera observing a planar target using continuous homography and inertial data[END_REF], one deduces that

0 = -p i R Rξ × p i = -p i R(I 3 + λ× )ξ × p i + O(| λ| 2 ) = p i R(ξ × p i ) × λ + p i Rp i× ξ + O(| λ| 2 ) = p i R( ξ × p i ) × λ + p i Rp i× ξ + O(| λ| 2 ) + O(| ξ|| λ|)
Therefore, by defining y as the vector in R n with zero entries, the output equation in ( 1) is satisfied with

C 1 =    p 1 R( ξ × p 1 ) × . . . p n R( ξ × p n ) ×    , C 2 =    p 1 Rp 1× . . . p n Rp n×   
As in the PnP case, one deduces that the innovation terms are

σ R = u 1 = -K 1 (y -C 2 ξ) and σ ξ = -K 2 (y -C 2 ξ).
The issue of uniform observability of (A(0, t), C(0, x 2 (t), t)), whose verification ensures the local exponential stability of the observer so obtained, is addressed in [START_REF] Hua | Relative pose estimation from bearing measurements of three unknown source points[END_REF]. In particular, uniform observability is proven in the case of three non-aligned source points provided that the camera's translational motion is sufficiently exciting.

IV. OBSERVER DESIGN FOR POSE AND LINEAR VELOCITY ESTIMATION FROM BEARING AND IMU

MEASUREMENTS

In this section, an extension of the PnP problem, addressing the combined estimation of the camera's pose and of the camera's linear velocity, is presented. The material is taken from [START_REF] Marco | Position, velocity, attitude and accelerometer-bias estimation from imu and bearing measurements[END_REF]. This extension is motivated by a large number of applications where sensors measuring this velocity directly are not available. It also contributes to demonstrate the applicability of the proposed observer design methodology for a broad range of estimation problems encountered in practice.

The so-called specific acceleration a B ∈ R 3 of the camera, i.e. the difference between the camera's linear acceleration and the gravity acceleration, both expressed in the bodyfixed frame, can be measured by a 3-axis accelerometer. By definition (and assuming that the accelerometer is fixed to the camera):

a B = V + Ω × V -gR e g ( 15
)
with g denoting the gravity constant and e g ∈ S 2 denoting the gravity direction expressed in the inertial frame {I}. In practice, the estimates of both gravity constant and gravity direction may be slightly erroneous. This leads to a bias between the gravity acceleration expressed in the inertial frame a g := ge g and its estimate a m g . From experience, a good estimate of this bias significantly improves the quality of the camera pose and velocity observer.

Let b g := R (a g -a m g ) denote the gravity bias expressed in the body-fixed frame {B}. The time-derivative of this vector satisfies the relation ḃg = -Ω × b g [START_REF] Mahony | Observers for kinematic systems with symmetry[END_REF] so that, in view of ( 15)

V = -Ω × V + R a m g + b g + a B Using the first-order approximation (11) of R (= R R) V = -Ω × V + R R a m g + b g + a B = -Ω × V + (I 3 -λ× ) R a m g + b g + a B + O(| λ| 2 ) = -Ω × V + bg + R a m g + a B + O(| λ| 2 ) ( 17
)
with bg := b g + ( R a m g ) × λ
Using [START_REF] Hua | Relative pose estimation from bearing measurements of three unknown source points[END_REF] and the first equation of ( 7) in the calculation of the time-derivative of the new bias bg , one verifies that

ḃg = -Ω × bg + ( R a m g ) × σ R + O(| λ| 2 ) + O(| λ||σ R |) (18)
In view of ( 5), ( 17) and ( 18), the proposed observer takes the following form:

           Ṙ = RΩ × -Rσ R× ξ = -Ω × ξ + V -σ ξ V = -Ω × V + bg + R a m g + a B -σ V ḃg = -Ω × bg + ( R a m g ) × σ R -σ b with initial conditions ( R(0), ξ(0), V (0), bg (0)) ∈ SO(3) × R 3 × R 3 × R 3 .
As for the innovation terms σ R , σ ξ , σ V , σ b ∈ R 3 , they are again derived from the observer design framework.

The bearing measurements of n source points whose inertial coordinates are known yield the output equations ( 13), as for the PnP problem addressed in subsection III-B.

Define the state variable

x := [x 1 , x 2 ] , with x 1 := λ ∈ B 3 2 and x 2 := [ξ , V , b g ] ∈ R 9
, and the output y := [y 1 , . . . , y n ] ∈ R 3n , with y i := Π pi R Pi . Then, from (12), ( 13) , ( 17), [START_REF] Markley | Attitude error representations for kalman filtering[END_REF] and the second equation of (5), one obtains the system equations (1) with

                           A 11 = -Ω × , A 21 = 0 3×9 , u 1 = σ R A 22 =    -Ω × I 3 0 3 0 3 -Ω × I 3 0 3 0 3 -Ω ×    , u 2 =    0 3×1 R a m g + a B ( R a m g ) × σ R    C 1 =     -Π p1 ( R P1 ) × . . . -Π pn ( R Pn ) ×     , C 2 =     Π p1 0 3 0 3 . . . . . . . . . Π pn 0 3 0 3    
The innovation terms, derived from ( 3) and ( 4), are

σ R = u 1 = -K 1 (y -C 2 x2 ) [σ ξ , σ V , σ b ] = -K 2 (y -C 2 x2 )
Establishing the uniform observability of the matrix pair (A(0, t), C(0, x 2 (t), t)), which in turn guarantees the (local) uniform exponential stability of the observer, is not a simple task. It is out of scope of the present paper. However, by analogy with the PnP problem, explicit conditions of nonuniform observability are established in [START_REF] Marco | Position, velocity, attitude and accelerometer-bias estimation from imu and bearing measurements[END_REF].

A. Experimental validation

In this subsection the performance of the proposed observer, when using a time-synchronised camera-IMU system, is tested experimentally. Details about the experimental setup, the observer implementation, and tuning measures are given in [START_REF] Marco | Position, velocity, attitude and accelerometer-bias estimation from imu and bearing measurements[END_REF]. In particular, point-feature detection and matching operations are performed with the ArUco library included in OpenCV.

The proposed observer is compared to the popular algebraic algorithm solvePnP() in OpenCV, and to the groundtruth data obtained from an Optitrack motion capture system MOCAP.

Figure 2 shows the time evolution of the estimated camera position obtained with the proposed Riccati observer (red), with solvePnP() (yellow) and with ground truth data (blue). One observes that the position estimates obtained with the proposed Riccati observer and the solvePnP() algorithm are very similar. These estimates are close to the ground-truth values except for the third component, when the distance between the camera and the target increases significantly during the time-interval (85 sec, 140 sec). We attribute this degradation of the estimates to the low resolution of the camera in relation to the ArUco points extraction algorithm.

On the other hand, one observes from Figure 3 showing the time evolution of the estimated and ground-truth attitudes (represented by roll, pitch and yaw Euler angles) that the proposed observer outperforms the solvePnP() algorithm, especially when the camera's pose changes quickly. More precisely, when the camera's motion combines a rapid displacement along the third axis (cf. third subplot of Figure 2) and a fast yaw rotation (cf. third subplot of Figure 3) during the time-interval (145 sec, 185 sec), the figure shows that the solvePnP() algorithm switches between two possible pitch angle solutions.

V. OBSERVER DESIGN FOR LINEAR VELOCITY, GRAVITY DIRECTION, AND DEPTH ESTIMATION OF A CAMERA

OBSERVING A PLANAR TARGET FROM OPTICAL FLOW AND IMU MEASUREMENTS This classical problem has been revisited in [START_REF] Hua | Attitude, linear velocity and depth estimation of a camera observing a planar target using continuous homography and inertial data[END_REF] by adopting the proposed observer design framework. A monocular camera and an IMU provide the measurements needed for the estimation process. More precisely, it is assumed that the camera images and IMU are perfectly time synchronised so that one can extract from consecutive images the optical flow φ := V /d, with d the distance between the camera and the observed planar scene, and also the flow divergence φ ⊥ := -ḋ/d (see, e.g., [START_REF] Hua | Attitude, linear velocity and depth estimation of a camera observing a planar target using continuous homography and inertial data[END_REF] for more explanations). It is further assumed that the measurements of the angular velocity Ω and of the specific acceleration a B provided by the IMU are available at a fast sampling rate.

By denoting s := 1/d, the equations of motion involved in the observer design are

     Ṙ = RΩ × V = -Ω × V + gR e 3 + a B ṡ = φ ⊥ s (19)
In view of the above equations, the proposed observer has the following form

     Ṙ = RΩ × -σ R× R V = -Ω × V + g R e 3 + a B -σ V ṡ = φ ⊥ ŝ -σ s
with initial conditions ( R(0), V (0), ŝ(0)) ∈ SO(3)×R 3 ×R 3 , and with σ R , σ V , σ s ∈ R 3 the innovation terms to be derived from the observer design framework, using the measurement of φ. The next step consists in transforming the considered system into the form (1). First, the following attitude error is considered (instead of ( 9)) R := R R whose time-derivative is given by (instead of ( 10)) Ṙ = Rσ R× By defining λ as twice the vector part of the unit quaternion associated with the attitude error matrix R, one verifies that

R = I 3 + λ× + O(| λ| 2 ) λ = σ R + O(| λ||σ R |) (20)
Rewriting the dynamics of V in (19) using the first-order approximation of R, one gets

V = -Ω × V + g R R e 3 + a B = -Ω × V + g R (I 3 -λ× )e 3 + a B + O(| λ| 2 ) = -Ω × V + g R e 3× λ + g R e 3 + a B + O(| λ| 2 ) = -Ω × V + g R e 2 λ1 -g R e 1 λ2 + g R e 3 + a B + O(| λ| 2 ) (21) Concerning the measurement of φ (= V d = V s), one has φ = ((V -V ) + V )((s -ŝ) + ŝ) = V s + (V -V )ŝ + O(|V -V ||s -ŝ|) = V s + (ŝI 3 )V -V ŝ + O(|V -V ||s -ŝ|)
By defining the system output as y := φ + V ŝ, the previous equality rewrites as

y = V s + (ŝI 3 )V + O(|V -V ||s -ŝ|) (22) Let us set x := [x 1 , x 2 ] , with x 1 := [ λ1 , λ2 ] ∈ B 2 2 and x 2 := [s, V ] ∈ R 4 .
In view of ( 21), ( 22), the third equation of ( 19) and the second equation of ( 20), one obtains the system equation ( 1) with

                   A 11 = 0 2 , A 21 = 0 0 g R e 2 -g R e 1 , u 1 = σ R A 22 = φ ⊥ 0 1×3 0 3×1 -Ω × , u 2 = 0 g R e 3 + a B C 1 = 0 3×1 0 3×1 , C 2 = V ŝI 3
The innovation terms, derived from ( 3) and ( 4), are

σ R = u 1 = -K 1 (y -C 2 x2 ) [σ s , σ V ] = -K 2 (y -C 2 x2 )
A persistent excitation condition, involving only the camera's translational motion, whose satisfaction ensures uniform observability of the matrix pair (A(0, t), C(0, x 2 (t), t)) and thus (local) uniform exponential stability of the observer, is given in [START_REF] Hua | Attitude, linear velocity and depth estimation of a camera observing a planar target using continuous homography and inertial data[END_REF].

A. Experimental validation

Experimental validations of the proposed observer has been carried out using a time-synchronised camera-IMU system. Details about the experimental setup, the observer implementation and tuning measures are reported in [START_REF] Hua | Attitude, linear velocity and depth estimation of a camera observing a planar target using continuous homography and inertial data[END_REF]. Ground-truth data are obtained from an Optitrack motion capture system. The experiment is performed with the camera looking downward to observe an horizontal textured planar pattern on the ground. A demo video of this experiment is available at: https://youtu.be/R09oTjr4s40

The time evolutions of the estimated ground-truth attitudes, linear velocities and depths are shown in Figures 456. During the first 60 seconds, depth, gravity direction (i.e. roll and pitch Euler angles) and linear velocity estimates approach the corresponding ground-truth values closely, due to the satisfaction of condition of persistent excitation. By contrast, the camera being kept motionless during the time interval (60 sec, 69 sec), the condition of persistent excitation is not satisfied and one can observe that the depth estimate slightly drifts away from the ground-truth value. At the same time, both gravity direction and linear velocity estimates remain close to the corresponding ground-truth values. Once the condition of persistent excitation is again satisfied on the time interval (69 sec, 89 sec), as a consequence of the camera motion, the depth estimate rejoins the ground-truth value. From Figure 4, the yaw angle estimate drifts away from the ground-truth value. This is expected, and not a shortcoming of the estimation of the state x. Indeed, the yaw angle error corresponds to the third component of λ which is not part of the state. Its estimate, obtained via simple integration of the gyros, is for this reason imperfect. To summarize, the reported experiment shows that (practical) convergence of the gravity direction and linear velocity estimates to the groundtruth values is always achieved, whereas the convergence of the depth estimate to the corresponding ground-truth value is obtained only when the condition of persistent excitation is satisfied.

VI. CONCLUSIONS

After briefly recalling a deterministic Riccati observer design framework proposed in [START_REF] Hamel | Riccati observers for the nonstationary pnp problem[END_REF], which exploits first-order approximations of a generic class of nonlinear systems, and formally generalises the multiplicative extended Kalman filter, the present paper addresses different estimation problems involving proprioceptive sensors and monocular vision to illustrate the applicability of the proposed observer design methodology for a range of estimation problems encountered in practice. The efficiency of the proposed observers for the last two problems are illustrated by means of experiments implementing a time-synchronised camera-IMU system. The proposed methodology is clearly applicable to many other practical problems, such as visual odometry (VO) and visual SLAM (VSLAM). This tutorial also points out the importance of verifying the satisfaction of adequate uniform observability conditions in order to ascertain the good conditioning and good performance of the obtained observers. In this respect, further work to simplify this verification and render it easier will be valuable.

Fig. 1 .

 1 Fig. 1. Intuitive representation of inertial coordinates Pi , planar projective coordinates p p i and spherical projective coordinates p i of the i th source point.

Fig. 2 .

 2 Fig. 2.Camera positions given by the proposed Riccati observer, solvePnP() algorithm and MOCAP ground truth.

Fig. 3 .

 3 Fig. 3. Camera attitudes (Euler angles) given by the proposed Riccati observer, solvePnP() algorithm and MOCAP ground truth.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Estimated and ground-truth attitudes represented by roll, pitch and yaw Euler angles (deg) versus time (s)

Acknowledgment: This work was supported by the ANR-ASTRID CONGRE and the ANR DACAR projects.