John S Richardson
email: john.richardson@ubc.ca

Eric Chauvet
email: eric.chauvet@univ-tlse3.fr

Consumer responses to resource patch size and architecture: leaf packs in streams

Keywords: decomposition, detritivores, detritus, invertebrates, microbial, refuge

Accumulations of leaf litter in freshwaters can vary from individual leaves to large leaf packs. Past studies have demonstrated that decomposition rates decrease as leaf pack size increases. We considered a set of hypotheses that lower breakdown in larger leaf packs occurs as a consequence of diffusion gradients of oxygen and nutrients, or lower accessibility to larger detritivores to leaf tissue in the middle of these leaf packs. We manipulated leaf pack size in a stream to quantify decomposition rates and the abundances of consumers relative to the amount of detrital mass available. Mass loss rates of beech leaves were lower as leaf pack size increased. We found no differences in fungal biomass across our treatment gradients, or when comparing leaves in the middle of the leaf pack versus those on the outside. The lower decomposition of larger leaf packs would thus not result from a lower quality of the resource based on fungal biomass. Invertebrates per unit mass of leaf packs declined exponentially with size of leaf pack. Smaller invertebrates were less abundant per unit of resource as leaf pack size increased, but abundances of larger invertebrates declined even more dramatically than that of smaller ones. The results are consistent with accessibility within leaf packs decreasing as leaf pack size increases, a factor that is important for the estimation of consumer-resource functions for this patchy resource.

Introduction

The size and shape of resource patches can determine resource use functions and predator prey interactions (e.g. [START_REF] Kareiva | Habitat fragmentation and the stability of predator-prey interactions[END_REF][START_REF] Fattebert | Structural habitat predicts functional dispersal habitat of a large carnivore: How leopards change spots[END_REF], along with other biological processes. Locating resource patches may be a function of size against the matrix of nonresource areas. A great deal of theory has considered the role of competition in consumer-resource dynamics, and body size is one factor that can affect use of resources (e.g. [START_REF] Pess | The influences of body size, habitat quality, and competition on the movement and survival of juvenile Coho Salmon during the early stages of stream recolonization[END_REF]. Predator prey relations may also be influenced by physical structure of the en-vironment, such as accumulations of leaves (e.g. [START_REF] Hildrew | Predators and prey in a patchy environment -a fresh-water study[END_REF][START_REF] Richardson | Food, microhabitat, or both? Macroinvertebrate use of leaf accumulations in a montane stream[END_REF].

In streams, leaf packs form a critical resource for a large number of consumers (e.g. [START_REF] Wallace | Effects of resource limitation on a detrital-based ecosystem[END_REF][START_REF] Richardson | Resource subsidy flows across freshwater-terrestrial boundaries and influence on processes linking adjacent ecosystems[END_REF]. Accumulations of leaves also provide habitat where they can find refuge from predators, from hydraulic forces and also collect fine particles that may serve as food (e.g. [START_REF] Richardson | Food, microhabitat, or both? Macroinvertebrate use of leaf accumulations in a montane stream[END_REF]). Leaf packs are patches of food resources against a matrix of physical habitats, and consumers aggregate on those patches [START_REF] Webster | Effects of forest clearcutting on leaf breakdown in a southern Appalachian stream[END_REF][START_REF] Dobson | A. test of resource limitation among shredding detritivores in low order streams in southern England[END_REF][START_REF] Gjerløv | Patchy resources in a heterogeneous environment: Effects of leaf litter and forest cover on colonisation patterns of invertebrates in a British Columbian stream[END_REF]). The spatial configuration of leaf patches in streams has important influences on consumer densities and species composition (e.g. [START_REF] Palmer | Streambed landscapes: Evidence that stream invertebrates respond to the type and spatial arrangement of patches[END_REF]. Moreover, consumers respond dynamically to short-term changes in resource abundances (e.g., [START_REF] Rowe | 129 Habitat structure and fish predation: Effects on invertebrate colonisation and breakdown of stream leaf packs[END_REF]. However, there are many questions about how resource patch size (leaf packs) influence consumer abundance and composition.

Rates of leaf litter breakdown vary with the amount of leaf litter in leaf packs, but are higher with size of leaf packs in some studies (e.g. [START_REF] Pozo | Is loss rate dependent on leaf-litter amount?[END_REF] and lower in others (e.g. [START_REF] Reice | Environmental patchiness and the breakdown of leaf litter in a woodland stream[END_REF]). However, most authors have found that decomposition is faster in smaller leaf packs [START_REF] Reice | Environmental patchiness and the breakdown of leaf litter in a woodland stream[END_REF][START_REF] Benfield | Influence of exposure technique on leaf breakdown rates in streams[END_REF][START_REF] Campbell | The influence of pack size and position, leaf type, and shredder access on the processing rate of Atherosperma moschatum leaves in an Australian cool temperate rainforest stream[END_REF][START_REF] Tsai | Effects of fish predators and litter pack size on leaf breakdown in a subtropical stream[END_REF]), but the mechanisms for why decomposition rates vary with leaf pack size have not been tested. [START_REF] Reice | Environmental patchiness and the breakdown of leaf litter in a woodland stream[END_REF] speculated that accessibility to leaves in leaf packs diminished with increased pack size, but did not have data to test that. Nor has the slope of the relationship been estimated in many instances, and some of these studies might predict decomposition rates to reach zero as leaf pack size increases based on their empirical results (e.g., [START_REF] Campbell | The influence of pack size and position, leaf type, and shredder access on the processing rate of Atherosperma moschatum leaves in an Australian cool temperate rainforest stream[END_REF][START_REF] Tsai | Effects of fish predators and litter pack size on leaf breakdown in a subtropical stream[END_REF].

In this study we sought to test the hypothesis that increasing leaf pack size would lead to increasing densities of consumers, but with a slope less than one relative to leaf abundance, and with a diminishing slope as other processes became limiting (e.g., [START_REF] Richardson | Limits to productivity in streams: evidence from studies of macroinvertebrates[END_REF]. We considered fungal biomass as a measure of the key decomposer on these leaf packs to determine if fungi could be limited by possible diffusion or position gradients within leaf packs of different sizes. We assessed the densities and types of detritivorous invertebrates on leaf packs of different sizes. Based on the literature we predicted a reduced decomposition rate and lower consumer densities per unit resource as leaf pack size increased.

Methods

The study site was in the Ruisseau de Peyreblanque (Alzeau River catchment; 43° 25 41.5 N, 2° 13 13.3 E) in southwest France. The stream is 1.5 to 2.0 m wide on average, with a gradient of about 3 %, at about 740 m elevation. Temperatures recorded from temperature data loggers ranged from 2 to 11 °C over the course of the study (17 th January to 13 th April 2007).

Senesced beech leaves (Fagus sylvatica L.) were collected in the Montagne Noire region east of Toulouse, France in the autumn and dried at ambient temperature in the laboratory. These leaves were used to construct leaf packs of different sizes as noted below. These leaf packs were placed into 10-mm mesh bags of approximately 15 × 10 cm fastened into a tetrahedron shape to avoid compressing leaves by the bag itself. Leaf packs of 0.5, 1, 2, 3, 4 and 5 g dry mass were prepared, with 8 replicates of each.

Leaves were placed in riffles in the stream on 17 th January 2007 using a complete block design, with subsequent collecth March (48 days after placement) and 13 th days after placement). Some bags had a substantial cover of sand, but most (~80 %) were uncovered and on the streambed surface. Some litter bags had some external litter on them, which was brushed off, and any material not cleared off was clearly distinguishable from the leaves in the treatments.

Laboratory procedures

Leaves were washed gently in the laboratory to remove invertebrates, which were saved on a sieve (250 µm mesh) and preserved in ethanol for later enumeration. Leaves for fungal assessments had 15 leaf disks of 12-mm diameter removed using a corer and frozen for later ergosterol determination following the method of [START_REF] Gessner | Ergosterol as a measure of fungal biomass[END_REF]. For leaf packs with 4 or 5 g (> 30 leaves per pack), leaves were divided into "outside" and "inside" leaves. Outside leaves were those that were clearly on the outside or had moved along the leaf pack so they were more-orless by themselves in the nets. The "outside" or "inside" leaves disks, which were processed for ergosterol determination.

Additional leaf disks (12-mm diameter) were taken from leaves to estimate the dry mass of each leaf disk, which was subsequently added back to obtain the overall mass of the leaves, accounting for those disks taken for ergosterol determioven before weighing, ashing (at 500 °C for at least 2 h), and determination of ash-free dry mass.

Invertebrates were sorted from debris under 10× magnification. Identification of invertebrates was based on the keys in [START_REF] Tachet | Invertébrés d'eau douce : systématique[END_REF], generally determined to genus or subfamily (chironomids). All invertebrates were enumerated. Trophic group was determined based on information in [START_REF] Tachet | Invertébrés d'eau douce : systématique[END_REF].

Statistics were run in SAS (version 9.1, SAS Inc, Cary, NC) or R using generalised mixed models. We analysed fungal biomass as a function of leaf position (inside [centre] or outside) from larger leaf packs and its interaction by date in a two-way remaining ash-free dry mass (AFDM) used as covariates. Per-nificant effect of block in our analyses for mass loss, and since there was no associated hypothesis for that term, we dropped it from subsequent analyses for simplicity. Numbers of invertebrates per mass of leaves were fitted using Sigmaplot's curvefitting procedures (SigmaPlot ver. 14).

Results

Leaf packs

There was a positive, but not significant relation between initial leaf pack mass and amount remainsize increased. There was a significant effect of date (p < 0.0001), but no interaction of collection date with the initial mass of leaf packs. When the first date of sampling (day 48) was analysed alone, there was a significant positive relation between initial leaf pack size and amount remaining (slope = 0.137, p < 0.0001, Fig. 1A). Decomposition coefficients (-k) for our beech leaves (at 48 d) ranged from 0.0048 d -1 for the 5 g leaf packs to 0.0101 d -1 for the 0.5 g leaf packs.

Fungal biomass

There was no significant effect of leaf pack size on ergosterol content (R 2 = 0.0053, p a significant effect of date (p = 0.25) or its interaction (p Fig. 2A). Comparison of fungal biomass (ergosterol) in the inside versus outside of 4 and 5 g leaf packs also showed no significant ef-1,27 = 0.39, p = 0.54) (Fig. 2B).

Invertebrates

The total numbers of invertebrates increased with leaf pack size, but numbers of invertebrates per g of remaining leaf litter AFDM fit a strong and negative function (p < 0.0001, R 2 = 0.78, Fig. 3A). Total numbers of invertebrates dropped off dramatically as leaf amount remaining exceeded ~1 g. As with total numbers of invertebrates, the relation between total num-bers of detritivorous invertebrates of all types and leaf pack size was positive, but as a function of numbers per g of leaf AFDM the relation was negative (R 2 = 0.51, p < 0.0001, Fig. 3B). In addition to the caddisflies and stoneflies mentioned below, detritivores here included Brillia sp. (Chironomidae) and Gammarus sp.

For shredding caddisflies only (mostly Limnephilidae [Potamophylax, Drusus] and Lepidostomatidae) there was a significant negative exponential fit per g of leaf litter AFDM as leaf pack size increased (R 2 p < 0.0001, Fig. 3C), with very low levels reached even at moderate leaf pack size.

Detritivorous stoneflies (mostly Nemouridae [Nemoura, Amphinemura, Protonemura] and Leuctridae) declined as with other detritivores (R 2 p = 0.0084, Fig. 3D). However, the percentage of when there was < 1 g (AFDM) to 47.3 % of total shredders for leaf remaining of > 1 g. Numbers per g leaf AFDM were also highly variable between leaf packs (Fig. 3D).

Discussion

We found a significant negative effect of leaf pack size on breakdown rate, although the slope of the relation- ship was very low, and only for the first collection date. This pattern of lower decomposition rate as leaf pack size increases has been demonstrated previously [START_REF] Tsai | Effects of fish predators and litter pack size on leaf breakdown in a subtropical stream[END_REF]. In [START_REF] Reice | Environmental patchiness and the breakdown of leaf litter in a woodland stream[END_REF], decomposition rates pending on season) across a range of pack sizes from 1 to 40 g dry mass. [START_REF] Tsai | Effects of fish predators and litter pack size on leaf breakdown in a subtropical stream[END_REF] found that decomposition rates declined with a slope approaching -1 across a range from 1 to 8 g dry mass, which would result in rates close to zero in packs slightly larger than 8 g. Likewise, Campbell et al.'s (1994) relation would have predicted zero decomposition at leaf pack sizes of 8.12 g. [START_REF] Reice | Environmental patchiness and the breakdown of leaf litter in a woodland stream[END_REF] found high rates of decomposition of Fraxinus americanus leaves during summer (breakdown range from smallest to largest packs: k = 0.024 to 0.011 d -1 ) even at 40 g (the largest). Even though the effect of leaf pack size on loss rate in our study was small, we used a small gradient compared to the authors noted above. The processes involved in reducing these rates with increasing leaf pack size are not clear, but it could involve oxygen gradients if diffusion is constrained within leaf packs or some other influence on consumers, such as accessibility. In our results we found no significant effects of leaf pack size on fungal biomass, even though one of our sampling dates likely coincided with the maximum mycelial accumulation on beech [START_REF] Gessner | Importance of stream microfungi in controlling breakdown rates of leaf litter[END_REF], thus optimizing the chance to detect an effect if one was present. Moreover, we found no difference between fungal biomass on leaves on the outside of a leaf pack versus those within. This would suggest that there was no diffusion gradient of oxygen or nutrients that limited fungal biomass in the middle of these leaf packs. When leaves get buried within the hyporheic zone of a stream, there can be a restriction in the rates of colonisation of leaves by fungal spores, and also in the types of fungi [START_REF] Cornut | Aquatic hyphomycete species are screened by the hyporheic zone of woodland streams[END_REF]. However, in our well-oxygenated stream, it is likely that fungal spores were able to colonize all leaf surfaces without impediment, and they were apparently not limited by diffusion of oxygen and nutrients (at least up to the size of our leaf packs).

Abundances of all invertebrates declined as a function of the size of the resource patch (leaf pack) in our experiment. Likewise, [START_REF] Tsai | Effects of fish predators and litter pack size on leaf breakdown in a subtropical stream[END_REF] found that invertebrate numbers per gram of leaf litter decreased as pack size increased (all with slope < 1.0 in their figure). We observed that large, cased caddisfly larvae declined most strongly in abundance per unit resource, presumably because they do not have access to the inner leaves in larger leaf packs, whereas smaller shredders do. This would mean that with increasing leaf pack size, a larger proportion of the pack is unavailable for these larger consumers. [START_REF] Reice | Environmental patchiness and the breakdown of leaf litter in a woodland stream[END_REF] had speculated that accessibility might be one hypoth- esis to support his observation of lower decomposition rates as leaf pack size increased. On the other hand, other consumers, including fungi and small-bodied detritivores, have access to leaves throughout the leaf pack. [START_REF] Tsai | Effects of fish predators and litter pack size on leaf breakdown in a subtropical stream[END_REF] found that the effect of leaf pack size was most pronounced for the cased caddisfly Anisocentropus, less strong for Nemouridae, and least for the leaf-boring chironomid Stenochironomus.

ference in biomass of detritivores across a gradient of leaf pack sizes similar to ours. Their most common shredder was Gammarus pseudolimnaeus (~81 % of invertebrates) which may have been able to get into the leaf packs more easily than cased caddisflies (Ru-We found no evidence that small-bodied shredders exhibited any compensation in density because of fewer larger-bodied species, and in fact, the slope of small-bodied invertebrate numbers versus leaf pack mass was negative and weak, although not significant. A smaller number of large-bodied detritivores per unit mass of leaves could have led to reduced facilitation of growth by smaller-bodied species, as demonstrated by [START_REF] Tonin | Interactions between large and small detritivores influence how biodiversity impacts litter decomposition[END_REF].

Another hypothesis that could explain lower densities of invertebrate consumers as leaf pack size increases is diminished concentrations of oxygen inside larger packs as microbes use up available oxygen. Leaf packs that are buried beneath the surface of a stream have lower oxygen concentrations, which apparently lead to lower fungal biomass and lower decomposition rates [START_REF] Cornut | Early stages of leaf decomposition are mediated by aquatic fungi in the hyporheic zone of woodland streams[END_REF]. It is also possible there could be a diffusion gradient of nutrients, but the biomass of fungi per biomass of leaf showed no decrease with leaf pack size, which would have been evidence for such a possibility. Thus, we do not see indication that any gradient in oxygen or nutrients could have been responsible for the lack of differences in fungal biomass throughout the leaf packs. Fragmentation of leaves at the outside of leaf packs through abrasion can be a possible source of variation, but we saw no visual evidence of that, and that would be inconsistent with the relationships with invertebrate densities. Moreover, the absence of evidence for variation in resource quality further supports accessibility of the interior of leaf packs as the explanation for lower densities per unit leaf mass of detritivores with increasing leaf size. The steeper slope of that diminishing curve of density per unit leaf mass for large-bodied detritivores is also consistent with reduced accessibility to resources deep within a larger leaf pack.

The quality of leaves in leaf packs varies considerably across species of leaf litter. Our leaf litter packs might have been avoided in preference to higher quality litter. However, in the stream we used, most litter was beech leaves, and thus the leaf packs we prepared would represent the natural leaf resource, or even a high-quality resource against the alternative, which was pine and spruce needles. Given that the leaves were of the same quality as most of the background detritus, leaf quality would not account for differential consumer use of differently sized leaf packs.

Leaf pack size and the architecture of these microhabitats clearly affect the kinds of consumers that have access to these resource patches. We may need to consider leaf pack size as another variable affecting decomposition rates. This is an added detail for consumer-resource dynamics consideration, and another reason measures of resource abundance or density per mass of resources should be included in studies of populations and community ecology of detrital-based streams.

  Fig. 1. lines) in the stream as a function of initial leaf pack mass.

Fig. 2 .

 2 Fig. 2. Leaf-associated fungal biomass based on ergosterol determinations. Fungal biomass per g leaf AFDM across the gradient of initial leaf pack mass (A), and fungal biomass based on samples from the inside or outside of (B).

Fig. 3 .

 3 Fig. 3. Numbers of invertebrates per g AFDM of beech leaf litter for total invertebrates (A), total shredders (B), cased caddisflies (C) and stoneflies only (D)

Acknowledgements

We appreciate the financial support of the University of British Columbia, Centre National de la Recherche Scientifique, l'Université de Toulouse, and the Natural Sciences and Engineering Research Council (Canada). Daphne Richardson and Debby Martin helped process leaf packs. Didier Lambrigot (EcoLab) kindly provided ergosterol measures for our leaf disks. We are grateful for the suggestions for improvements from the reviewers and editor.


Ide-Flore Kenmogne

Vincent Drevelle
email: vincent.drevelle@irisa.fr

Eric Marchand

Cooperative Localization of Drones by using Interval Methods

Keywords: cooperative localization, intervals, vision, constraint propagation

In this article we address the problem of cooperative pose estimation in a group of unmanned aerial vehicles (UAVs) in a bounded-error context. The UAVs are equipped with cameras to track landmarks, and with a communication and ranging system to cooperate with their neighbors. Measurements are represented by intervals, and constraints are expressed on the robots poses (positions and orientations). Pose domains subpavings are obtained by using set inversion via interval analysis. Each robot of the group first computes a pose domain using only its sensors measurements. Then, through position boxes exchanges, the positions are cooperatively refined by constraint propagation in the group. Results with real robot data are presented, and show that the position accuracy is improved thanks to cooperation.

Introduction

Unmanned aerial vehicles (UAVs) have significantly attracted the attention and interest in several applications areas involving multi-robot tasks, such as search and coverage missions [START_REF] Maza | Multiple UAV cooperative searching operation using polygon area decomposition and efficient coverage algorithms[END_REF][START_REF] Bevacqua | Mixed-initiative planning and execution for multiple drones in search and rescue missions[END_REF], exploration [START_REF] Fox | Distributed multirobot exploration and mapping[END_REF], cooperative manipulation [START_REF] Michael | Cooperative manipulation and transportation with aerial robots[END_REF][START_REF] Gioioso | The flying hand: A formation of UAVs for cooperative aerial tele-manipulation[END_REF], control [START_REF] Jiang | Shared-control for a UAV operating in the 3D space[END_REF][START_REF] Schiano | Bearing rigidity maintenance for formations of quadrotor UAVs[END_REF], etc. In each of these cases, one problem to be solved is the cooperative localization (CL) in a group of robots [START_REF] Roumeliotis | Distributed multirobot localization[END_REF]. CL consists in improving the positioning capacity of each robot through information exchanges (e.g. sensors measurements) with other robots in the group.

Several methods exist for solving CL, such as the Extended Kalman Filter (EKF) for a centralized system [START_REF] Huang | Observability-based consistent EKF estimators for multi-robot cooperative localization[END_REF][START_REF] Qu | Cooperative localization against gps signal loss in multiple uavs flight[END_REF][START_REF] Davis | Applying cooperative localization to swarm UAVs using an extended Kalman filter[END_REF] or, if the computation is decentralized and the communication is unreliable, other techniques like Covariance Intersection [START_REF] Uhlmann | General decentralized data fusion with covariance intersection[END_REF] or Interleaved Update [START_REF] Bahr | Cooperative localization for autonomous underwater vehicles[END_REF]. Approaches that assume bounded errors using polytopes and linear programming algorithms have also been proposed in [START_REF] Spletzer | A bounded uncertainty approach to multi-robot localization[END_REF][START_REF] Camillo | A bounded uncertainty approach to cooperative localization using relative bearing constraints[END_REF].

In a bounded-error measurements context, interval set-membership methods [START_REF] Moore | Interval analysis[END_REF] allow rigorous errors bounds propagation in non-linear estimation problems.

Many applications of these methods are found in practical areas like mobile robots localization [START_REF] Colle | Mobile robot localization by multiangulation using set inversion[END_REF] or positioning integrity for autonomous cars [START_REF] Drevelle | Localization confidence domains via set inversion on short-term trajectory[END_REF], and as part of the cooperative localization of underwater robots with sonar [START_REF] Bethencourt | Cooperative localization of underwater robots with unsynchronized clocks[END_REF] and vehicle networks with GNSS [START_REF] Lassoued | Cooperative localization with reliable confidence domains between vehicles sharing GNSS pseudoranges errors with no base station[END_REF]. Interval methods also enable to derive efficient particle filter implementations for highly non-linear localization problems [START_REF] Merlinge | A box regularized particle filter for terrain navigation with highly non-linear measurements[END_REF]. In [START_REF] Kenmogne | Image-based UAV localization using interval methods[END_REF], the authors have shown that interval methods are an effective tool for vision-based boundederror positioning of a drone.

In this paper, we propose to use interval set-membership methods as a tool to solve the problem of cooperative localization of a fleet of drones. Each drone in the group is equipped with a camera able to track georeferenced landmarks, and a communication and ranging system providing each robot a way to exchange data and measure distances with its neighbors. Communication and ranging are also available with a base station of known position. The aim is to calculate a pose domain for each robot from constraint propagation and set inversion techniques. The proposed method computes, for each drone, a set that contains all its feasilble poses. The solution sets take into account image measurements, distance measurements and the georeferenced landmarks positions; all being uncertain with bounded errors. The computed pose domain of each robot is guaranteed to contain the solution as long as the measurements error bounds are not violated. The paper is structured as follows: Section 2 presents the problem statement, then the proposed estimation method is presented in Section 3. Finally, experimental results are reported in Section 4.

Problem statement

Let us consider the global reference frame F w , in which evolves a fleet of n drones R k (k ∈ {1 . . . n}). A reference frame F r k (also known as body frame) is attached to the center of gravity of each robot. The pose of a drone R k is given by r k = (x k , y k , z k , φ k , θ k , ψ k ) in the global reference frame.

In the sequel, we consider that the drones are identical in their architecture. They are equipped with calibrated onboard cameras (the intrinsic parameters are known), and the rigid transformation between the camera frame and the robot body frame is known by calibration and represented by the homogeneous rotation and translation matrix c T r . We also suppose, as in [START_REF] Schiano | Bearing rigidity maintenance for formations of quadrotor UAVs[END_REF], that each robot flies in nearhovering regime, thus allowing the precise measurement of roll and pitch angles. The neighborhood of a robot R k , i.e. the set of robots it can communicate with, is denoted by N (R k ).

Scenario description

The navigation environment consists of a base station B (see Figure 1) able to communicate with the drones, and m georeferenced landmarks (for which the positions are known). In order to accomplish its navigation task, each robot of the group must be located relative to F w . Each drone R k is equipped with the following

R 1 R 2 B d 1 d 2 d 1,3 Landmarks R 3 d 3 d 1,2 d 2,3
Figure 1: Cooperative localization with camera and range measurements embedded sensors: camera, altimeter and inertial measurement unit (IMU) with a compass. This enables the following measurements:

• the altitude z k ;

• the roll φ k and pitch θ k angles, with good precision;

• the yaw angle ψ k (with rough precision due to magnetic disturbances);

• and the 2-D points xk 1 , . . . , xk m in the image I k , corresponding to the observable 3-D landmarks w X k 1 , . . . , w X k m in the environment.

In addition to these on-board sensor measurements, each robot R k is able to:

• measure its distance d k,j to a neighboring robot R j (R j ∈ N (R k )),
• measure its distance d k to the base station B (b = (x B , y B , z B ) in F w )

• communicate and exchange information with each robot in its neighborhood.

This can be done by using for instance an IEEE 802.15.4.a Ultrawide Band (UWB) communication system [START_REF] Sahinoglu | Ranging in the IEEE 802.15.4a standard[END_REF] which allows time-of-flight ranging.

Under the assumption of bounded-error measurements, we want to compute the domains of all the feasible 6-DOF poses for each drone in the global reference frame F w . The components of the pose are those of the position vector p k = (x k , y k , z k ) and the attitude (roll, pitch and yaw) q k = (φ k , θ k , ψ k ). Given that three of the six degrees of freedoms are measured with low uncertainty (z k , φ k , θ k ), our attention will be focused on estimating the horizontal position (x k , y k ) and the yaw angle ψ k of each robot. The knowledge on the altitude, the roll and the pitch measured for each robot will be integrated into the problem in the form of constraints. The set of observation equations allows us to define, for each robot, a set of constraints described in the following section.

Set of constraints related to the measurements

To solve the problem described in the previous scenario, let us express it as a constraint satisfaction problem (CSP). We will then be able to use the SIVIA [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF] branch-and-bound algorithm with contractors (see Section 2.3) to characterize the domain of feasible poses for each robot. For this, each measurement is related to the state of the robot by a constraint in the CSP describing the problem.

2D image measurements constraints

The pinhole camera model describes the projection of a 3D point in the image plane. In homogeneous coordinates, a world point w X i = (X, Y, Z, 1) is projected in the image at the pixel coordinates xk i = (u, v, 1) given by:

xk i = K Π c T w (r k ) w X i , (1)
where K is the camera intrinsic parameters matrix (principal point coordinates u 0 , v 0 and focal length to pixel size ratios p x and p y ) and Π is the perspective projection matrix:

K =   p x 0 u 0 0 p y v 0 0 0 1   , Π =   1 0 0 0 0 1 0 0 0 0 1 0   , (2)
and c T w (r k ) is the world to camera rigid transformation matrix, which depends on the robot pose r k and the known robot to camera transformation c T r .

Let us denote by

( c X k , c Y k , c Z k , 1) = c T w (r k ) w X i
the coordinates of a landmark in the coordinate frame attached to the camera of the robot R k . We can now define the set of constraints related to measurements in the image:

• The perspective projection constraint of a point in the image C k proj :

C k proj,i :

u = p x x k i + u 0 , v = p y y k i + v 0 , with x k i = c X k c Z k and y k i = c Y k c Z k . (3)
• The front looking camera constraint C k f ront , ensuring that the landmark is in the front half-space of the camera:

C k f ront,i : c Z k i > 0 . (4)
The pose of each drone R k will have to verify the constraints C k proj,i and C k f ront,i for all i corresponding to the observed landmarks.

Base station distance constraint

Considering a measurement d k of the distance to the base station B, the constraint C k base expresses the fact that the Euclidean distance between the robot R k and B is equal to the measured distance. Recalling that the position part of the robot pose is p k = (x k , y k , z k ) , we have :

C k base : {d k = p k -b 2 } . ( 5
)
This constraint is used to reduce the robot's position domain. Indeed, as the position of the base station b is known, the knowledge of d k constrains the position r k of the drone to be located on a sphere S 2 (b, d k ) as shown in Figure 2.

Inter-distance measurements constraints

Assuming that the ranging system antenna coincides with the center of mass of the drone, distance measurements are independent of the robot attitude. The constraint related to inter-distance measurements between a drone R k and its neighbor R j with R j ∈ N (R k ) is defined by:

d k,j = p k -p j 2 with (k, j) in {1 . . . n} 2 and k = j, (6)
C r k ,rj : {d k,j = p k -p j 2 }. (7)
The additional knowledge of d k,j by the robot R k restricts the possible configurations. For a given candidate position of neighbor R j , the eligible positions of R k are limited to a circle (intersection of the two spheres S 2 (b, d k ) and S 2 (p k , d k,j )).

Measurements of distances d k and d k,j provide constraints on the robot positions, but there are still 3 degrees of freedom (on 6DoF) corresponding to all rotations of the formation around the base station.

Altitude measurement constraint

Let us consider the altitude z meas Using only distance and altitude measurements leads to an ambiguity on the position of the robot, dependent on the orientation and corresponding to the rotations around the vertical axis of the base station B. Adding the constraints from the image measurements will enable to remove the ambiguity.

Set inversion via interval analysis with contractors

The set inversion via interval analysis (SIVIA) algorithm [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF] is a branch-andbound method, which enables to compute an outer approximation of the solution of a CSP. The algorithm starts from an arbitrarily big box [x 0 ] that must contain the solution set. The solution is then enclosed in an outer subpaving S, i.e. a set of non-overlapping boxes, which covers the solution. A parameter controls the precision of the obtained solution approximation. SIVIA can be implemented with box reduction operators called contractors [START_REF] Chabert | Contractor programming[END_REF]. A contractor for a CSP takes an input box, and returns a smaller included box without losing any solution. Assuming a contractor C is available for the CSP, using for instance interval constraint propagation techniques [START_REF] Benhamou | Revising hull and box consistency[END_REF], a very straightforward implementation of SIVIA is given by the following algorithm (see the book [START_REF] Jaulin | Applied interval analysis: with examples in parameter and state estimation, robust control and robotics[END_REF] for more detailed algorithms):

Algorithm 1 Outer SIVIA with contractor. Input: [x 0 ]: initial box, C: contractor, : subpaving box width Result: S: outer solution subpaving

1: S ← ∅ 2: L ← [x 0 ]
Stack of working boxes initialized with [x 0 ]. 3: while L is not empty do 4:

[x] ← pop(L)
Get the next working box from the stack.

5:

[x] ← C([x]) Contract the working box.

S ← S ∪ [x]

Otherwise, add the box to the solution set.

11:

end if 12: end while

Cooperative localization via position domain exchange

From the set of constraints related to the measurements defined in Section 2.2, it is possible to define a CSP which will allow the effective computation of the robot poses domains by propagating the uncertainties on the measurements. We propose a distributed approach that consists in performing a data fusion based on position domains sharing.

Each robot computes its own pose domain only. In this way, whatever the number n of robots, the number of bisections dimensions in SIVIA remains unchanged (equal to 3). Position estimation is done in two steps (Figure 3). Firstly, each drone R k (k in 1 . . . n) estimates its pose domain by using image measurements and distance to the base station (measurements that depends on robot R k only). Secondly, each drone exchanges its resulting position domain with all its neighbors and, by using inter-distance measurements, refines its position thanks to the constraints of distances between each robot R k and its neighbors R j , j ∈ N (k).

To solve the two steps, we define two CSPs for which we characterize a solution set using SIVIA with the HC4 contractor [START_REF] Benhamou | Revising hull and box consistency[END_REF].

CSP related to vision and distance to the base station

At each time step, R k first computes an outer approximation S k img+base of the solution set of the CSP H img+base defined from the constraints of Section 2.2 and given by:

H img+base :       r k ∈ [r k ], xk i ∈ [x k i ], w X k i ∈ [ w X k i ], d k ∈ [d k ], {C k proj,i , C k f ront,i , C k base }, i ∈ 1 . . . m       . (8)
Adding the distance measurement to the base station makes it possible to refine the position of the robot. We can observe this tightening of the bounds through the result of Figure 4 where a drone observes five landmarks and has a measurement of its distance to the base station. Figure 4a shows the subpaving result for the image measurements only. Figure 4b shows the result with measurement in the image and distance to the base station.

CSP related to inter-distance measurements for cooperation

After computing the set S k img + base of poses that are compatible with the distance to the base station and the image, R k computes the bounding box [p k ] of its position domain:

[p k ] = proj p (S k img+base )
where is the bounding box operator and proj p the projection of the pose domain on the position space: y,z,φ,θ,ψ) → (x,y,z) .

proj p : SE(3) → R 3 (x,
The domain [p k ] is then shared to all the neighbors R j , j ∈ N (k) of the robot R k , and the distances d k,j between R k and R j are simultaneously measured.

Pose domain contraction

After receiving information from its neighbors (position boxes [p j ] and distance measurements [d k,j ]), each drone R k tries to refine the current bounds (S k img + base ) of its pose domain. It propagates the information on new inter-distance measurements by locally contracting the CSP H inter defined as follows:

H inter :       r k ∈ S k img+base p k = proj p (r k ), p j ∈ [p j ], j ∈ N (k) d k,j ∈ [d k,j ], j ∈ N (k) d k,j = p k -p j 2 j ∈ N (k)       (9)
where p k = (x k , y k , z k ) is the position of R k . The pose domain S k img + base is thus reduced by using interval constraint propagation to contract H inter and provide the final solution set S k .

Propagation of the position domain improvement

The constraint network formed by the group of robots can contain cycles covering several robots. The contraction of the local CSPs H inter must be propagated again across the network to improve the reduction of the feasible pose domain of each robot.

If, after solving H inter the bounding box of the robot's position domain [p k ] is reduced, R k therefore re-transmits the update of [p k ] to its neighborhood. This process is iterated until a fixed point is reached, i.e. when there is no significant improvement on the position domains.

The proposed method has been implemented and tested using real data and results are presented in the following section.

Experimental results

The proposed method has been tested for a fleet of n = 4 drones, with data acquired on a Parrot AR-Drone2 UAV. The environment contains m = 5 landmarks, which are boxes on which are placed AprilTag markers [START_REF] Wang | AprilTag 2: Efficient and robust fiducial detection[END_REF]. A Vicon motion capture system has been employed to determine the landmarks coordinates, and to provide the ground truth position and orientation of the drone. Figure 5 illustrates the views of the onboard cameras for each drone at time t = 8 s. Error bounds in the image are equal to ±0.5 px and the distance measurement error is assumed to be less than ±5 cm.

Influence of cooperation on the size of the pose domain

The acquired dataset enables to test our method for a different number of drones. Figure 6 shows in green the position subpaving obtained for the drone R 1 at time t = 6.3 s in the different cases, from n = 1 (R 1 alone, no cooperation) to n = 4 (the full fleet cooperates). Figure 6b shows the position subpaving of the drones R 1 and R 2 , when n = 2. We notice the significant reduction of the position domain of R 1 following the cooperation with R 2 , although R 2 is not very well located (able to observe only two landmarks). More domain reduction occurs in the cases where n = 3 and n = 4. This can be observed in Figure 6c and Figure 6d.

These different figures show that cooperation allows a better localization of each robot, even in the case of full landmark visibility loss (case of the robot R 1 ). This can be confirmed by the statistics of domain widths and horizontal position errors of the 4 robots for groups of different sizes, computed over the whole dataset and presented in Table 1. The average width of the position domain of the drone R 1 is 1.29 m when the latter is located without cooperation. Thanks to the cooperation, the width of its domain can be reduced and reach a value of 33 cm (in the case of a group of 4 drones), which is a 74% decrease. The average horizontal position error for R 1 , using the center of the subpaving bounding box as a point estimate, improves from 22.6 cm to less than 5 cm (first column of Table 1b ).

Influence of cooperation in case of reduced visibility

At certain timestamps of the experiment, the tracking of landmarks in the image fails, thus leading to violations of the error bounds.

Figure 7 presents the subpaving results of our cooperative approach for each of the four drones. The first sub-figure (top-left) presents the case of complete visibility, where each of the drones is able to track the 5 landmarks. The other three sub-figures correspond to cases of reduced visibility where some of the 4 robots were not able to track any point in the image. Black outlines represent the subpavings obtained before cooperation. In case of inconsistency in the image measurements, they are discarded and only the distance to the base station is used to compute the black subpavings. The colored subpavings represent the results obtained for each robot after a fixed point has been reached at the cooperation stage.

R 1 R 2 R 3 R 4
These results show that if one of the drones is well located, the cooperation step allows to find an acceptable position domain for all its neighbors.

Conclusion

In this article, an interval analysis based approach to solve the problem of cooperative localization for a group of unmanned aerial vehicles has been proposed. The fleet of drones use cameras to track visual landmarks, and a communication and ranging system for cooperation. The approach is distributed and based on position domains exchanges between robots. We have shown that using cooperation and domain exchanges makes the approach extensible with respect to the number of robots in the group without a significant computational cost in the estimation process.

Experimental results with real image data have been presented for a group of 4 robots, with an average computation time for each of the drones equal to 350 ms without having implemented code optimization. Additionally, the results show that cooperation between robots is an asset for group operation when many robots have reduced visibility or are completely blind.

The proposed method can deal with detectable erroneous measurements in the images, i.e. outliers leading to an empty-set solution. It then only relies on distances measurements and data exchange with other robots for position determination. This basic fault detection scheme however does not prevent the propagation of non-detectable image measurements faults to the whole fleet, nor the mitigation of faulty distance measurements. Future work will aim towards the development of adding more fault-tolerance and robustness to interval-based distributed cooperative localization systems.

Figure 2 :

 2 Figure 2: Distance constraint to the base station

k

  of each robot measured by the altimeter, we have the constraint z k = z meas k which places R k on a plane. It allows to constrain the sphere S 2 (b, d k ) formed by the distance to the base station B and the robot R k on a horizontal circle resulting from the intersection of S 2 (b, d k ) and the plane.It also allows to constrain the sphere S 2 (p k , d k,j ) solution of the measurement of the inter-distances d k,j . For a known position of R j , R k has only two possible positions corresponding to the intersection of the horizontal position circle of R k with the sphere S 2 (p k , d k,j ).

Figure 3 :

 3 Figure 3: Distributed set-membership pose estimation method.

  (a) (x, y)-plane projection of one robot pose domain with only camera measurements. (b) (x, y)-plane projection of one robot pose domain with camera and distance measurement to the base station B.

Figure 4 :

 4 Figure 4: Single drone pose domain computation (horizontal projection).

Figure 5 :

 5 Figure 5: Onboard cameras views at t = 8 s. Landmarks are boxes with AprilTag printed pattern.

  (a) (x, y)-plane projection of the pose domain for the drone R 1 (in green) at time t = 6.3s (b) (x, y)-plane of the pose domains for drones R 1 (in green) and R 2 (in red) at time t = 6.3s (c) (x, y)-plane of the pose domains for drones R 1 , R 2 and R 3 (in blue) at time t = 6.3 s (d) (x, y)-plane of the pose domains for drones R 1 , R 2 , R 3 and R 4 (in pink ) at time t = 6.3 s

Figure 6 :

 6 Figure 6: Horizontal projection of the pose domains computed for an increasing number of robots in the group, at time t = 6.3 s.

4

 4 33 0.37 0.78 0.61 (a) Mean width of the horizontal position domain bounding box (in meters) UAVs 0.04 0.05 0.25 0.19 (b) Mean horizontal position error, considering the center of the bounding box (in meters)

  (a) (x, y)-projection of the subpavings of the 4 drones in the case of full visibility at t = 0 s. (b) (x, y)-projection of the subpavings of the 4 drones in the case of reduced visibility (R 3 , in blue, is blind) at t = 14.8 s. (c) Subpavings of the 4 drones in the case of reduced visibility, R 1 (in green) and R 2 (in red) are blind at t = 6.4 s. (d) Subpavings of the 4 drones in the case of reduced visibility, R 1 (in green) ,R 2 (in red) and R 3 (in blue) are blind at t = 12.2 s.

Figure 7 :

 7 Figure 7: Position domains of the 4 drones. Black outlines: subpavings before robot cooperation. Colored domains: subpavings obtained after cooperative localization.

Table 1 :

 1 Horizontal position results statistics, for groups of 1 to 4 drones. The reported figures are mean values over the whole experiment.


Laurent Lisnard

Lise-Marie Chamoreau

Yanling Li

Solvothermal synthesis of oxamate-based helicate: temperature dependence of the hydrogen-bond-structuring in the solid

Solvothermal synthesis has been successfully tested as a new synthetic method towards the formation of oxamate-based coordination compounds. The reaction of 1,3benzenedioxamic acid (H4mpba) with divalent metal ions has afforded the dinuclear compounds TBA2[Co2(H2mpba)3].2DMF.5H2O (1a), (HNEt3)2[Co2(H2mpba)3].6DMF.5H2O (1b), TBA2[Ni2(H2mpba)3].2DMF.2.5H2O (2) and (HNEt3)2[Co2(H2mpba)3] (3). Although the 3:2 ligand to metal ratio is known for the (mpba) 4-ligand under bench conditions these complexes are the first examples of oxamate-based helicate. Furthermore, crystalographic studies show a temperaturedependent hydrogen bond structuring that leads to racemic or chiral hexagonal 3D networks.

INTRODUCTION

The synthesis of molecular building blocks that self-assemble or are capable of connexion with another component to form molecular objects or network of defined topology or geometry is one of the major subject in crystal engineering [1][2] and in supramolecular chemistry [3][4][5] . The assembly process relies on weak chemical interactions such as H-bond, electrostatic interactions or coordination bonding. However, due to subtle equilibria between these weak interactions there are still a lot of surprises in the outcome and the final product can be quite different from the expected one. These unexpected results often open up new opportunities and contribute greatly to the beauty of synthetic chemistry. Among all the research work on the crystal engineering of nets, the synthesis of metal-organic frameworks has increased exponentially. 6 The reason for this success is related to the chemical flexibility of these materials owing to the presence of tunable organic and inorganic parts which allows to scale up the system or to introduce new properties.

Moreover MOFs have a lot of potential applications in catalysis but also in hydrogen 7 and CO2 8 storage which are two of the most pressing technological social needs. MOFs are also highly interesting for their physical properties such as luminescence [9][10] or ferroelectricity 11 and magnestism 12 . Regarding magnetic MOFs, the goal is the obtention of magnets and more precisely porous magnets, so in addition to the control over atom arrangement, the ability of organic ligands to propagate the electronic interaction between the magnetic ions is of paramount importance. At first sight, large porosity seems incompatible with magnetic ordering due to the exponential decrease of the interaction between the magnetic ions with growing metal-metal distances [13][14] inevitably leading to a dramatic drop in the ordering temperature value when the system is scaled up with longer organic linkers. To tackle this problem, we have designed over the past few years new polytopic oxamate-based ligands able to efficiently transmit electronic interaction across extended aromatic bridges via spin polarization mechanism. Starting from these ligands, we were able to obtain homometallic polynuclear complexes which effectively exhibit relatively strong ferromagnetic interaction in spite of large metal-metal separation [15][16][17][18] . When reacted with additional metal ions, some of these complexes give 2D or 3D magnetic MOFs [19][20][21] with magnetic ordering in the 6.5 K to 22.5 K temperature range. The metal-metal distance through the bridging oxamate ligand is not tunable but with the help of serendipity large octogonal pores have been obtained. 21 Furthermore these networks are resizable by changing the aromatic parts while retaining a significant interaction. Nevertheless the final geometry of the nets depends on subtle chemical equilibria and to fully exploit this family of ligand we have decided to explore their reactivity in solvothermal conditions. This method imposes strong constraints both in pressure and in temperature to the reaction media and allows to work beyond the solvent's boiling point. 22 Consequently the solubility and the mobility of the reactants are increased and reactions that could not be carried or would not yield crystals under bench conditions become feasible. 23 This method has convincingly proved its efficiency for the synthesis of multidimensional materials 24- 25 as well as the synthesis of coordination clusters 26 and represents therefore a convenient tool to investigate polyoxamate's reactivity.

To start our study we have been focusing on the 1,3-benzenedioxamate ligand (Figure 1a) since it is known to promote ferromagnetic exchange between paramagnetic ions through spin polarisation mechanism. 15 We have thus reacted this ligand with cobalt(II) or nickel (II) Crystallographic studies reveal that all the complexes adopt a helicate geometry which has never been observed to date in oxamate-based complexes. X-ray diffraction studies also show that the supramolecular hydrogen-bond arrangement of the complexes in the solid is strongly dependent of the reaction temperature: reactions performed at 100°C have afforded a racemic hexagonal 3D structure in contrast with a chiral phase that forms at 150°C.

EXPERIMENTAL SECTION

Materials and Methods. All reagents were used as purchased with no further purification.

Diethyl N,N′-1,3-benzenedioxamate, H2Et2mpba, was prepared according to the literature procedure. 27 1 H and 13 C NMR spectra were collected on a 400 MHz Bruker Avance spectrometer at 298 K. ATR/FT-IR spectra were collected on a Bruker TENSOR 27 equipped with a simple reflexion ATR diamond plate of the Harrick MPV2 series. Magnetic measurements in dc mode were performed on a Quantum Design MPMS SQUID on polycrystalline samples restrained in a plastic film. Data were corrected for the diamagnetism contributions of the samples using Pascal constants. The sample holder diamagnetism was measured and subtracted from the raw data.

Synthesis.

Synthesis of H4mpba.3H2O. Hydrate N,N′-1,3-benzenedioxamic acid was prepared as follows:

aqueous NaOH (6.22g, 1.55 mol, 100 mL) was added dropwise to a suspension of H2Et2mpba (12g, 38.9 mmol) in water (800 mL). The solution was stirred for 30 min and filtered. Addition of 4M HCl (38.9 mL, 1.55 mol) to the resulting solution leads to the formation of a white precipitate.

Crystallographic studies.

Data were collected on a Bruker Kappa-APEX II CCD diffractometer for 1b and 3 (MoK , = 0.71069 Å). Crystals were mounted on a Hamilton cryoloop using Paratone-N oil and placed in the cold flow produced with an Oxford Cryocooling device. Partial hemispheres of data -predefined with the APEX II software 28a -were collected using ϕ and scans (110 s/frame for 1b, 100 s/frame for 3). Integrated intensities were obtained with SAINT+ 28a and were corrected for absorption with SADABS 28b,c ; structure solution and refinement was performed with the SHELXTL-package. 28a The structures were solved by direct methods and completed by iterative cycles of ΔF syntheses and full-matrix least-squares refinement against F 2 . 1a, 1b and 2 are poorly diffracting and despite numerous efforts the best data set obtained, for 1b, still gives some large reliability factors and restraints have been used to model counter-ions and solvent molecules. Crystallographic data and refinements parameters for 1b and 3 are given in Table 1.

X-ray powder diffraction patterns for 1a, 1b and 2 are presented in the Figure S1 of the supplementary information. Diagrams were collected on a Philips X'pert Pro diffractometer using CuK 1 monochromatised radiation ( =1.54060 Å) and equipped with a X'celerator linear detector.

γ [°] 120 90
V [Å 3 ] 12359(3) 2378,5(4) Z 6 2
T [K] 200(2) 200(2) λ [Å] 0.71073 0.71073 ρcalc [g cm -3 ] 1.
€ R 1 = F o -F c / F o ∑ ∑ € ωR 2 = ω F o 2 -F c 2 ( ) 2 $ % & ' ( ) ∑ ω F o 2 ( ) 2 $ % & ' ( ) ∑ + , - . / 0 1 2

RESULTS AND DISCUSSION

Compounds 1a, 1b and 2 are isostructural and crystallise in the R-3c centrosymetric space group with an asymmetric unit that consists of one sixth of the bimetallic complex. Each complex is made of two divalent ions in octahedral geometry bridged by three partially deprotonated (H2mpba) 2- ligands. The metal ions bind two oxygen atoms from the oxamate groups leaving one carbonyl and one amido function uncoordinated. Such a binding mode arises from the non-deprotonation of the amido groups -independently of the strength of the base used in the reaction -and results

in the formation of a triple-stranded chiral helicate complex where the oxamate groups lie in the plane of the benzyl rings (Figure 1b). This binding mode of the (Hxmbpa) (4-x)-ligand is encountered for the first time. Indeed, under bench conditions and in aqueous media, the fully deprotonated form of the ligand (mbpa) 4-also binds octahedral ions in a 3:2 fashion but affords a heterochiral triple-stranded mesocate bimetallic complex [Co2(mpba)3] 8-where the oxamate groups are perpendicular to the benzyl rings (Figure 3a). 20 This difference in the geometrical arrangement in 1b and in [Co2(mpba)3] 8-changes the relative orientation of the coordination polyhedra between the two halves of the dinuclear complexes. In 1b the two polyhedra have a staggered conformation when viewing along the Co-Co axis whereas in [Co2(mpba)3] 8-the two polyhedra have an eclipsed conformation (Figure 3b and3c). Consequently these conformations lead to different networks. In 1b, the double H-bonds between adjacent complexes lead to a stacking of staggered hexagonal networks (Figure 4a) with the same topology than diamond or sphalerite structures i.e. "dia" net in RCSR database 34 whereas in Li5[Li3Co2(mpba)3(H2O)6]•31H2O, the coordination of the two cis carbonyl-oxygen atoms of the oxamato groups to Li + ions leads to 3D honeycomb framework with hexagonal channels and is topologically like the lonsdaleite or wurtzite structures i.e. "lon" in RCSR database. By contrast, the staggered conformation of the helicate does not lead to large pore in 1b. Compound 3 crystallises in the P21 chiral space group and a whole complex is found in the asymmetric unit. The bimetallic complex is identical to the previously described compounds however there are drastic differences in the 3D layout since the H-bond-driven arrangement of the bimetallic species leads to a homochiral framework (Figure 5). The H-bond network is far more complex than in the previous cases with each ligand of the same molecular unit displaying a different H-bonding mode to two or three adjacent molecules (Figure 6). Triethylammonium cations also form hydrogen bonds with the carbonyl groups of the ligand and are located in the cavities of the distorted hexagonal architecture defined by the metallic complexes (Figure S1 Magnetic properties. Magnetic measurements in DC mode were performed on all compounds (Figure 7). At room temperature, the χM.T values, approx. 6.5 cm 3 .K.mol  -1 for 1a, 1b and 3, are close to the expected one for two isolated Co(II) ions. Upon cooling, the χM.T products steadily decrease and reach 3.7 cm 3 .K.mol  -1 at 2K. In the 300-2 K temperature range, the experimental curves can be modelled using the theoretical behaviour of two isolated distorted octahedral cobalt(II) ions. 35 The best fit of the data obtained by full-matrix diagonalisation of the appropriate spin Hamiltonian gave λ = -128 cm -1 , D =-391 cm -1 and α = 0.99 for 1a ; λ=-179 cm -1 , D = -565 cm - 1 and α = 0.76 for 1b and λ = -168 cm -1 , D = -494 cm -1 and α = 0.79 for 3 where λ, D and α are the spin orbit coupling constant, the distortion parameter and reduction parameter of spin orbit coupling respectively. The theoretical curves reproduce quite well the experimental data over the whole range of temperature showing that there is no interaction between the cobalt ions within the helicates. For the bis-nickel helicate 2, the χM.T value remains constant in the 300-100 K temperature range and is equal to 2.2 cm 3 .K.mol  -1 which is close to the expected one for two magnetically isolated Ni(II) ions. Below 100 K, the value of χM.T slowly decreases to reach a value of 1.9 cm 3 .K.mol  -1 at 2K. This behaviour could be interpreted either by a very weak interaction between the two Ni(II) ions within the helicate or by the effect of the zero field splitting of the individual Ni(II) ions. The best data fits gave g = 2.1, J = -0.6 cm -1 for the first model and gNi = 2.1 and DNi = -6.5 cm -1 for the second one. The agreement factor R =  [(χMT)exp -(χMT)calcd]2/ [(χMT)exp] 2 is slightly better for the first model with values equal to R=10 -4 and 2. 10 -4 respectively. However, the lack of interaction in the cobalt-based helicates leads us to favour the second model.

The lack of interaction in these helicates contrasts with the results obtained for the cobalt and nickel mesocates -in which the (mpba) 4-ligand is deprotonated -where a weak but noticeable ferromagnetic interaction is observed with J = 1 cm -1 and J = 3.6 cm -1 respectively. 20,36 This difference in the magnetic behaviour is explained by the relative orientation of oxamate groups with the benzene rings. In the helicate family the oxamate groups are almost in the same plane as the benzene cycles. As a consequence the interaction between the magnetic orbitals of the cobalt or nickel ions and the π orbitals of the benzene rings is very weak and there is no efficient pathway with the bonds of the benzene rings. On the contrary, in the mesocates the oxamate groups are perpendicular to the benzene ring, this conformation allows a σ overlap between the eg magnetic orbitals and the lone pair of the deprotonated amide nitrogen atom which are in strong interaction with the extended π conjugated bond system of the phenylene spacers. 37 This situation leads to a quite large spin polarization effect through the π orbitals of the benzene rings and despite a metalmetal distance that goes up 6.85 Å -which is actually longer than in the case of the helicatesthere is a magnetic interaction in the mesocate family. The simple protonation or deprotonation of the mpba ligands completely change the magnetic properties of the dinuclear compounds from ferromagnetic coupling to an uncoupled system and this is a perfect illustration that the important parameter in the exchange interaction is not primarily the metal-metal distance but the existence of molecular orbitals able to transmit the interaction. Discussion. Unexpectedly, solvothermal synthesis using polyoxalamide ligands has led to new coordination compounds with ligands bearing double H-bonds sites. These complexes are built from metal ions and partially protonated ligands comprising oxamide anions. Oxamide anions are known to be flexible building blocks for hydrogen-bonded architectures. 38 Indeed, the compounds presented in this paper are interesting examples of networks built simultaneously from coordination complexes and complementary hydrogen bonding sites. [39][40][41] In inorganic crystal engineering this strategy is, in theory, very effective. The geometrical preferences of the coordination sphere of the metal ions (square planar, octahedral, etc…) induce a precise orientation of hydrogen-bonding groups that can also propagate geometrical information which would lead, in principle, to predictable results. In practice, the situation is less clear. For instance, with 2,2-bisimidazolato tris-chelate complex building blocks the directions of the H-bonds are coplanar and separated by an angle of 120° so the obvious construction is a honeycomb network and some results confirm this expectation. 42 However, there are several examples of more complicated structures which includes interpenetrating nets 42 , achiral 43 , racemate 44 and chiral 3D structures. 45 Even, 0D, 1D zig-zag and 1D helical structures were obtained. 43,46 Switching from one structure to another depends on other subtle intermolecular interactions, such as the role of the counter ions or the control of the chirality of the metal-ions coordination sphere. Therefore, whatever the type of connection used to form a network either by hydrogen bonds or coordination bonds, if tris-chelate complex building blocks appear as a reliable way to obtain self-organized 2D or 3D nets, the final dimensionality remains difficult to predict. This is especially true for 3D nets as they seem difficult to achieve although it is not completely unexpected since a basic unit bearing six connecting points is necessary to build a 3D net by translation. [47][48] Consequently, a tris-chelate building block with only three connecting points does not meet this requirement. Since the basic unit must be connected to six others, it is then compulsory to have at least four tris-chelate in the basic unit in order to build a 3 connected 3D net, i.e. a tetranuclear complex as building block (scheme 1). In the case of monomeric metal complexes, only an octahedral complex with six bismonodentate ligands like hexacyanometallates is able to build a 3D network. [49][50] For dinuclear building blocks it is necessary to have a connectivity of four at each node to obtain 3D networks and the helicates described in this paper meet this requirement. Actually, these new helicates complexes bearing six double H-bonds sites illustrate nicely the "recipe" A.F. Wells proposed in the fifties to build a 3D 4 connected net, [51][START_REF] Wells | Structural Inorganic Chemistry[END_REF] « The simplest planar 3-connected net is the hexagonal net with 2 points in the repeat unit. By connecting alternate points to points in layers above and below there arises the simplest three-dimensional 4connected net ». This description corresponds to the exact structure of compound 1b.

Furthermore, owing to the structure of the helicate, the resulting net in 1b was perfectly predictable. However, it should be noted that the structure of 3 with the entanglement of hydrogen bonds that results from a temperautue effect was not anticipated even though the same initial synthon is present. To summarize, the dinuclear triple-stranded helicates or mesocates are perfect building blocks to obtain 3D networks because they offer the six required connecting points. It is worth noting that the polynuclear complexes {Zn4O(RCO2)6} and {Cr3O(RCO2)6} which are used as SBU (Secondary Building Units) in 3D MOF 5 [START_REF] Li | [END_REF] and MIL 101 54 also possess 6 connecting points.

CONCLUSION

Solvothermal synthesis has successfully led to novel chiral helicate oxamate-based dinuclear complexes where supramolecular H-bond networking in the solid influenced by the synthesis temperature leads to 3D diamond like or 3D chiral structures. From these results obtained fortuitously, it is possible to conclude that, as in the synthesis of MOF, the use of polynuclear metal complexes as building blocks is a good strategy to build 3D networks in a predictable and controlled way. To the best of our knowledge, the strategic use of dicrete polymeatllic coordination compounds bearing hydrogen bond sites specifically designed to construct networks rationnaly has not been explored so far in crystal engineering and the present results open a synthetic route that is worth following. There had been no example of oxamate-based coordination compounds obtained solvothermally and this preliminary study is thus promising.

Indeed the use of solvothermal conditions combined to the rich family of ligands that oxamate derivatives constitute clearly opens a new synthetic route towards multidimensional molecular magnetic materials.

SYNOPSIS TOC

Solvothermal synthesis has been successfully tested as a new synthetic method towards oxamate-based coordination compounds. Novel bimetallic helicate complexes have been obtained with divalent metal ions and benzenedioxamic acid. Crystallographic studies have also shown a temperature-dependent H-bond networking that leads to hetero or homochiral 3D architectures.

  salts in DMF varying the temperature of the solvothermal treatment. Reactions carried at 100°C in the presence of TBAOH or NEt3 have yielded the bimetallic compounds TBA2[Co2(H2mpba)3].2DMF.5H2O (1a), (HNEt3)2[Co2(H2mpba)3].6DMF.5H2O (1b) and TBA2[Ni2(H2mpba)3].2DMF.2.5H2O (2) whereas the compound (HNEt3)2[Co2(H2mpba)3] (3) has been obtained at 150°C in the presence of triethylamine.

Figure 1 .

 1 Figure 1. a) representation of the 1,3-benzenedioxamic acid; b) structure of the anionic complex in 1b 1a, with legend as an inset, H atoms are omitted for clarity; c) Representation of the N(-H)•••O bond (orange dotted lines) between adjacent molecules in 1b; H atoms are omitted for clarity; the dinuclear complex is schemed with an intramolecular Co-Co bound pictured in purple.The formation of multiple hydrogen bonds between the amide (proton donor) and carbonyl (proton acceptor) functions of oxamic acids from adjacent complexes leads to a 3D framework composed of 2D hexagonal layers and pillared by the helicoidal complexes (Fgure 1c and 2). The

Figure 2 .

 2 Figure 2. Crystal packing of 1b showing the H-bond structuring (orange dotted lines) into a heterochiral hexagonal 3D network; H atoms are omitted for clarity; dinuclear complexes are schemed with an intramolecular Co-Co bound pictured in purple in the (bc) plane; benzyl rings are omitted for clarity in the representation of the (ab) plane.

Figure 3 .

 3 Figure 3. a) structure of the anionic complex [Co2(mpba)3] 8-20 b) Stick model of 1b along the Co-Co axis, c) Stick model of [Co2(mpba)3] 8-along the Co-Co axis.

Figure 4 .

 4 Figure 4. Staking of hexagonal networks in a) 1b and b) in [Li3Co2(mpba)3(H2O)6] 5-; benzyl rings are omitted for clarity.

  in the Supporting Information). The intermolecular N(-H)•••O distances found in the structure range from 2.8708(2) to 3.0886(3) Å with a mean value of 2.97 Å. One can suppose that performing the reaction at 150 instead of 100°C leads to a higher autogenous pressure during the solvothermal treatment which in turn constrains the system to stabilise into a denser phase and hence this more complex H-bond networking. The consequence of this densification is the lack of one of the two enantiomers of the bis-cobalt(II) helicate in the hexagonal lattice. Although distorted, the hexagonal lattice is still visible in Figure5with vertices alternatively defined by cobalt ocathedra and the entanglement of hydrogen bonds leading then to an enantiopure network." No signals were found in circular dichroism measurements performed on the bulk material indicating that both enantiomers (HNEt3)2[Δ-CoΔ-Co(Hmpba)3] and (HNEt3)2[ -Co -Co(Hmpba)3] crystallise out from the solvothermal reaction. Attempts to measure circular dichroism data on a single crystal diluted in KBr proved unsuccessful whereas measurement in solution were made impossible since compound 3 is insoluble in most common solvents, including DMSO or DMF.

Figure 5 .

 5 Figure 5. Crystal packing of 3 showing the H-bond structuring (orange dotted lines) into a homochiral 3D network; H atoms are omitted for clarity; dinuclear complexes are schemed with an intramolecular Co-Co bound pictured in purple in the (bc) plane; benzyl rings are omitted for clarity in the representation of the (ab) plane.

Figure 6 .

 6 Figure 6. Schematic view of the N(-H)###O bonds (orange dotted lines) between adjacent molecules in 3; H atoms are omitted for clarity; the dinuclear complex is schemed with an intramolecular Co-Co bound pictured in purple.

Figure 7 .

 7 Figure7. a) Plots of χMT vs. T measured from 300 to 2 K at 1 kOe for 1a (!), 1b ("), 2 (!) and 3 (") and respective fits in solid line: 1a-blue, 1b-green, 2-red, 3-orange; b) plots of M vs. H measured from 0 to 7 T at 2 K for 1a (!), 1b ("), 2 (!) and 3 (").

Scheme 1 .

 1 Scheme 1. Simplest building block providing the 6 necessary connecting points to build 3D nets: (a) mononuclear, (b) dinuclear, (c) and (d) tetranuclear.

  Co 10.98, C 47.02, H 4.69, N 10.44. Found: Co 10.66, C 47.56, H 4.64, N 10.66.

	1062 s; 1031 s; 988 s; 935 s; 894 s; 837 s; 802 m; 762 s; 739 s; 690 s; 660 s; 599 s; 540 s; 517 s;
	499 s; 483 s; 462 s, 446 s; 406 s; 360 s, 318 m; 274 m. Elemental analysis (%) calculated for
	C68H109N10Ni2O22.5 (Mr=1544 g mol -1 ): Ni 7.60, C 52.89, H 7.11, N 9.07. Found: Ni 8,10, C 50.74, H
	6.75, N 9.6.
	Synthesis of (HNEt3)2[Co2(H2mpba)3], 3 : CoCl2.6H2O (0.119 g, 0.5 mmol), H4mpba.3H20 (0.153 g,
	0.5 mmol) and NEt3 (99%, 0.280 mL, 2 mmol) were placed in a 23 mL Teflon-lined autoclave filled
	with DMF (9 mL) and heated at 150°C for 12 hours. After slow cooling to room temperature,
	purple crystals of 3 were collected, washed by sonication in DMF and dried in air. Yield: 0.135 g
	(50.3%, based on Co). IR (ATR, cm -1 ): 3228 s; 1631 S; 1598 S; 1540 m; 1475 m; 1456 m; 1361 S;
	1290 m; 1233 s; 1161 s; 1017 s; 979 s; 934 s; 885 s; 800 S; 707 s; 684 m; 630 s; 604 s; 545 m; 509
	m; 439 m; 407 s; 358 s; 296 m; 273 S; 243 S. Elemental analysis (%) calculated for C42H50Co2N8O18
	(Mr=1072.75 g mol -1 ):
	After

Table 1

 1 Crystallographic data for compounds 1b and 3

		1b	3
	Formula a	C51H91Co2N11O31	C42H50Co2N8O18
	FW [g mol -1 ]	1472.21	1072.75
	Crystal system	Trigonal	Monoclinic
	Space group	R-3c	P21
	a [Å]	17.527(7)	10.3375(6)
	b [Å]	17.527(7)	15.2574(15)
	c [Å]	46.452(7)	15.0870(15)
	α [°]	90	90
	β [°]	90	91.745(7)

ACKNOWLEDGMENT

This work was supported by the Ministère de la Recherche et de l'Enseignement Supérieur (MRES) and the Centre National de la Recherche Scientifique (CNRS).

SUPPORTING INFORMATION AVAILABLE

X-ray crystallographic files for 1b and 3 in CIF format; figures showing: (i) Experimental X-ray powder diffraction patterns for 1a, 1b and 2 and calculated patterns for 1a and 1b, (ii) intermolecular hydrogen bonds between the bimetallic complex and its HNEt3 + counter-ions in 3. This information is available free of charge via the Internet at http://pubs.acs.org.

Supplementary information for:

Solvothermal synthesis of oxamate-based helicate: temperature dependence of the hydrogen bond structuring in the solid.