arXiv subjects

Dario Izzo

Publications and source records attributed to Dario Izzo.

At least 37 records · Page 2Linked to original sources

arXiv2025

EclipseNETs: Learning Irregular Small Celestial Body Silhouettes

Accurately predicting eclipse events around irregular small bodies is crucial for spacecraft navigation, orbit determination, and spacecraft systems management. This paper introduces a novel approach leveraging neural implicit representations to model eclipse conditions efficiently and reliably. We propose neural network architectures that capture the complex silhouettes of asteroids and comets with high precision. Tested on four well-characterized bodies - Bennu, Itokawa, 67P/Churyumov-Gerasimenko, and Eros - our method achieves accuracy comparable to traditional ray-tracing techniques while offering orders of magnitude faster performance. Additionally, we develop an indirect learning framework that trains these models directly from sparse trajectory data using Neural Ordinary Differential Equations, removing the requirement to have prior knowledge of an accurate shape model. This approach allows for the continuous refinement of eclipse predictions, progressively reducing errors and improving accuracy as new trajectory data is incorporated.

astro-ph.EP↗

arXiv2025

High-order expansion of Neural Ordinary Differential Equations flows

Artificial neural networks, widely recognised for their role in machine learning, are now transforming the study of ordinary differential equations (ODEs), bridging data-driven modelling with classical dynamical systems and enabling the development of infinitely deep neural models. However, the practical applicability of these models remains constrained by the opacity of their learned dynamics, which operate as black-box systems with limited explainability, thereby hindering trust in their deployment. Existing approaches for the analysis of these dynamical systems are predominantly restricted to first-order gradient information due to computational constraints, thereby limiting the depth of achievable insight. Here, we introduce Event Transition Tensors, a framework based on high-order differentials that provides a rigorous mathematical description of neural ODE dynamics on event manifolds. We demonstrate its versatility across diverse applications: characterising uncertainties in a data-driven prey-predator control model, analysing neural optimal feedback dynamics, and mapping landing trajectories in a three-body neural Hamiltonian system. In all cases, our method enhances the interpretability and rigour of neural ODEs by expressing their behaviour through explicit mathematical structures. Our findings contribute to a deeper theoretical foundation for event-triggered neural differential equations and provide a mathematical construct for explaining complex system dynamics.

math.OC↗

arXiv2024

Closing the Gap Between SGP4 and High-Precision Propagation via Differentiable Programming

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.

cs.LG↗

arXiv2024

Nonlinear Propagation of Non-Gaussian Uncertainties

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 Poincaré 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.

physics.space-ph↗

arXiv2024

Asteroid Mining: ACT&Friends' Results for the GTOC 12 Problem

In 2023, the 12th edition of Global Trajectory Competition was organised around the problem referred to as "Sustainable Asteroid Mining". This paper reports the developments that led to the solution proposed by ESA's Advanced Concepts Team. Beyond the fact that the proposed approach failed to rank higher than fourth in the final competition leader-board, several innovative fundamental methodologies were developed which have a broader application. In particular, new methods based on machine learning as well as on manipulating the fundamental laws of astrodynamics were developed and able to fill with remarkable accuracy the gap between full low-thrust trajectories and their representation as impulsive Lambert transfers. A novel technique was devised to formulate the challenge of optimal subset selection from a repository of pre-existing optimal mining trajectories as an integer linear programming problem. Finally, the fundamental problem of searching for single optimal mining trajectories (mining and collecting all resources), albeit ignoring the possibility of having intra-ship collaboration and thus sub-optimal in the case of the GTOC12 problem, was efficiently solved by means of a novel search based on a look-ahead score and thus making sure to select asteroids that had chances to be re-visited later on.

astro-ph.EP↗

arXiv2024

Certifying Guidance & Control Networks: Uncertainty Propagation to an Event Manifold

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.

eess.SY↗

arXiv2024

EclipseNETs: a differentiable description of irregular eclipse conditions

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.

cs.LG↗

arXiv2024

Breaking traditions: introducing a surrogate Primer Vector in non Keplerian dynamics

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.

physics.space-ph↗

arXiv2024

Stochastic Continuation of Trajectories in the Circular Restricted Three-Body Problem via Differential Algebra

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.

physics.space-ph↗

arXiv2024

Global Optimization for Trajectory Design via Invariant Manifolds in the Earth-Moon Circular Restricted Three-Body Problem

This study addresses optimal impulsive trajectory design within the Circular Restricted Three-Body Problem (CR3BP), presenting a global optimization-based approach to identify minimum $ΔV$ 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 $ΔV$ 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 $ΔV$ solutions and showcases the algorithm's effectiveness across various mission scenarios.

physics.space-ph↗

arXiv2024

Computing low-thrust transfers in the asteroid belt, a comparison between astrodynamical manipulations and a machine learning approach

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.

astro-ph.EP↗

arXiv2024

NeuralODEs for VLEO simulations: Introducing thermoNET for Thermosphere Modeling

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.

astro-ph.EP↗

arXiv2024

Guidance and Control Networks with Periodic Activation Functions

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.

cs.LG↗

arXiv2024

Closing the gap: Optimizing Guidance and Control Networks through Neural ODEs

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.

cs.LG↗

arXiv2023

End-to-end Reinforcement Learning for Time-Optimal Quadcopter Flight

Aggressive time-optimal control of quadcopters poses a significant challenge in the field of robotics. The state-of-the-art approach leverages reinforcement learning (RL) to train optimal neural policies. However, a critical hurdle is the sim-to-real gap, often addressed by employing a robust inner loop controller -an abstraction that, in theory, constrains the optimality of the trained controller, necessitating margins to counter potential disturbances. In contrast, our novel approach introduces high-speed quadcopter control using end-to-end RL (E2E) that gives direct motor commands. To bridge the reality gap, we incorporate a learned residual model and an adaptive method that can compensate for modeling errors in thrust and moments. We compare our E2E approach against a state-of-the-art network that commands thrust and body rates to an INDI inner loop controller, both in simulated and real-world flight. E2E showcases a significant 1.39-second advantage in simulation and a 0.17-second edge in real-world testing, highlighting end-to-end reinforcement learning's potential. The performance drop observed from simulation to reality shows potential for further improvement, including refining strategies to address the reality gap or exploring offline reinforcement learning with real flight data.

cs.RO↗

arXiv2023

Totimorphic structures for space application

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.

cond-mat.mtrl-sci↗

arXiv2023

Tracking Particles Ejected From Active Asteroid Bennu With Event-Based Vision

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.

cs.CV↗

arXiv2023

Small Celestial Body Exploration with CubeSat Swarms

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.

cs.RO↗