Search

We perform uncertainty propagation on an event manifold for Guidance & Control Networks (G&CNETs), aiming to enhance the certification tools for neural networks in this field. This work utilizes three previously solved optimal control problems with varying levels of dynamics nonlinearity and event manifold complexity. The G&CNETs are trained to represent the optimal control policies of a time-optimal interplanetary transfer, a mass-optimal landing on an asteroid and energy-optimal drone racing, respectively. For each of these problems, we describe analytically the terminal conditions on an event manifold with respect to initial state uncertainties. Crucially, this expansion does not depend on time but solely on the initial conditions of the system, thereby making it possible to study the robustness of the G&CNET at any specific stage of a mission defined by the event manifold. Once this analytical expression is found, we provide confidence bounds by applying the Cauchy-Hadamard theorem and perform uncertainty propagation using moment generating functions. While Monte Carlo-based (MC) methods can yield the results we present, this work is driven by the recognition that MC simulations alone may be insufficient for future certification of neural networks in guidance and control applications.

In the field of spaceflight mechanics and astrodynamics, determining eclipse regions is a frequent and critical challenge. This determination impacts various factors, including the acceleration induced by solar radiation pressure, the spacecraft power input, and its thermal state all of which must be accounted for in various phases of the mission design. This study leverages recent advances in neural image processing to develop fully differentiable models of eclipse regions for highly irregular celestial bodies. By utilizing test cases involving Solar System bodies previously visited by spacecraft, such as 433 Eros, 25143 Itokawa, 67P/Churyumov--Gerasimenko, and 101955 Bennu, we propose and study an implicit neural architecture defining the shape of the eclipse cone based on the Sun's direction. Employing periodic activation functions, we achieve high precision in modeling eclipse conditions. Furthermore, we discuss the potential applications of these differentiable models in spaceflight mechanics computations.

This paper presents a novel approach for propagating uncertainties in dynamical systems building on high-order Taylor expansions of the flow and moment-generating functions (MGFs). Unlike prior methods that focus on Gaussian distributions, our approach leverages the relationship between MGFs and distribution moments to extend high-order uncertainty propagation techniques to non-Gaussian scenarios. This significantly broadens the applicability of these methods to a wider range of problems and uncertainty types. High-order moment computations are performed one-off and symbolically, reducing the computational burden of the technique to the calculation of Taylor series coefficients around a nominal trajectory, achieved by efficiently integrating the system's variational equations. Furthermore, the use of the proposed approach in combination with event transition tensors, allows for accurate propagation of uncertainties at specific events, such as the landing surface of a celestial body, the crossing of a predefined Poincar\'e section, or the trigger of an arbitrary event during the propagation. Via numerical simulations we demonstrate the effectiveness of our method in various astrodynamics applications, including the unperturbed and perturbed two-body problem, and the circular restricted three-body problem, showing that it accurately propagates non-Gaussian uncertainties both at future times and at event manifolds.

We introduce a novel neural architecture termed thermoNET, designed to represent thermospheric density in satellite orbital propagation using a reduced amount of differentiable computations. Due to the appearance of a neural network on the right-hand side of the equations of motion, the resulting satellite dynamics is governed by a NeuralODE, a neural Ordinary Differential Equation, characterized by its fully differentiable nature, allowing the derivation of variational equations (hence of the state transition matrix) and facilitating its use in connection to advanced numerical techniques such as Taylor-based numerical propagation and differential algebraic techniques. Efficient training of the network parameters occurs through two distinct approaches. In the first approach, the network undergoes training independently of spacecraft dynamics, engaging in a pure regression task against ground truth models, including JB-08 and NRLMSISE-00. In the second paradigm, network parameters are learned based on observed dynamics, adapting through ODE sensitivities. In both cases, the outcome is a flexible, compact model of the thermosphere density greatly enhancing numerical propagation efficiency while maintaining accuracy in the orbital predictions., Comment: Paper presented and published in the 29th ISSFD Conference, Darmstadt, Germany

Low-thrust trajectories play a crucial role in optimizing scientific output and cost efficiency in asteroid belt missions. Unlike high-thrust transfers, low-thrust trajectories require solving complex optimal control problems. This complexity grows exponentially with the number of asteroids visited due to orbital mechanics intricacies. In the literature, methods for approximating low-thrust transfers without full optimization have been proposed, including analytical and machine learning techniques. In this work, we propose new analytical approximations and compare their accuracy and performance to machine learning methods. While analytical approximations leverage orbit theory to estimate trajectory costs, machine learning employs a more black-box approach, utilizing neural networks to predict optimal transfers based on various attributes. We build a dataset of about 3 million transfers, found by solving the time and fuel optimal control problems, for different time of flights, which we also release open-source. Comparison between the two methods on this database reveals the superiority of machine learning, especially for longer transfers. Despite challenges such as multi revolution transfers, both approaches maintain accuracy within a few percent in the final mass errors, on a database of trajectories involving numerous asteroids. This work contributes to the efficient exploration of mission opportunities in the asteroid belt, providing insights into the strengths and limitations of different approximation strategies., Comment: Paper presented and published in the proceedings of the 29th ISSFD Conference, Darmstadt, Germany, 2024

This study addresses optimal impulsive trajectory design within the Circular Restricted Three-Body Problem (CR3BP), presenting a global optimization-based approach to identify minimum $\Delta V$ transfers between periodic orbits, including heteroclinic connections. By combining a Monotonic Basin Hopping (MBH) algorithm with a sequential quadratic solver in a parallel optimization framework, a wide range of minimum $\Delta V$ transfers are efficiently found. To validate this approach, known connections from the literature are reproduced. Consequently, three-dimensional periodic orbits are explored and a systematic search for minimum propellant trajectories is conducted within a selected interval of Jacobi constants and a maximum time of flight. Analysis of the results reveals the presence of very low $\Delta V$ solutions and showcases the algorithm's effectiveness across various mission scenarios., Comment: Paper presented and published in the proceedings of the 29th ISSFD conference (2024, Darmstadt, Germany)

Numerical continuation techniques are powerful tools that have been extensively used to identify particular solutions of nonlinear dynamical systems and enable trajectory design in chaotic astrodynamics problems such as the Circular Restricted Three-Body Problem. However, the applicability of equilibrium points and periodic orbits may be questionable in real-world applications where the uncertainties of the initial conditions of the spacecraft and dynamical parameters of the problem (e.g., mass ratio parameter) are taken into consideration. Due to uncertain parameters and initial conditions, the spacecraft might not follow the reference periodic orbit owing to growing uncertainties that cause the satellite to deviate from its nominal path. Hence, it is crucial to keep track of the probability of finding the spacecraft in a given region. Building on previous work, we extend numerical continuation to moments of the distribution (i.e., stochastic continuation) by directly continuing moments of the probability density function of the spacecraft state. Only assuming normality of the initial conditions, and leveraging moment-generating functions, Isserlis' theorem, and the algebra of truncated polynomials, we propagate the distribution of the spacecraft state at consecutive surface of section crossings while retaining a symbolic map of the final moments of the distribution that depend on the initial mean and covariance matrix only. The goal of the work is to offer a differential algebra-based general framework to continue 3D periodic orbits in the presence of uncertain dynamical systems. The proposed approach is compared against traditional Monte Carlo simulations to validate the uncertainty propagation approach and demonstrate the advantages of the proposed in terms of uncertainty propagation computational burden and access to higher-dimensional problems., Comment: Paper presented and published in the proceedings of the 2024 ISSFD Conference. (Darmstadt, Germany)

In this study, we investigate trajectories involving multiple impulses within the framework of a generic spacecraft dynamics. Revisiting the age-old query of "How many impulses?", we present novel manipulations heavily leveraging on the properties ofthe state transition matrix. Surprisingly, we are able to rediscover classical results leading to the introduction of a primer vector, albeit not making use of Pontryagin Maximum Principle as in the original developments by Lawden. Furthermore, our mathematical framework exhibits great flexibility and enables the introduction of what we term a "surrogate primer vector" extending a well known concept widely used in mission design. This enhancement allows to derive new simple optimality conditions that provide insights into the possibility to add and/or move multiple impulsive manoeuvres and improve the overall mass budget. This proves especially valuable in scenarios where a baseline trajectory arc is, for example, limited to a single impulse-an instance where traditional primer vector developments become singular and hinder conclusive outcomes. In demonstrating the practical application of the surrogate primer vector, we examine a specific case involving the four-body dynamics of a spacecraft within an Earth-Moon-Sun system. The system is characterized by the high-precision and differentiable VSOP2013 and ELP2000 ephemerides models. The focal point of our investigation is a reference trajectory representing a return from Mars, utilizing the weak stability boundary (WSB) of the Sun-Earth-Moon system. The trajectory incorporates two consecutive lunar flybys to insert the spacecraft into a lunar distant retrograde orbit (DRO). Conventionally, this trajectory necessitates a single maneuver at the DRO injection point., Comment: Paper presented and part of the proceedings of ISSFD 2024. (29th edition, Darmstadt, Germany)

Inspired by the versatility of sinusoidal representation networks (SIRENs), we present a modified Guidance & Control Networks (G&CNETs) variant using periodic activation functions in the hidden layers. We demonstrate that the resulting G&CNETs train faster and achieve a lower overall training error on three different control scenarios on which G&CNETs have been tested previously. A preliminary analysis is presented in an attempt to explain the superior performance of the SIREN architecture for the particular types of tasks that G&CNETs excel on.

We improve the accuracy of Guidance & Control Networks (G&CNETs), trained to represent the optimal control policies of a time-optimal transfer and a mass-optimal landing, respectively. In both cases we leverage the dynamics of the spacecraft, described by Ordinary Differential Equations which incorporate a neural network on their right-hand side (Neural ODEs). Since the neural dynamics is differentiable, the ODEs sensitivities to the network parameters can be computed using the variational equations, thereby allowing to update the G&CNET parameters based on the observed dynamics. We start with a straightforward regression task, training the G&CNETs on datasets of optimal trajectories using behavioural cloning. These networks are then refined using the Neural ODE sensitivities by minimizing the error between the final states and the target states. We demonstrate that for the orbital transfer, the final error to the target can be reduced by 99% on a single trajectory and by 70% on a batch of 500 trajectories. For the landing problem the reduction in error is around 98-99% (position) and 40-44% (velocity). This step significantly enhances the accuracy of G&CNETs, which instills greater confidence in their reliability for operational use. We also compare our results to the popular Dataset Aggregation method (DaGGER) and allude to the strengths and weaknesses of both methods.

The Simplified General Perturbations 4 (SGP4) orbital propagation method is widely used for predicting the positions and velocities of Earth-orbiting objects rapidly and reliably. Despite continuous refinement, SGP models still lack the precision of numerical propagators, which offer significantly smaller errors. This study presents dSGP4, a novel differentiable version of SGP4 implemented using PyTorch. By making SGP4 differentiable, dSGP4 facilitates various space-related applications, including spacecraft orbit determination, state conversion, covariance transformation, state transition matrix computation, and covariance propagation. Additionally, dSGP4's PyTorch implementation allows for embarrassingly parallel orbital propagation across batches of Two-Line Element Sets (TLEs), leveraging the computational power of CPUs, GPUs, and advanced hardware for distributed prediction of satellite positions at future times. Furthermore, dSGP4's differentiability enables integration with modern machine learning techniques. Thus, we propose a novel orbital propagation paradigm, ML-dSGP4, where neural networks are integrated into the orbital propagator. Through stochastic gradient descent, this combined model's inputs, outputs, and parameters can be iteratively refined, surpassing SGP4's precision. Neural networks act as identity operators by default, adhering to SGP4's behavior. However, dSGP4's differentiability allows fine-tuning with ephemeris data, enhancing precision while maintaining computational speed. This empowers satellite operators and researchers to train the model using specific ephemeris or high-precision numerical propagation data, significantly advancing orbital prediction capabilities.

We propose to use a recently introduced Totimorphic metamaterial for constructing morphable space structures. As a first step to investigate the feasibility of this concept, we present a method for morphing such structures autonomously between different shapes using physically plausible actuations, guaranteeing that the material traverses through valid configurations only while morphing. With this work, we aim to lay a foundation for exploring a promising and novel class of multi-functional, reconfigurable space structures., Comment: 4 pages, 2 figures, presented at the XXVII Italian Association of Aeronautics and Astronautics (AIDAA) Congress, 4-7 September 2023, Padova Italy

Early detection and tracking of ejecta in the vicinity of small solar system bodies is crucial to guarantee spacecraft safety and support scientific observation. During the visit of active asteroid Bennu, the OSIRIS-REx spacecraft relied on the analysis of images captured by onboard navigation cameras to detect particle ejection events, which ultimately became one of the mission's scientific highlights. To increase the scientific return of similar time-constrained missions, this work proposes an event-based solution that is dedicated to the detection and tracking of centimetre-sized particles. Unlike a standard frame-based camera, the pixels of an event-based camera independently trigger events indicating whether the scene brightness has increased or decreased at that time and location in the sensor plane. As a result of the sparse and asynchronous spatiotemporal output, event cameras combine very high dynamic range and temporal resolution with low-power consumption, which could complement existing onboard imaging techniques. This paper motivates the use of a scientific event camera by reconstructing the particle ejection episodes reported by the OSIRIS-REx mission in a photorealistic scene generator and in turn, simulating event-based observations. The resulting streams of spatiotemporal data support future work on event-based multi-object tracking., Comment: 6 pages, 3 figures, presented at the XXVII Italian Association of Aeronautics and Astronautics (AIDAA) Congress, 4-7 September 2023, Padova Italy

This work presents a large-scale simulation study investigating the deployment and operation of distributed swarms of CubeSats for interplanetary missions to small celestial bodies. Utilizing Taylor numerical integration and advanced collision detection techniques, we explore the potential of large CubeSat swarms in capturing gravity signals and reconstructing the internal mass distribution of a small celestial body while minimizing risks and Delta V budget. Our results offer insight into the applicability of this approach for future deep space exploration missions.

Context. Current constraints on distributed matter in the innermost Galactic Centre (such as a cluster of faint stars and stellar remnants, Dark Matter or a combination thereof) based on the orbital dynamics of the visible stars closest to the central black hole, typically assume simple functional forms for the distributions. Aims. We take instead a general model agnostic approach in which the form of the distribution is not constrained by prior assumptions on the physical composition of the matter. This approach yields unbiased - entirely observation driven - fits for the matter distribution and places constraints on our ability to discriminate between different density profiles (and consequently between physical compositions) of the distributed matter. Methods. We construct a spherical shell model with the flexibility to fit a wide variety of physically reasonable density profiles by modelling the distribution as a series of concentric mass shells. We test this approach in an analysis of mock observations of the star S2. Results. For a sufficiently large and precise data set, we find that it is possible to discriminate between several physically motivated density profiles. However, for data coming from current and expected next generation observational instruments, the potential for profile distinction will remain limited by the precision of the instruments. Future observations will still be able to constrain the overall enclosed distributed mass within the apocentre of the probing orbit in an unbiased manner. We interpret this in the theoretical context of constraining the secular versus non-secular orbital dynamics., Comment: 8 pages, 4 figures, submitted to Astronomy & Astrophysics

An event-based camera outputs an event whenever a change in scene brightness of a preset magnitude is detected at a particular pixel location in the sensor plane. The resulting sparse and asynchronous output coupled with the high dynamic range and temporal resolution of this novel camera motivate the study of event-based cameras for navigation and landing applications. However, the lack of real-world and synthetic datasets to support this line of research has limited its consideration for onboard use. This paper presents a methodology and a software pipeline for generating event-based vision datasets from optimal landing trajectories during the approach of a target body. We construct sequences of photorealistic images of the lunar surface with the Planet and Asteroid Natural Scene Generation Utility at different viewpoints along a set of optimal descent trajectories obtained by varying the boundary conditions. The generated image sequences are then converted into event streams by means of an event-based camera emulator. We demonstrate that the pipeline can generate realistic event-based representations of surface features by constructing a dataset of 500 trajectories, complete with event streams and motion field ground truth data. We anticipate that novel event-based vision datasets can be generated using this pipeline to support various spacecraft pose reconstruction problems given events as input, and we hope that the proposed methodology would attract the attention of researchers working at the intersection of neuromorphic vision and guidance navigation and control.

Recent advances in modeling density distributions, so-called neural density fields, can accurately describe the density distribution of celestial bodies without, e.g., requiring a shape model - properties of great advantage when designing trajectories close to these bodies. Previous work introduced this approach, but several open questions remained. This work investigates neural density fields and their relative errors in the context of robustness to external factors like noise or constraints during training, like the maximal available gravity signal strength due to a certain distance exemplified for 433 Eros and 67P/Churyumov-Gerasimenko. It is found that both models trained on a polyhedral and mascon ground truth perform similarly, indicating that the ground truth is not the accuracy bottleneck. The impact of solar radiation pressure on a typical probe affects training neglectable, with the relative error being of the same magnitude as without noise. However, limiting the precision of measurement data by applying Gaussian noise hurts the obtainable precision. Further, pretraining is shown as practical in order to speed up network training. Hence, this work demonstrates that training neural networks for the gravity inversion problem is appropriate as long as the gravity signal is distinguishable from noise. Code and results are available at https://github.com/gomezzz/geodesyNets, Comment: Will be presented as a poster at the GNC 2023 conference

Spacecraft and drones aimed at exploring our solar system are designed to operate in conditions where the smart use of onboard resources is vital to the success or failure of the mission. Sensorimotor actions are thus often derived from high-level, quantifiable, optimality principles assigned to each task, utilizing consolidated tools in optimal control theory. The planned actions are derived on the ground and transferred onboard where controllers have the task of tracking the uploaded guidance profile. Here we argue that end-to-end neural guidance and control architectures (here called G&CNets) allow transferring onboard the burden of acting upon these optimality principles. In this way, the sensor information is transformed in real time into optimal plans thus increasing the mission autonomy and robustness. We discuss the main results obtained in training such neural architectures in simulation for interplanetary transfers, landings and close proximity operations, highlighting the successful learning of optimality principles by the neural model. We then suggest drone racing as an ideal gym environment to test these architectures on real robotic platforms, thus increasing confidence in their utilization on future space exploration missions. Drone racing shares with spacecraft missions both limited onboard computational capabilities and similar control structures induced from the optimality principle sought, but it also entails different levels of uncertainties and unmodelled effects. Furthermore, the success of G&CNets on extremely resource-restricted drones illustrates their potential to bring real-time optimal control within reach of a wider variety of robotic systems, both in space and on Earth.

Reaching fast and autonomous flight requires computationally efficient and robust algorithms. To this end, we train Guidance & Control Networks to approximate optimal control policies ranging from energy-optimal to time-optimal flight. We show that the policies become more difficult to learn the closer we get to the time-optimal 'bang-bang' control profile. We also assess the importance of knowing the maximum angular rotor velocity of the quadcopter and show that over- or underestimating this limit leads to less robust flight. We propose an algorithm to identify the current maximum angular rotor velocity onboard and a network that adapts its policy based on the identified limit. Finally, we extend previous work on Guidance & Control Networks by learning to take consecutive waypoints into account. We fly a 4x3m track in similar lap times as the differential-flatness-based minimum snap benchmark controller while benefiting from the flexibility that Guidance & Control Networks offer.

Developing optimal controllers for aggressive high-speed quadcopter flight poses significant challenges in robotics. Recent trends in the field involve utilizing neural network controllers trained through supervised or reinforcement learning. However, the sim-to-real transfer introduces a reality gap, requiring the use of robust inner loop controllers during real flights, which limits the network's control authority and flight performance. In this paper, we investigate for the first time, an end-to-end neural network controller, addressing the reality gap issue without being restricted by an inner-loop controller. The networks, referred to as G\&CNets, are trained to learn an energy-optimal policy mapping the quadcopter's state to rpm commands using an optimal trajectory dataset. In hover-to-hover flights, we identified the unmodeled moments as a significant contributor to the reality gap. To mitigate this, we propose an adaptive control strategy that works by learning from optimal trajectories of a system affected by constant external pitch, roll and yaw moments. In real test flights, this model mismatch is estimated onboard and fed to the network to obtain the optimal rpm command. We demonstrate the effectiveness of our method by performing energy-optimal hover-to-hover flights with and without moment feedback. Finally, we compare the adaptive controller to a state-of-the-art differential-flatness-based controller in a consecutive waypoint flight and demonstrate the advantages of our method in terms of energy optimality and robustness., Comment: 11 pages, 16 figures

The development and adoption of artificial intelligence (AI) technologies in space applications is growing quickly as the consensus increases on the potential benefits introduced. As more and more aerospace engineers are becoming aware of new trends in AI, traditional approaches are revisited to consider the applications of emerging AI technologies. Already at the time of writing, the scope of AI-related activities across academia, the aerospace industry and space agencies is so wide that an in-depth review would not fit in these pages. In this chapter we focus instead on two main emerging trends we believe capture the most relevant and exciting activities in the field: differentiable intelligence and on-board machine learning. Differentiable intelligence, in a nutshell, refers to works making extensive use of automatic differentiation frameworks to learn the parameters of machine learning or related models. Onboard machine learning considers the problem of moving inference, as well as learning, onboard. Within these fields, we discuss a few selected projects originating from the European Space Agency's (ESA) Advanced Concepts Team (ACT), giving priority to advanced topics going beyond the transposition of established AI techniques and practices to the space domain.

The term ``neuromorphic'' refers to systems that are closely resembling the architecture and/or the dynamics of biological neural networks. Typical examples are novel computer chips designed to mimic the architecture of a biological brain, or sensors that get inspiration from, e.g., the visual or olfactory systems in insects and mammals to acquire information about the environment. This approach is not without ambition as it promises to enable engineered devices able to reproduce the level of performance observed in biological organisms -- the main immediate advantage being the efficient use of scarce resources, which translates into low power requirements. The emphasis on low power and energy efficiency of neuromorphic devices is a perfect match for space applications. Spacecraft -- especially miniaturized ones -- have strict energy constraints as they need to operate in an environment which is scarce with resources and extremely hostile. In this work we present an overview of early attempts made to study a neuromorphic approach in a space context at the European Space Agency's (ESA) Advanced Concepts Team (ACT).

Event sensing is a major component in bio-inspired flight guidance and control systems. We explore the usage of event cameras for predicting time-to-contact (TTC) with the surface during ventral landing. This is achieved by estimating divergence (inverse TTC), which is the rate of radial optic flow, from the event stream generated during landing. Our core contributions are a novel contrast maximisation formulation for event-based divergence estimation, and a branch-and-bound algorithm to exactly maximise contrast and find the optimal divergence value. GPU acceleration is conducted to speed up the global algorithm. Another contribution is a new dataset containing real event streams from ventral landing that was employed to test and benchmark our method. Owing to global optimisation, our algorithm is much more capable at recovering the true divergence, compared to other heuristic divergence estimators or event-based optic flow methods. With GPU acceleration, our method also achieves competitive runtimes., Comment: Accepted in the ECCV 2022 workshop on AI for Space, 18 pages, 6 figures

Interpretable regression models are important for many application domains, as they allow experts to understand relations between variables from sparse data. Symbolic regression addresses this issue by searching the space of all possible free form equations that can be constructed from elementary algebraic functions. While explicit mathematical functions can be rediscovered this way, the determination of unknown numerical constants during search has been an often neglected issue. We propose a new multi-objective memetic algorithm that exploits a differentiable Cartesian Genetic Programming encoding to learn constants during evolutionary loops. We show that this approach is competitive or outperforms machine learned black box regression models or hand-engineered fits for two applications from space: the Mars express thermal power estimation and the determination of the age of stars by gyrochronology.

Dyson spheres are hypothetical megastructures encircling stars in order to harvest most of their energy output. During the 11th edition of the GTOC challenge, participants were tasked with a complex trajectory planning related to the construction of a precursor Dyson structure, a heliocentric ring made of twelve stations. To this purpose, we developed several new approaches that synthesize techniques from machine learning, combinatorial optimization, planning and scheduling, and evolutionary optimization effectively integrated into a fully automated pipeline. These include a machine learned transfer time estimator, improving the established Edelbaum approximation and thus better informing a Lazy Race Tree Search to identify and collect asteroids with high arrival mass for the stations; a series of optimally-phased low-thrust transfers to all stations computed by indirect optimization techniques, exploiting the synodic periodicity of the system; and a modified Hungarian scheduling algorithm, which utilizes evolutionary techniques to arrange a mass-balanced arrival schedule out of all transfer possibilities. We describe the steps of our pipeline in detail with a special focus on how our approaches mutually benefit from each other. Lastly, we outline and analyze the final solution of our team, ACT&Friends, which ranked second at the GTOC 11 challenge.

We present a novel approach for the detection of events in systems of ordinary differential equations. The new method combines the unique features of Taylor integrators with state-of-the-art polynomial root finding techniques to yield a novel algorithm ensuring strong event detection guarantees at a modest computational overhead. Detailed tests and benchmarks focused on problems in astrodynamics and celestial mechanics (such as collisional N-body systems, spacecraft dynamics around irregular bodies accounting for eclipses, computation of Poincare' sections, etc.) show how our approach is superior in both performance and detection accuracy to strategies commonly employed in modern numerical integration works. The new algorithm is available in our open source Taylor integration package heyoka., Comment: Accepted for publication in MNRAS

We train neural models to represent both the optimal policy (i.e. the optimal thrust direction) and the value function (i.e. the time of flight) for a time optimal, constant acceleration low-thrust rendezvous. In both cases we develop and make use of the data augmentation technique we call backward generation of optimal examples. We are thus able to produce and work with large dataset and to fully exploit the benefit of employing a deep learning framework. We achieve, in all cases, accuracies resulting in successful rendezvous (simulated following the learned policy) and time of flight predictions (using the learned value function). We find that residuals as small as a few m/s, thus well within the possibility of a spacecraft navigation $\Delta V$ budget, are achievable for the velocity at rendezvous. We also find that, on average, the absolute error to predict the optimal time of flight to rendezvous from any orbit in the asteroid belt to an Earth-like orbit is small (less than 4\%) and thus also of interest for practical uses, for example, during preliminary mission design phases.

Numerical simulations are at the center of predicting the space debris environment of the upcoming decades. In light of debris generating events, such as continued anti-satellite weapon tests and planned mega-constellations, accurate predictions of the space debris environment are critical to ensure the long-term sustainability of critical satellite orbits. Given the computational complexity of accurate long-term trajectory propagation for a large number of particles, numerical models usually rely on Monte-Carlo approaches for stochastic conjunction assessment. On the other hand, deterministic methods bear the promise of higher accuracy and can serve to validate stochastic approaches. However, they pose a substantial challenge to computational feasibility. In this work, we present the architecture and proof of concept results for a numerical simulation capable of modeling the long term debris evolution over decades with a deterministic conjunction tracking model. For the simulation, we developed an efficient propagator in modern C++ accounting for Earth's gravitational anomalies, solar radiation pressure, and atmospheric drag. We utilized AutoPAS, a sophisticated particle container, which automatically selects the most efficient data structures and algorithms. We present results from a simulation of 16 024 particles in low-Earth orbit over 20 years. Overall, conjunctions are tracked for predicted collisions and close encounters to allow a detailed study of both. We analyze the runtime and computational cost of the simulation in detail. In summary, the obtained results show that modern computational tools finally enable deterministic conjunction tracking and can serve to validate prior results and build higher-fidelity numerical simulations of the long-term debris environment.

Autonomous vision-based spaceborne navigation is an enabling technology for future on-orbit servicing and space logistics missions. While computer vision in general has benefited from Machine Learning (ML), training and validating spaceborne ML models are extremely challenging due to the impracticality of acquiring a large-scale labeled dataset of images of the intended target in the space environment. Existing datasets, such as Spacecraft PosE Estimation Dataset (SPEED), have so far mostly relied on synthetic images for both training and validation, which are easy to mass-produce but fail to resemble the visual features and illumination variability inherent to the target spaceborne images. In order to bridge the gap between the current practices and the intended applications in future space missions, this paper introduces SPEED+: the next generation spacecraft pose estimation dataset with specific emphasis on domain gap. In addition to 60,000 synthetic images for training, SPEED+ includes 9,531 hardware-in-the-loop images of a spacecraft mockup model captured from the Testbed for Rendezvous and Optical Navigation (TRON) facility. TRON is a first-of-a-kind robotic testbed capable of capturing an arbitrary number of target images with accurate and maximally diverse pose labels and high-fidelity spaceborne illumination conditions. SPEED+ is used in the second international Satellite Pose Estimation Challenge co-hosted by SLAB and the Advanced Concepts Team of the European Space Agency to evaluate and compare the robustness of spaceborne ML models trained on synthetic images.

Asteroids and other small bodies in the solar system tend to have irregular shapes, owing to their low gravity. This irregularity does not only apply to the topology, but also to the underlying geology, potentially containing regions of different densities and materials. The topology can be derived from optical observations, while the mass density distribution of an object is only observable, to some extent, in its gravitational field. In a companion paper, we presented geodesyNets, a neural network approach to infer the mass density distribution of an object from measurements of its gravitational field. In the present work, we apply this approach to the asteroid Bennu using real data from the Osiris Rex mission. The mission measured the trajectories of not only the Osiris Rex spacecraft itself, but also of numerous pebble-sized rock particles which temporarily orbited Bennu. From these trajectory data, we obtain a representation of Bennu's mass density and validate it by propagating, in the resulting gravity field, multiple pebbles not used in the training process. The performance is comparable to that of a polyhedral gravity model of uniform density, but does not require a shape model. As little additional information is needed, we see this as a step towards autonomous on-board inversion of gravitational fields.

We present a novel approach based on artificial neural networks, so-called geodesyNets, and present compelling evidence of their ability to serve as accurate geodetic models of highly irregular bodies using minimal prior information on the body. The approach does not rely on the body shape information but, if available, can harness it. GeodesyNets learn a three-dimensional, differentiable, function representing the body density, which we call neural density field. The body shape, as well as other geodetic properties, can easily be recovered. We investigate six different shapes including the bodies 101955 Bennu, 67P Churyumov-Gerasimenko, 433 Eros and 25143 Itokawa for which shape models developed during close proximity surveys are available. Both heterogeneous and homogeneous mass distributions are considered. The gravitational acceleration computed from the trained geodesyNets models, as well as the inferred body shape, show great accuracy in all cases with a relative error on the predicted acceleration smaller than 1\% even close to the asteroid surface. When the body shape information is available, geodesyNets can seamlessly exploit it and be trained to represent a high-fidelity neural density field able to give insights into the internal structure of the body. This work introduces a new unexplored approach to geodesy, adding a powerful tool to consolidated ones based on spherical harmonics, mascon models and polyhedral gravity.

In advanced mission concepts with high levels of autonomy, spacecraft need to internally model the pose and shape of nearby orbiting objects. Recent works in neural scene representations show promising results for inferring generic three-dimensional scenes from optical images. Neural Radiance Fields (NeRF) have shown success in rendering highly specular surfaces using a large number of images and their pose. More recently, Generative Radiance Fields (GRAF) achieved full volumetric reconstruction of a scene from unposed images only, thanks to the use of an adversarial framework to train a NeRF. In this paper, we compare and evaluate the potential of NeRF and GRAF to render novel views and extract the 3D shape of two different spacecraft, the Soil Moisture and Ocean Salinity satellite of ESA's Living Planet Programme and a generic cube sat. Considering the best performances of both models, we observe that NeRF has the ability to render more accurate images regarding the material specularity of the spacecraft and its pose. For its part, GRAF generates precise novel views with accurate details even when parts of the satellites are shadowed while having the significant advantage of not needing any information about the relative pose.

We present heyoka, a new, modern and general-purpose implementation of Taylor's integration method for the numerical solution of ordinary differential equations. Detailed numerical tests focused on difficult high-precision gravitational problems in astrodynamics and celestial mechanics show how our general-purpose integrator is competitive with and often superior to state-of-the-art specialised symplectic and non-symplectic integrators in both speed and accuracy. In particular, we show how Taylor methods are capable of satisfying Brouwer's law for the conservation of energy in long-term integrations of planetary systems over billions of dynamical timescales. We also show how close encounters are modelled accurately during simulations of the formation of the Kirkwood gaps and of Apophis' 2029 close encounter with the Earth (where heyoka surpasses the speed and accuracy of domain-specific methods). heyoka can be used from both C++ and Python, and it is publicly available as an open-source project.

We present a new generic method for shadow-aware multi-view satellite photogrammetry of Earth Observation scenes. Our proposed method, the Shadow Neural Radiance Field (S-NeRF) follows recent advances in implicit volumetric representation learning. For each scene, we train S-NeRF using very high spatial resolution optical images taken from known viewing angles. The learning requires no labels or shape priors: it is self-supervised by an image reconstruction loss. To accommodate for changing light source conditions both from a directional light source (the Sun) and a diffuse light source (the sky), we extend the NeRF approach in two ways. First, direct illumination from the Sun is modeled via a local light source visibility field. Second, indirect illumination from a diffuse light source is learned as a non-local color field as a function of the position of the Sun. Quantitatively, the combination of these factors reduces the altitude and color errors in shaded areas, compared to NeRF. The S-NeRF methodology not only performs novel view synthesis and full 3D shape estimation, it also enables shadow detection, albedo synthesis, and transient object filtering, without any explicit shape supervision., Comment: Accepted to CVPR2021 - EarthVision

Spacecraft collision avoidance procedures have become an essential part of satellite operations. Complex and constantly updated estimates of the collision risk between orbiting objects inform the various operators who can then plan risk mitigation measures. Such measures could be aided by the development of suitable machine learning models predicting, for example, the evolution of the collision risk in time. In an attempt to study this opportunity, the European Space Agency released, in October 2019, a large curated dataset containing information about close approach events, in the form of Conjunction Data Messages (CDMs), collected from 2015 to 2019. This dataset was used in the Spacecraft Collision Avoidance Challenge, a machine learning competition where participants had to build models to predict the final collision risk between orbiting objects. This paper describes the design and results of the competition and discusses the challenges and lessons learned when applying machine learning methods to this problem domain., Comment: Pre-print

One of the main and largely unexplored challenges in evolving the weights of neural networks using genetic algorithms is to find a sensible crossover operation between parent networks. Indeed, naive crossover leads to functionally damaged offspring that do not retain information from the parents. This is because neural networks are invariant to permutations of neurons, giving rise to multiple ways of representing the same solution. This is often referred to as the competing conventions problem. In this paper, we propose a two-step safe crossover(SC) operator. First, the neurons of the parents are functionally aligned by computing how well they correlate, and only then are the parents recombined. We compare two ways of measuring relationships between neurons: Pairwise Correlation (PwC) and Canonical Correlation Analysis (CCA). We test our safe crossover operators (SC-PwC and SC-CCA) on MNIST and CIFAR-10 by performing arithmetic crossover on the weights of feed-forward neural network pairs. We show that it effectively transmits information from parents to offspring and significantly improves upon naive crossover. Our method is computationally fast,can serve as a way to explore the fitness landscape more efficiently and makes safe crossover a potentially promising operator in future neuroevolution research and applications.

We consider the Earth-Venus mass-optimal interplanetary transfer of a low-thrust spacecraft and show how the optimal guidance can be represented by deep networks in a large portion of the state space and to a high degree of accuracy. Imitation (supervised) learning of optimal examples is used as a network training paradigm. The resulting models are suitable for an on-board, real-time, implementation of the optimal guidance and control system of the spacecraft and are called G&CNETs. A new general methodology called Backward Generation of Optimal Examples is introduced and shown to be able to efficiently create all the optimal state action pairs necessary to train G&CNETs without solving optimal control problems. With respect to previous works, we are able to produce datasets containing a few orders of magnitude more optimal trajectories and obtain network performances compatible with real missions requirements. Several schemes able to train representations of either the optimal policy (thrust profile) or the value function (optimal mass) are proposed and tested. We find that both policy learning and value function learning successfully and accurately learn the optimal thrust and that a spacecraft employing the learned thrust is able to reach the target conditions orbit spending only 2 permil more propellant than in the corresponding mathematically optimal transfer. Moreover, the optimal propellant mass can be predicted (in case of value function learning) within an error well within 1%. All G&CNETs produced are tested during simulations of interplanetary transfers with respect to their ability to reach the target conditions optimally starting from nominal and off-nominal conditions.

Optimal control holds great potential to improve a variety of robotic applications. The application of optimal control on-board limited platforms has been severely hindered by the large computational requirements of current state of the art implementations. In this work, we make use of a deep neural network to directly map the robot states to control actions. The network is trained offline to imitate the optimal control computed by a time consuming direct nonlinear method. A mixture of time optimality and power optimality is considered with a continuation parameter used to select the predominance of each objective. We apply our networks (termed G\&CNets) to aggressive quadrotor control, first in simulation and then in the real world. We give insight into the factors that influence the `reality gap' between the quadrotor model used by the offline optimal control method and the real quadrotor. Furthermore, we explain how we set up the model and the control structure on-board of the real quadrotor to successfully close this gap and perform time-optimal maneuvers in the real world. Finally, G\&CNet's performance is compared to state-of-the-art differential-flatness-based optimal control methods. We show, in the experiments, that G\&CNets lead to significantly faster trajectory execution due to, in part, the less restrictive nature of the allowed state-to-input mappings., Comment: 6 pages, 8 figures

Reliable pose estimation of uncooperative satellites is a key technology for enabling future on-orbit servicing and debris removal missions. The Kelvins Satellite Pose Estimation Challenge aims at evaluating and comparing monocular vision-based approaches and pushing the state-of-the-art on this problem. This work is based on the Satellite Pose Estimation Dataset, the first publicly available machine learning set of synthetic and real spacecraft imageries. The choice of dataset reflects one of the unique challenges associated with spaceborne computer vision tasks, namely the lack of spaceborne images to train and validate the developed algorithms. This work briefly reviews the basic properties and the collection process of the dataset which was made publicly available. The competition design, including the definition of performance metrics and the adopted testbed, is also discussed. The main contribution of this paper is the analysis of the submissions of the 48 competitors, which compares the performance of different approaches and uncovers what factors make the satellite pose estimation problem especially challenging., Comment: Accepted to IEEE Transactions on Aerospace and Electronic Systems

The ability to design complex neural network architectures which enable effective training by stochastic gradient descent has been the key for many achievements in the field of deep learning. However, developing such architectures remains a challenging and resourceintensive process full of trial-and-error iterations. All in all, the relation between the network topology and its ability to model the data remains poorly understood. We propose to encode neural networks with a differentiable variant of Cartesian Genetic Programming (dCGPANN) and present a memetic algorithm for architecture design: local searches with gradient descent learn the network parameters while evolutionary operators act on the dCGPANN genes shaping the network architecture towards faster learning. Studying a particular instance of such a learning scheme, we are able to improve the starting feed forward topology by learning how to rewire and prune links, adapt activation functions and introduce skip connections for chosen regression tasks. The evolved network architectures require less space for network parameters and reach, given the same amount of time, a significantly lower error on average., Comment: a short version of this was accepted as poster paper at GECCO 2019

ESA's PROBA-V Earth observation satellite enables us to monitor our planet at a large scale, studying the interaction between vegetation and climate and provides guidance for important decisions on our common global future. However, the interval at which high resolution images are recorded spans over several days, in contrast to the availability of lower resolution images which is often daily. We collect an extensive dataset of both, high and low resolution images taken by PROBA-V instruments during monthly periods to investigate Multi Image Super-resolution, a technique to merge several low resolution images to one image of higher quality. We propose a convolutional neural network that is able to cope with changes in illumination, cloud coverage and landscape features which are challenges introduced by the fact that the different images are taken over successive satellite passages over the same region. Given a bicubic upscaling of low resolution images taken under optimal conditions, we find the Peak Signal to Noise Ratio of the reconstructed image of the network to be higher for a large majority of different scenes. This shows that applied machine learning has the potential to enhance large amounts of previously collected earth observation data during multiple satellite passes., Comment: To appear in Special Issue on Applications of Artificial Intelligence in Aerospace Engineering in the Journal "Astrodynamics"

A number of applications to interplanetary trajectories have been recently proposed based on deep networks. These approaches often rely on the availability of a large number of optimal trajectories to learn from. In this paper we introduce a new method to quickly create millions of optimal spacecraft trajectories from a single nominal trajectory. Apart from the generation of the nominal trajectory, no additional optimal control problems need to be solved as all the trajectories, by construction, satisfy Pontryagin's minimum principle and the relevant transversality conditions. We then consider deep feed forward neural networks and benchmark three learning methods on the created dataset: policy imitation, value function learning and value function gradient learning. Our results are shown for the case of the interplanetary trajectory optimization problem of reaching Venus orbit, with the nominal trajectory starting from the Earth. We find that both policy imitation and value function gradient learning are able to learn the optimal state feedback, while in the case of value function learning the optimal policy is not captured, only the final value of the optimal propellant mass is.

Optimal state-feedback controllers, capable of changing between different objective functions, are advantageous to systems in which unexpected situations may arise. However, synthesising such controllers, even for a single objective, is a demanding process. In this paper, we present a novel and straightforward approach to synthesising these policies through a combination of trajectory optimisation, homotopy continuation, and imitation learning. We use numerical continuation to efficiently generate optimal demonstrations across several objectives and boundary conditions, and use these to train our policies. Additionally, we demonstrate the ability of our policies to effectively learn families of optimal state-feedback controllers, which can be used to change objective functions online. We illustrate this approach across two trajectory optimisation problems, an inverted pendulum swingup and a spacecraft orbit transfer, and show that the synthesised policies, when evaluated in simulation, produce trajectories that are near-optimal. These results indicate the benefit of trajectory optimisation and homotopy continuation to the synthesis of controllers in dynamic-objective contexts., Comment: Accepted to the 59th IEEE Conference on Decision and Control (CDC)

In the design of multitarget interplanetary missions, there are always many options available, making it often impractical to optimize in detail each transfer trajectory in a preliminary search phase. Fast and accurate estimation methods for optimal transfers are thus of great value. In this paper, deep feed-forward neural networks are employed to estimate solutions to three types of optimization problems: the transfer time of time-optimal low-thrust transfers, fuel consumption of fuel-optimal low-thrust transfers, and the total dv of minimum-dv J2-perturbed multi-impulse transfers. To generate the training data, low-thrust trajectories are optimized using the indirect method and J2-perturbed multi-impulse trajectories are optimized using J2 homotopy and particle swarm optimization. The hyper-parameters of our deep networks are searched by grid search, random search, and the tree-structured Parzen estimators approach. Results show that deep networks are capable of estimating the final mass or time of optimal transfers with extremely high accuracy; resulting into a mean relative error of less than 0.5% for low-thrust transfers and less than 4% for multi-impulse transfers. Our results are also compared with other off-the-shelf machine-learning algorithms and investigated with respect to their capability of predicting cases well outside of the training data.

Imitation learning is a control design paradigm that seeks to learn a control policy reproducing demonstrations from expert agents. By substituting expert demonstrations for optimal behaviours, the same paradigm leads to the design of control policies closely approximating the optimal state-feedback. This approach requires training a machine learning algorithm (in our case deep neural networks) directly on state-control pairs originating from optimal trajectories. We have shown in previous work that, when restricted to low-dimensional state and control spaces, this approach is very successful in several deterministic, non-linear problems in continuous-time. In this work, we refine our previous studies using as a test case a simple quadcopter model with quadratic and time-optimal objective functions. We describe in detail the best learning pipeline we have developed, that is able to approximate via deep neural networks the state-feedback map to a very high accuracy. We introduce the use of the softplus activation function in the hidden units of neural networks showing that it results in a smoother control profile whilst retaining the benefits of rectifiers. We show how to evaluate the optimality of the trained state-feedback, and find that already with two layers the objective function reached and its optimal value differ by less than one percent. We later consider also an additional metric linked to the system asymptotic behaviour - time taken to converge to the policy's fixed point. With respect to these metrics, we show that improvements in the mean absolute error do not necessarily correspond to better policies.

The rapid developments of Artificial Intelligence in the last decade are influencing Aerospace Engineering to a great extent and research in this context is proliferating. We share our observations on the recent developments in the area of Spacecraft Guidance Dynamics and Control, giving selected examples on success stories that have been motivated by mission designs. Our focus is on evolutionary optimisation, tree searches and machine learning, including deep learning and reinforcement learning as the key technologies and drivers for current and future research in the field. From a high-level perspective, we survey various scenarios for which these approaches have been successfully applied or are under strong scientific investigation. Whenever possible, we highlight the relations and synergies that can be obtained by combining different techniques and projects towards future domains for which newly emerging artificial intelligence techniques are expected to become game changers.

Recent work have shown how the optimal state-feedback, obtained as the solution to the Hamilton-Jacobi-Bellman equations, can be approximated for several nonlinear, deterministic systems by deep neural networks. When imitation (supervised) learning is used to train the neural network on optimal state-action pairs, for instance as derived by applying Pontryagin's theory of optimal processes, the resulting model is referred here as the guidance and control network. In this work, we analyze the stability of nonlinear and deterministic systems controlled by such networks. We then propose a method utilising differential algebraic techniques and high-order Taylor maps to gain information on the stability of the neurocontrolled state trajectories. We exemplify the proposed methods in the case of the two-dimensional dynamics of a quadcopter controlled to reach the origin and we study how different architectures of the guidance and control network affect the stability of the target equilibrium point and the stability margins to time delay. Moreover, we show how to study the robustness to initial conditions of a nominal trajectory, using a Taylor representation of the neurocontrolled neighbouring trajectories.