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To date, the most widespread scenario is that the Earth originated without water and was brought to the planet mainly due to impacts by wet asteroids coming from further out in space. However, many uncertainties remain regarding the exact processes that supply water to inner terrestrial planets. This article explores a new mechanism that would allow water to be efficiently transported to planets without impacts. We propose that primordial asteroids were icy and that when the ice sublimated, it formed a gaseous disk that could then reach planets and deliver water. We have developed a new model that follows the sublimation of asteroids and evolves the subsequent gas disk using a viscous diffusion code. We can then quantify the amount of water that can be accreted onto each planet in a self-consistent manner. We find that this new disk-delivery mechanism can explain the water content on Earth as well as on other planets. Our model shows most of the water being delivered between 20 and 30 Myr after the birth of the Sun. Our scenario implies the presence of a gaseous water disk with substantial mass for 100s Myr, which could be one of the key tracers of this mechanism. We show that such a watery disk could be detected in young exo-asteroid belts with ALMA. We propose that viscous water transport is inevitable and more generic than the impact scenario. We also suggest it is a universal process that may also occur in extrasolar systems. The conditions required for this scenario to unfold are indeed expected to be present in most planetary systems: an opaque proto-planetary disk that is initially cold enough for ice to form in the exo-asteroid belt region, followed by a natural outward-moving snow line that allows this initial ice to sublimate after the dissipation of the primordial disk, creating a viscous secondary gas disk and leading to the accretion of water onto the exoplanets., Comment: 26 pages, 11 figures, accepted for publication in A&A. Abstract shortened

In this paper, we consider functionals of the form $H_\alpha(u)=F(u)+\alpha G(u)$ with $\alpha\in[0,+\infty)$, where $u$ varies in a set $U\neq\emptyset$ (without further structure). We first revisit a result stating that, excluding at most countably many values of $\alpha$, we have $\inf_{H_\alpha^\star}G= \sup_{H_\alpha^\star}G$, where $H_\alpha^\star := \arg\min_UH_\alpha$, which is assumed to be non-empty. Then, we prove a stronger result that concerns the invariance of the limiting value of the functional $G$ along minimizing sequences for $H_\alpha$, which extends the above Principle to the case $H_\alpha^\star= \emptyset$. Moreover, we show to what extent these findings generalize to multi-regularized functionals and -- in the presence of an underlying differentiable structure -- to critical points. Finally, the main result implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of $\alpha$, it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent., Comment: 20 pages, extension to multi-regularization and to critical points, differentiability of the value function

This letter proposes a new approach for Inertial Measurement Unit (IMU) preintegration, a fundamental building block that can be leveraged in different optimization-based Inertial Navigation System (INS) localization solutions. Inspired by recent advances in equivariant theory applied to biased INSs, we derive a discrete-time formulation of the IMU preintegration on ${\mathbf{Gal}(3) \ltimes \mathfrak{gal}(3)}$, the left-trivialization of the tangent group of the Galilean group $\mathbf{Gal}(3)$. We define a novel preintegration error that geometrically couples the navigation states and the bias leading to lower linearization error. Our method improves in consistency compared to existing preintegration approaches which treat IMU biases as a separate state-space. Extensive validation against state-of-the-art methods, both in simulation and with real-world IMU data, implementation in the Lie++ library, and open-source code are provided.

Consensus-based optimization (CBO) is a versatile multi-particle optimization method for performing nonconvex and nonsmooth global optimizations in high dimensions. Proofs of global convergence in probability have been achieved for a broad class of objective functions in unconstrained optimizations. In this work we adapt the algorithm for solving constrained optimizations on compact and unbounded domains with boundary by leveraging emerging reflective boundary conditions. In particular, we close a relevant gap in the literature by providing a global convergence proof for the many-particle regime comprehensive of convergence rates. On the one hand, for the sake of minimizing running cost, it is desirable to keep the number of particles small. On the other hand, reducing the number of particles implies a diminished capability of exploration of the algorithm. Hence numerical heuristics are needed to ensure convergence of CBO in the few-particle regime. In this work, we also significantly improve the convergence and complexity of CBO by utilizing an adaptive region control mechanism and by choosing geometry-specific random noise. In particular, by combining a hierarchical noise structure with a multigrid finite element method, we are able to compute global minimizers for a constrained $p$-Allen-Cahn problem with obstacles, a very challenging variational problem.

Surface porosity has been found to be an important property for small bodies. Some asteroids and comets can exhibit an extremely high surface porosity in the first millimeter layer. This layer may be produced by various processes and maintained by the lack of an atmosphere. However, the influence of porosity on the spectro-photometric properties of small body surfaces is not yet fully understood. In this study, we looked into the effect of porosity on the spectro-photometric properties of Phobos regolith spectroscopic simulants; created by mixing the simulants with ultra-pure water, producing ice-dust particles, and then sublimating the water. The reflectance spectroscopic properties in the visible and near-infrared (0.5-4.2 $\mu$m) show no strong variations between the porous and compact samples. However, one simulant exhibits a bluing of the slope after increasing porosity, providing possible insights into the differences between the blue and red units observed on Phobos. In the mid-infrared range, a contrast increase of the 10-$\mu$m emissivity plateau due to silicates is observed. Photometry reveals a modification in the phase reddening behavior between the compact powder and the sublimation residue for both simulants. However, the observed behavior is different between the simulants, suggesting that the phase reddening may be dependent on the composition of the simulants. The phase curve also appears to be modified by the addition of porosity, with a higher contribution of forward scattering observed for the sublimation residue. The derivation of the Hapke parameters indicates an increase in roughness for the porous sample, but no significant modification of the opposition effect. This study aims to provide new insights into the understanding of porosity by using two Phobos simulants in the context of the upcoming JAXA/Martian Moons eXploration mission., Comment: Submitted to Icarus

Respecting the geometry of the underlying system and exploiting its symmetry have been driving concepts in deriving modern geometric filters for inertial navigation systems (INSs). Despite their success, the explicit treatment of inertial measurement unit (IMU) biases remains challenging, unveiling a gap in the current theory of filter design. In response to this gap, this dissertation builds upon the recent theory of equivariant systems to address and overcome the limitations in existing methodologies. The goal is to identify new symmetries of inertial navigation systems that include a geometric treatment of IMU biases and exploit them to design filtering algorithms that outperform state-of-the-art solutions in terms of accuracy, convergence rate, robustness, and consistency. This dissertation leverages the semi-direct product rule and introduces the tangent group for inertial navigation systems as the first equivariant symmetry that properly accounts for IMU biases. Based on that, we show that it is possible to derive an equivariant filter (EqF) algorithm with autonomous navigation error dynamics. The resulting filter demonstrates superior to state-of-the-art solutions. Through a comprehensive analysis of various symmetries of inertial navigation systems, we formalized the concept that every filter can be derived as an EqF with a specific choice of symmetry. This underlines the fundamental role of symmetry in determining filter performance. This dissertation advances the understanding of equivariant symmetries in the context of inertial navigation systems and serves as a basis for the next generation of equivariant estimators, marking a significant leap toward more reliable navigation solutions., Comment: PhD thesis

This study aims to determine the size, albedo and rotational period of (98943) 2001 CC21, target of the Hayabusa2 extended mission, using thermal data from the Spitzer Space telescope and ground based observations. The Spitzer data were acquired with the Infrared Spectrograph in the 6-38 micron range, reduced using the Spitzer pipeline and modeled with the Near Earth Asteroid Thermal Modeling to determine the asteroid size and albedo. The absolute magnitude and the rotational period were determined thanks to new observations carried out at the 3.5m New Technology Telescope, at the 1.2m Observatoire de Haute Provence, and at the 0.7m Abastumani telescope. Three complete lightcurves were obtained in 2023-2024 at the last mentioned telescope. We determine an absolute magnitude of H=18.94$\pm$0.05, and a rotational period of 5.02124$\pm$0.00001 hours, with a large lightcurve amplitude of $\sim$ 0.8 mag. at a phase angle of 22$^o$, indicating a very elongated shape with estimated a/b semiaxis ratio $\geq$ 1.7, or a close-contact binary body. The emissivity of 2001 CC21 is consistent with that of silicates, and its albedo is 21.6$\pm$1.6 %. Finally, the spherical-equivalent diameter of 2001 CC21 is 465$\pm$15 m. The albedo value and emissivity here determined, coupled with results from polarimetry and spectroscopy from the literature, confirm that 2001 CC21 is an S-complex asteroid, and not a L-type, as previously suggested. The size of 2001 CC21 is less than 500 m, which is smaller than its first size estimation ($\sim$ 700 m). These results are relevant in preparation of the observing strategy of 2001 CC21 by Hayabusa2 extended mission., Comment: 7 pages, 9 figures, accepted for publication in A&A letter

Approximation of solutions to partial differential equations (PDE) is an important problem in computational science and engineering. Using neural networks as an ansatz for the solution has proven a challenge in terms of training time and approximation accuracy. In this contribution, we discuss how sampling the hidden weights and biases of the ansatz network from data-agnostic and data-dependent probability distributions allows us to progress on both challenges. In most examples, the random sampling schemes outperform iterative, gradient-based optimization of physics-informed neural networks regarding training time and accuracy by several orders of magnitude. For time-dependent PDE, we construct neural basis functions only in the spatial domain and then solve the associated ordinary differential equation with classical methods from scientific computing over a long time horizon. This alleviates one of the greatest challenges for neural PDE solvers because it does not require us to parameterize the solution in time. For second-order elliptic PDE in Barron spaces, we prove the existence of sampled networks with $L^2$ convergence to the solution. We demonstrate our approach on several time-dependent and static PDEs. We also illustrate how sampled networks can effectively solve inverse problems in this setting. Benefits compared to common numerical schemes include spectral convergence and mesh-free construction of basis functions., Comment: 16 pages, 15 figures

Previous observations of Phobos and Deimos, the moons of Mars, have improved our understanding of these small bodies. However, their formation and composition remain poorly constrained. Physical and spectral properties suggest that Phobos may be a weakly thermal-altered captured asteroid but the dynamical properties of the martian system suggest a formation by giant collision similar to the Earth moon. In 2027, the JAXA's MMX mission aims to address these outstanding questions. We undertook measurements with a new simulant called OPPS (Observatory of Paris Phobos Simulant) which closely matches Phobos spectra in the visible to the mid-infrared range. The simulant was synthesized using a mixture of olivine, saponite, anthracite, and coal. Since observation geometry is a crucial aspect of planetary surface remote sensing exploration, we evaluated the parameters obtained by modeling the phase curves -- obtained through laboratory measurements -- of two different Phobos simulants (UTPS-TB and OPPS) using Hapke IMSA model. Our results show that the photometric properties of Phobos simulants are not fully consistent with those of Tagish Lake, Allende, or the NWA 4766 shergottite. We also investigated the detection of volatiles/organic compounds and hydrated minerals, as the presence of such components is expected on Phobos in the hypothesis of a captured primitive asteroid. The results indicate that a significant amount of organic compounds is required for the detection of C-H bands at 3.4 $\mu$m. In contrast, the 2.7 $\mu$m absorption band, due to hydrated minerals, is much deeper and easier to detect than C-H organic features at the same concentration levels. Posing limits on detectability of some possible key components of Phobos surface will be pivotal to prepare and interpret future observations of the MIRS spectrometer onboard MMX mission., Comment: 58 pages, 19 figures, submitted to Icarus

This study aims to analyze Phobos' photometric properties using Mars Express mission observations to support the Martian Moons eXploration mission. We analyzed resolved images of Phobos acquired between 2004 and 2022 by the HRSC and the SRC cameras on board the Mars Express spacecraft. We performed photometric analysis using the Hapke model for both integrated and disk-resolved data. The Phobos phase function has a strong opposition effect due to shadow hiding, with an amplitude and a half-width of the opposition surge of 2.28$\pm$0.03 and 0.0573$\pm$0.0001, respectively. Overall, the surface of Phobos is dark, with a geometric albedo of 6.8 % in the green filter and backscattering. We also found a surface porosity of 87\%, indicating the presence of a thick dust mantle or of fractal aggregates on the top surface. The SSA maps revealed high reflectance variability, with the blue unit area in the northeast Stickney rim being up to 65\% brighter than average, while the Stickney floor is among the darkest regions, with reflectance 10 to 20% lower than average. Photometric modeling of the regions of interest selected in the red and blue units indicates that red unit terrains have a stronger opposition effect and a smaller SSA value than the blue ones, but they have similar porosity and backscattering properties. The HRSC data provide a unique investigation of the Phobos phase function and opposition surge, which is valuable information for the MMX observational planning. The Phobos opposition surge, surface porosity, phase integral, and spectral slope are very similar to the values observed for the comet 67P and for Jupiter family comets in general. Based on these similarities, we formulate a hypothesis that the Mars satellites might be the results of a binary or bilobated comet captured by Mars., Comment: 30 pages, 16 figures, accepted for publication in Astron. & Astroph

Consensus-based optimization (CBO) is an agent-based derivative-free method for non-smooth global optimization that has been introduced in 2017, leveraging a surprising interplay between stochastic exploration and Laplace principle. In addition to its versatility and effectiveness in handling high-dimensional, non-convex, and non-smooth optimization problems, this approach lends itself well to theoretical analysis. Indeed, its dynamics is governed by a degenerate nonlinear Fokker--Planck equation, whose large time behavior explains the convergence of the method. Recent results provide guarantees of convergence under the restrictive assumption of a unique global minimizer for the objective function. In this work, we propose a novel and simple variation of CBO to tackle non-convex optimization problems with multiple global minimizers. Despite the simplicity of this new model, its analysis is particularly challenging because of its nonlinearity and nonlocal nature. We prove the existence of solutions of the corresponding nonlinear Fokker--Planck equation and we show exponential concentration in time to the set of minimizers made of multiple smooth, convex, and compact components. Our proofs require combining several ingredients, such as delicate geometrical arguments, new variants of a quantitative Laplace principle, ad hoc regularizations and approximations, and regularity theory for parabolic equations. Ultimately, this result suggests that the corresponding CBO algorithm, formulated as an Euler-Maruyama discretization of the underlying empirical stochastic process, tends to converge to multiple global minimizers., Comment: 46 pages

This letter re-visits the problem of visual-inertial navigation system (VINS) and presents a novel filter design we dub the multi state constraint equivariant filter (MSCEqF, in analogy to the well known MSCKF). We define a symmetry group and corresponding group action that allow specifically the design of an equivariant filter for the problem of visual-inertial odometry (VIO) including IMU bias, and camera intrinsic and extrinsic calibration states. In contrast to state-of-the-art invariant extended Kalman filter (IEKF) approaches that simply tack IMU bias and other states onto the $\mathbf{SE}_2(3)$ group, our filter builds upon a symmetry that properly includes all the states in the group structure. Thus, we achieve improved behavior, particularly when linearization points largely deviate from the truth (i.e., on transients upon state disturbances). Our approach is inherently consistent even during convergence phases from significant errors without the need for error uncertainty adaptation, observability constraint, or other consistency enforcing techniques. This leads to greatly improved estimator behavior for significant error and unexpected state changes during, e.g., long-duration missions. We evaluate our approach with a multitude of different experiments using three different prominent real-world datasets., Comment: Accepted for publication in the IEEE Robotics and Automation Letters (RA-L), 2023

The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. We delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches: 1. Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials. 2. Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces. 3. Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation. We furnish explicit and quantitative bounds on generalization errors for each of these solutions. We leverage the theory of metric Sobolev spaces and we combine it with techniques of optimal transport, variational calculus, and large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. These networks undergo training using diverse methodologies. This approach allows us to obtain approximating functions that can be rapidly evaluated after training. Our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude. This allows evaluations over large datasets several times faster, including training, than traditional optimal transport algorithms. Our analytically designed deep learning architecture slightly outperforms the test error of state-of-the-art CNN architectures on datasets of images., Comment: Revised version

Consensus-based optimization (CBO) is a versatile multi-particle metaheuristic optimization method suitable for performing nonconvex and nonsmooth global optimizations in high dimensions. It has proven effective in various applications while at the same time being amenable to a theoretical convergence analysis. In this paper, we explore a variant of CBO, which incorporates truncated noise in order to enhance the well-behavedness of the statistics of the law of the dynamics. By introducing this additional truncation in the noise term of the CBO dynamics, we achieve that, in contrast to the original version, higher moments of the law of the particle system can be effectively bounded. As a result, our proposed variant exhibits enhanced convergence performance, allowing in particular for wider flexibility in choosing the noise parameter of the method as we confirm experimentally. By analyzing the time-evolution of the Wasserstein-$2$ distance between the empirical measure of the interacting particle system and the global minimizer of the objective function, we rigorously prove convergence in expectation of the proposed CBO variant requiring only minimal assumptions on the objective function and on the initialization. Numerical evidences demonstrate the benefit of truncating the noise in CBO., Comment: 24pages, accepted by European Journal of Applied Mathematics

In this work, we explore the recent advances in equivariant filtering for inertial navigation systems to improve state estimation for uncrewed aerial vehicles (UAVs). Traditional state-of-the-art estimation methods, e.g., the multiplicative Kalman filter (MEKF), have some limitations concerning their consistency, errors in the initial state estimate, and convergence performance. Symmetry-based methods, such as the equivariant filter (EqF), offer significant advantages for these points by exploiting the mathematical properties of the system - its symmetry. These filters yield faster convergence rates and robustness to wrong initial state estimates through their error definition. To demonstrate the usability of EqFs, we focus on the sensor-fusion problem with the most common sensors in outdoor robotics: global navigation satellite system (GNSS) sensors and an inertial measurement unit (IMU). We provide an implementation of such an EqF leveraging the semi-direct product of the symmetry group to derive the filter equations. To validate the practical usability of EqFs in real-world scenarios, we evaluate our method using data from all outdoor runs of the INSANE Dataset. Our results demonstrate the performance improvements of the EqF in real-world environments, highlighting its potential for enhancing state estimation for UAVs., Comment: Pre-print version, accepted to IEEE ICAR 2023

This paper investigates the problem of inertial navigation system (INS) filter design through the lens of symmetry. The extended Kalman filter (EKF) and its variants, have been the staple of INS filtering for 50 years; however, recent advances in inertial navigation systems have exploited matrix Lie group structure to design stochastic filters and state observers that have been shown to display superior performance compared to classical solutions. In this work we consider the case where a vehicle has an inertial measurement unit (IMU) and a global navigation satellite system (GNSS) receiver. We show that all the modern variants of the EKF for these sensors can be interpreted as the recently proposed Equivariant Filter (EqF) design methodology applied to different choices of symmetry group for the INS problem. This leads us to propose two new symmetries for the INS problem that have not been considered in the prior literature, and provide a discussion of the relative strengths and weaknesses of all the different algorithms. We believe the collection of symmetries that we present here capture all the sensible choices of symmetry for this problem and sensor suite, and that the analysis provided is indicative of the relative real-world performance potential of the different algorithms., Comment: Submitted to Automatica

This paper introduces UVIO, a multi-sensor framework that leverages Ultra Wide Band (UWB) technology and Visual-Inertial Odometry (VIO) to provide robust and low-drift localization. In order to include range measurements in state estimation, the position of the UWB anchors must be known. This study proposes a multi-step initialization procedure to map multiple unknown anchors by an Unmanned Aerial Vehicle (UAV), in a fully autonomous fashion. To address the limitations of initializing UWB anchors via a random trajectory, this paper uses the Geometric Dilution of Precision (GDOP) as a measure of optimality in anchor position estimation, to compute a set of optimal waypoints and synthesize a trajectory that minimizes the mapping uncertainty. After the initialization is complete, the range measurements from multiple anchors, including measurement biases, are tightly integrated into the VIO system. While in range of the initialized anchors, the VIO drift in position and heading is eliminated. The effectiveness of UVIO and our initialization procedure has been validated through a series of simulations and real-world experiments.

The connection between Residual Neural Networks (ResNets) and continuous-time control systems (known as NeurODEs) has led to a mathematical analysis of neural networks which has provided interesting results of both theoretical and practical significance. However, by construction, NeurODEs have been limited to describing constant-width layers, making them unsuitable for modeling deep learning architectures with layers of variable width. In this paper, we propose a continuous-time Autoencoder, which we call AutoencODE, based on a modification of the controlled field that drives the dynamics. This adaptation enables the extension of the mean-field control framework originally devised for conventional NeurODEs. In this setting, we tackle the case of low Tikhonov regularization, resulting in potentially non-convex cost landscapes. While the global results obtained for high Tikhonov regularization may not hold globally, we show that many of them can be recovered in regions where the loss function is locally convex. Inspired by our theoretical findings, we develop a training method tailored to this specific type of Autoencoders with residual connections, and we validate our approach through numerical experiments conducted on various examples., Comment: 35 pages, 11 figures. Minor adjustments and new bibliographical references

In this paper we provide a novel analytical perspective on the theoretical understanding of gradient-based learning algorithms by interpreting consensus-based optimization (CBO), a recently proposed multi-particle derivative-free optimization method, as a stochastic relaxation of gradient descent. Remarkably, we observe that through communication of the particles, CBO exhibits a stochastic gradient descent (SGD)-like behavior despite solely relying on evaluations of the objective function. The fundamental value of such link between CBO and SGD lies in the fact that CBO is provably globally convergent to global minimizers for ample classes of nonsmooth and nonconvex objective functions, hence, on the one side, offering a novel explanation for the success of stochastic relaxations of gradient descent. On the other side, contrary to the conventional wisdom for which zero-order methods ought to be inefficient or not to possess generalization abilities, our results unveil an intrinsic gradient descent nature of such heuristics. This viewpoint furthermore complements previous insights into the working principles of CBO, which describe the dynamics in the mean-field limit through a nonlinear nonlocal partial differential equation that allows to alleviate complexities of the nonconvex function landscape. Our proofs leverage a completely nonsmooth analysis, which combines a novel quantitative version of the Laplace principle (log-sum-exp trick) and the minimizing movement scheme (proximal iteration). In doing so, we furnish useful and precise insights that explain how stochastic perturbations of gradient descent overcome energy barriers and reach deep levels of nonconvex functions. Instructive numerical illustrations support the provided theoretical insights., Comment: 38 pages, 4 figures

We present the most extensive catalog of exposures of volatiles on the 67P/Churyumov-Gerasimenko nucleus generated from observations acquired with the OSIRIS cameras on board the Rosetta mission. We identified more than 600 volatile exposures on the comet. Bright spots are found isolated on the nucleus or grouped in clusters, usually at the bottom of cliffs, and most of them are small, typically a few square meters or smaller. Several of them are clearly correlated with the cometary activity. We note a number of peculiar exposures of volatiles with negative spectral slope values in the high-resolution post-perihelion images, which we interpret as the presence of large ice grains ($>$ 1000 $\mu$m) or local frosts condensation. We observe a clear difference both in the spectral slope and in the area distributions of the bright spots pre- and post-perihelion, with these last having lower average spectral slope values and a smaller size, with a median surface of 0.7 m$^2$, even if the size difference is mainly due to the higher resolution achieved post-perihelion. The minimum duration of the bright spots shows three clusters: an area-independent cluster dominated by short-lifetime frosts; an area-independent cluster with lifetime of 0.5--2 days, probably associated with the seasonal fallout of dehydrated chunks; and an area-dependent cluster with lifetime longer than 2 days consistent with water-driven erosion of the nucleus. Even if numerous bright spots are detected, the total surface of exposed water ice is less than 0.1% of the total 67P nucleus surface, confirming that the 67P surface is dominated by refractory dark terrains, while exposed ice occupies only a tiny fraction. Moreover, the abundance of volatile exposures is six times less in the small lobe than in the big lobe, adding additional evidence to the hypothesis that comet 67P is composed of two distinct bodies., Comment: 24 pages, 19 Figures; paper accepted for publication in Astron. and Astrophysics on February 2023

We present new results of the observing program which is a part of the NEOROCKS project aimed to improve knowledge on physical properties of near-Earth Objects (NEOs) for planetary defense. Photometric observations were performed using the 1.2m telescope at the Haute-Provence observatory (France) in the BVRI filters of the Johnson-Cousins photometric systems between June 2021 and April 2022. We obtained new surface colors for 42 NEOs. Based on the measured colors we classified 20 objects as S-complex, 9 as C-complex, 9 as X-complex, 2 as D-type, one object as V-type, and one object remained unclassified. For all the observed objects we estimated their absolute magnitudes and diameters. Combining these new observations with the previously acquired data within the NEOROCKS project extended our dataset to 93 objects. The majority of objects in the dataset with diameters D<500m belongs to a group of silicate bodies, which could be related to observational bias. Based on MOID and $\Delta$V values we selected 14 objects that could be accessible by a spacecraft. Notably, we find D-type asteroid (163014) 2001 UA5 and A-type asteroid 2017 SE19 to be of particular interest as possible space mission targets.

Artificial neural networks are functions depending on a finite number of parameters typically encoded as weights and biases. The identification of the parameters of the network from finite samples of input-output pairs is often referred to as the \emph{teacher-student model}, and this model has represented a popular framework for understanding training and generalization. Even if the problem is NP-complete in the worst case, a rapidly growing literature -- after adding suitable distributional assumptions -- has established finite sample identification of two-layer networks with a number of neurons $m=\mathcal O(D)$, $D$ being the input dimension. For the range $D

For real-world applications, autonomous mobile robotic platforms must be capable of navigating safely in a multitude of different and dynamic environments with accurate and robust localization being a key prerequisite. To support further research in this domain, we present the INSANE data sets - a collection of versatile Micro Aerial Vehicle (MAV) data sets for cross-environment localization. The data sets provide various scenarios with multiple stages of difficulty for localization methods. These scenarios range from trajectories in the controlled environment of an indoor motion capture facility, to experiments where the vehicle performs an outdoor maneuver and transitions into a building, requiring changes of sensor modalities, up to purely outdoor flight maneuvers in a challenging Mars analog environment to simulate scenarios which current and future Mars helicopters would need to perform. The presented work aims to provide data that reflects real-world scenarios and sensor effects. The extensive sensor suite includes various sensor categories, including multiple Inertial Measurement Units (IMUs) and cameras. Sensor data is made available as raw measurements and each data set provides highly accurate ground truth, including the outdoor experiments where a dual Real-Time Kinematic (RTK) Global Navigation Satellite System (GNSS) setup provides sub-degree and centimeter accuracy (1-sigma). The sensor suite also includes a dedicated high-rate IMU to capture all the vibration dynamics of the vehicle during flight to support research on novel machine learning-based sensor signal enhancement methods for improved localization. The data sets and post-processing tools are available at: https://sst.aau.at/cns/datasets

Between 2014 December 31 and 2015 March 17, the OSIRIS cameras on Rosetta documented the growth of a 140m wide and 0.5m deep depression in the Hapi region on Comet 67P/Churyumov-Gerasimenko. This shallow pit is one of several that later formed elsewhere on the comet, all in smooth terrain that primarily is the result of airfall of coma particles. We have compiled observations of this region in Hapi by the microwave instrument MIRO on Rosetta, acquired during October and November 2014. We use thermophysical and radiative transfer models in order to reproduce the MIRO observations. This allows us to place constraints on the thermal inertia, diffusivity, chemical composition, stratification, extinction coefficients, and scattering properties of the surface material, and how they evolved during the months prior to pit formation. The results are placed in context through long-term comet nucleus evolution modelling. We propose that: 1) MIRO observes signatures that are consistent with a solid-state greenhouse effect in airfall material; 2) CO2 ice is sufficiently close to the surface to have a measurable effect on MIRO antenna temperatures, and likely is responsible for the pit formation in Hapi observed by OSIRIS; 3) the pressure at the CO2 sublimation front is sufficiently strong to expel dust and water ice outwards, and to compress comet material inwards, thereby causing the near-surface compaction observed by CONSERT, SESAME, and groundbased radar, manifested as the "consolidated terrain" texture observed by OSIRIS., Comment: 32 pages, 41 figures. This is a pre-copyedited, author-produced PDF of an article accepted for publication in MNRAS following peer review

Thanks to Rosetta orbiter's and Philae lander's data our knowledge of cometary nuclei composition has experienced a great advancement. The properties of 67P/CG nucleus are discussed and compared with other comets explored in the past by space missions. Cometary nuclei are made by a collection of ices, minerals, organic matter, and salts resulting in very dark and red-colored surfaces. When far from the Sun, exposed water and carbon dioxide ices are found only in few locations of 67P/CG where the exposure of pristine subsurface layers or the recondensation of volatile species driven by the solar heating and local terrain morphology can sustain their temporary presence on the surface. The nucleus surface appears covered by a dust layer of variable thickness. Dust grains appear mostly dehydrated and are made by an assemblage of minerals, organic matter, and salts. Spectral analysis shows that the mineral phase is dominated by silicates, fine-grained opaques and ammoniated salts. Aliphatic and aromatic groups, with the presence of the strong hydroxyl group, are identified within the organic matter. The surface composition of cometary nuclei evolves with heliocentric distance and seasonal cycling: approaching perihelion the increase of the solar flux boosts the activity through the sublimation of volatiles which in turn causes the erosion of surface layers, the exposure of ices, the activity in cliffs and pits, the collapse of overhangs and walls, and the mobilization and redistribution of dust. The evolution of color, composition, and texture changes occurring across different morphological regions of the nucleus are correlated with these processes. In this chapter we discuss 67P/CG nucleus composition and evolutionary processes as observed by Rosetta mission in the context of other comets previously explored by space missions or observed from Earth., Comment: 46 page, 34 figures, 4 tables, Chapter in press for the book Comets III, edited by K. Meech and M. Combi, University of Arizona Press

Stochastic filters for on-line state estimation are a core technology for autonomous systems. The performance of such filters is one of the key limiting factors to a system's capability. Both asymptotic behavior (e.g.,~for regular operation) and transient response (e.g.,~for fast initialization and reset) of such filters are of crucial importance in guaranteeing robust operation of autonomous systems. This paper introduces a new generic formulation for a gyroscope aided attitude estimator using N direction measurements including both body-frame and reference-frame direction type measurements. The approach is based on an integrated state formulation that incorporates navigation, extrinsic calibration for all direction sensors, and gyroscope bias states in a single equivariant geometric structure. This newly proposed symmetry allows modular addition of different direction measurements and their extrinsic calibration while maintaining the ability to include bias states in the same symmetry. The subsequently proposed filter-based estimator using this symmetry noticeably improves the transient response, and the asymptotic bias and extrinsic calibration estimation compared to state-of-the-art approaches. The estimator is verified in statistically representative simulations and is tested in real-world experiments., Comment: to be published in Robotics and Automation Letters

We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric-Sobolev space $H^{1,p}(X,\mathsf{d},\mathfrak{m})$ associated with a positive and finite Borel measure $\mathfrak{m}$ in a separable and complete metric space $(X,\mathsf{d})$. We then provide a relevant application to the case of the algebra of cylinder functions in the Wasserstein Sobolev space $H^{1,2}(\mathcal{P}_2(\mathbb{M}),W_{2},\mathfrak{m})$ arising from a positive and finite Borel measure $\mathfrak{m}$ on the Kantorovich-Rubinstein-Wasserstein space $(\mathcal{P}_2(\mathbb{M}),W_{2})$ of probability measures in a finite dimensional Euclidean space, a complete Riemannian manifold, or a separable Hilbert space $\mathbb{M}$. We will show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure $\mathfrak{m}$ so that the resulting Cheeger energy is a Dirichlet form. We will eventually provide an explicit characterization for the corresponding notion of $\mathfrak{m}$-Wasserstein gradient, showing useful calculus rules and its consistency with the tangent bundle and the $\Gamma$-calculus inherited from the Dirichlet form., Comment: 53 pages - final version

We probed the surface environment of large ($>$80 km in diameter) T-type asteroids, a taxonomic type relatively ill-constrained as an independent group, and discussed their place of origin. We performed spectroscopic observations of two T-type asteroids, (96) Aegle and (570) Kythera, over 2.8--4.0 $\mu$m using the Subaru telescope. With other T-types' spectra available in the literature and survey datasets, we strove to find commonalities and global trends in this group. We also utilised the asteroids' polarimetric data and meteorite spectra to constrain their surface texture and composition. Our targets exhibit red $L$-band continuum slopes similar to (1) Ceres and 67P/Churyumov-Gerasimenko, and have an OH-absorption feature with band centres $<$2.8 $\mu$m. (96) Aegle hints at a shallow N--H band near 3.1 $\mu$m and C--H band of organic materials over 3.4--3.6 $\mu$m, whereas no diagnostic bands of water ice and other volatiles exceeding the noise of the data were seen for both asteroids. The large T-type asteroids but (596) Scheila display similar spectral shapes to our targets. $\sim$50 \% of large T-types contain an absorption band near 0.6--0.65 $\mu$m likely associated with hydrated minerals. For T-type asteroids (except Jupiter Trojans) of all sizes, we found a weak correlation: the smaller the diameter and the closer the Sun, the redder the visible slope. The 2.9-$\mu$m band depths of large T-types suggest that they might have experienced aqueous alteration comparable to Ch-types but more intense than most of the main-belt asteroids. The polarimetric phase curve of the T-types is well described by a particular surface structure and their 0.5--4.0 $\mu$m reflectance spectra appear most similar to CI chondrites with grain sizes of $\sim$25--35 $\mu$m. Taken as a whole, we propose that large T-type asteroids might be dislodged roughly around 10 au in the early solar system., Comment: Accepted for publication in Astronomy & Astrophysics

Inertial Navigation Systems (INS) are a key technology for autonomous vehicles applications. Recent advances in estimation and filter design for the INS problem have exploited geometry and symmetry to overcome limitations of the classical Extended Kalman Filter (EKF) approach that formed the mainstay of INS systems since the mid-twentieth century. The industry standard INS filter, the Multiplicative Extended Kalman Filter (MEKF), uses a geometric construction for attitude estimation coupled with classical Euclidean construction for position, velocity and bias estimation. The recent Invariant Extended Kalman Filter (IEKF) provides a geometric framework for the full navigation states, integrating attitude, position and velocity, but still uses the classical Euclidean construction to model the bias states. In this paper, we use the recently proposed Equivariant Filter (EqF) framework to derive a novel observer for biased inertial-based navigation in a fully geometric framework. The introduction of virtual velocity inputs with associated virtual bias leads to a full equivariant symmetry on the augmented system. The resulting filter performance is evaluated with both simulated and real-world data, and demonstrates increased robustness to a wide range of erroneous initial conditions, and improved accuracy when compared with the industry standard Multiplicative EKF (MEKF) approach.

In this paper we study anisotropic consensus-based optimization (CBO), a multi-agent metaheuristic derivative-free optimization method capable of globally minimizing nonconvex and nonsmooth functions in high dimensions. CBO is based on stochastic swarm intelligence, and inspired by consensus dynamics and opinion formation. Compared to other metaheuristic algorithms like particle swarm optimization, CBO is of a simpler nature and therefore more amenable to theoretical analysis. By adapting a recently established proof technique, we show that anisotropic CBO converges globally with a dimension-independent rate for a rich class of objective functions under minimal assumptions on the initialization of the method. Moreover, the proof technique reveals that CBO performs a convexification of the optimization problem as the number of agents goes to infinity, thus providing an insight into the internal CBO mechanisms responsible for the success of the method. To motivate anisotropic CBO from a practical perspective, we further test the method on a complicated high-dimensional benchmark problem, which is well understood in the machine learning literature., Comment: 18 pages, 4 figures. arXiv admin note: text overlap with arXiv:2103.15130

In this paper we consider a measure-theoretical formulation of the training of NeurODEs in the form of a mean-field optimal control with $L^2$-regularization of the control. We derive first order optimality conditions for the NeurODE training problem in the form of a mean-field maximum principle, and show that it admits a unique control solution, which is Lipschitz continuous in time. As a consequence of this uniqueness property, the mean-field maximum principle also provides a strong quantitative generalization error for finite sample approximations. Our derivation of the mean-field maximum principle is much simpler than the ones currently available in the literature for mean-field optimal control problems, and is based on a generalized Lagrange multiplier theorem on convex sets of spaces of measures. The latter is also new, and can be considered as a result of independent interest.