Giacomo Acciarini

Postgraduate Research Student

Academic and research departments

Astrodynamics, Surrey Space Centre.


My research project


Laurent Beauregard, Dario Izzo, Giacomo Acciarini (2024)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 of the 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.Prior to implementing the surrogate primer vector, a local optimization of the trajectory is performed. Upon application of the surrogate primer vector, we successfully identify potential maneuver injection points, strategically reducing the overall mission cost. The introduction of these additional maneuvers, followed by local optimization, validates that the revised trajectory indeed incurs a lower cost compared to the original configuration.

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

Dario Izzo, Giacomo Acciarini, Francesco Biscani (2024)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.

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.

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. Usually, the robustness of a candidate trajectory is tested via a two-step approach, whereby trajectories are first designed in a deterministic scenario, and then Monte Carlo methods are a posteriori used to check their robustness. While this strategy is ubiquitous in preliminary mission design, it can however lead to time-consuming and potentially not robust solutions, meaning that the found tra-jectories are not designed to account for uncertainties. Instead, the robustness of the determin-istic optimal solutions is usually ensured. 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. While the technique is only valid for initial Gaussian distributions, it does not assume that the distribution maintains its normality throughout the integration. The symbolic Poincaré map can then be directly used to evaluate the final moments of an initial distribution , as a function of the initial mean and covariance. This can therefore be used to accelerate the evolving step in the stochastic continuation procedure. 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.

Giacomo Acciarini, Edward Brown, Tom Berger, Madhulika Guhathakurta, James Parr, Christopher Paul Bridges, Atılım Güneş Baydin (2024)Improving Thermospheric Density Predictions in Low‐Earth Orbit With Machine Learning, In: Space weather22(2)e2023SW003652

Abstract Thermospheric density is one of the main sources of uncertainty in the estimation of satellites' position and velocity in low‐Earth orbit. This has negative consequences in several space domains, including space traffic management, collision avoidance, re‐entry predictions, orbital lifetime analysis, and space object cataloging. In this paper, we investigate the prediction accuracy of empirical density models (e.g., NRLMSISE‐00 and JB‐08) against black‐box machine learning (ML) models trained on precise orbit determination‐derived thermospheric density data (from CHAMP, GOCE, GRACE, SWARM‐A/B satellites). We show that by using the same inputs, the ML models we designed are capable of consistently improving the predictions with respect to state‐of‐the‐art empirical models by reducing the mean absolute percentage error (MAPE) in the thermospheric density estimation from the range of 40%–60% to approximately 20%. As a result of this work, we introduce Karman: an open‐source Python software package developed during this study. Karman provides functionalities to ingest and preprocess thermospheric density, solar irradiance, and geomagnetic input data for ML readiness. Additionally, it facilitates developing and training ML models on the aforementioned data and benchmarking their performance at different altitudes, geographic locations, times, and solar activity conditions. Through this contribution, we offer the scientific community a comprehensive tool for comparing and enhancing thermospheric density models using ML techniques. Plain Language Summary Accurately modeling the density of the thermosphere is pivotal for spacecraft operations such as collision avoidance, re‐entry prediction, and orbital lifetime analysis. In this study, our aim is twofold. First, we want to study and compare the performance of data‐driven machine learning (ML) models in predicting thermospheric density data against standard empirical models used in the field, which are used as baseline. By training ML models using precise orbit determination‐derived satellite data, we show that they can achieve significant performance improvement compared to empirical models, with a reduction of 61% in the mean absolute percentage error. Second, we also provide the community with a shared software framework that supports the ingestion of solar irradiance, geomagnetic, and thermospheric density data, as well as a training and benchmarking framework to develop ML models. This framework allows researchers and operators to both train their ML models and to compare them at different periods of the solar cycle, geomagnetic storm conditions, geographical locations, and times. Key Points Machine learning (ML) models can significantly outperform existing physics‐based thermospheric neutral density models on exactly the same inputs ML models can improve over NRLMSISE‐00 and JB‐08 empirical density models by 61% and 39% respectively in the mean absolute percentage error The software allows the creation of ML‐ready data for training and benchmarking new models, supporting solar irradiance and geomagnetic data

Gonzalo Mateo-Garcia, Cesar Aybar, Giacomo Acciarini, Vit Ruzicka, Gabriele Meoni, Nicolas Longepe, Luis Gomez-Chova (2023)Onboard Cloud Detection and Atmospheric Correction with Deep Learning Emulators

This paper introduces DTACSNet, a Convolutional Neural Network (CNN) model specifically developed for efficient onboard atmospheric correction and cloud detection in optical Earth observation satellites. The model is developed with Sentinel-2 data. Through a comparative analysis with the operational Sen2Cor processor, DTACSNet demonstrates a significantly better performance in cloud scene classification (F2 score of 0.89 for DTACSNet compared to 0.51 for Sen2Cor v2.8) and a surface reflectance estimation with average absolute error below 2% in reflectance units. Moreover, we tested DTACSNet on hardware-constrained systems similar to recent deployed missions and show that DTACSNet is 11 times faster than Sen2Cor with a significantly lower memory consumption footprint. These preliminary results highlight the potential of DTACSNet to provide enhanced efficiency, autonomy, and responsiveness in onboard data processing for Earth observation satellite missions.

Shreshth Malik, James Walsh, Giacomo Acciarini, Thomas E Berger, Atılım Günes Baydin (2023)High-Cadence Thermospheric Density Estimation enabled by Machine Learning on Solar Imagery

Accurate estimation of thermospheric density is critical for precise modeling of satellite drag forces in low Earth orbit (LEO). Improving this estimation is crucial to tasks such as state estimation, collision avoidance, and re-entry calculations. The largest source of uncertainty in determining thermospheric density is modeling the effects of space weather driven by solar and geomagnetic activity. Current operational models rely on ground-based proxy indices which imperfectly correlate with the complexity of solar outputs and geomagnetic responses. In this work, we directly incorporate NASA's Solar Dynamics Observatory (SDO) extreme ultraviolet (EUV) spectral images into a neural thermospheric density model to determine whether the predictive performance of the model is increased by using space-based EUV imagery data instead of, or in addition to, the ground-based proxy indices. We demonstrate that EUV imagery can enable predictions with much higher temporal resolution and replace ground-based proxies while significantly increasing performance relative to current operational models. Our method paves the way for assimilating EUV image data into operational thermospheric density forecasting models for use in LEO satellite navigation processes.

Giacomo Acciarini, Cristian Greco, Massimiliano Vasile (2023)Uncertainty Propagation in Orbital Dynamics via Galerkin Projection of the Fokker-Planck Equation, In: Advances in Space Research Elsevier

The Fokker-Planck equation is a partial differential equation that describes how the probability density function of an object state varies, when subject to deterministic and random forces. The solution to this equation is crucial in many space applications, such as space debris trajectory tracking and prediction, guidance navigation and control under uncertainties, space situational awareness, and mission analysis and planning. However, no general closed-form solutions are known and several methods exist to tackle its solution. In this work, we use a known technique to transform this equation into a set of linear ordinary differential equations in the context of orbital dynamics. In particular, we show the advantages of the applied methodology, which allows to decouple the time and state-dependent components and to retain the entire shape of the probability density function through time, in the presence of both deterministic and stochastic dynamics. With this approach, the probability density function values at future times and for different initial conditions can be computed without added costs, provided that some time-independent integrals are solved offline. We showcase the efficacy and use of this method on some orbital dynamics example, by also leveraging the use of automatic differentiation for efficiently computing the involved derivatives.

Giacomo Acciarini, Nicola Baresi, David Lloyd, Dario Izzo (2023)Stochastic Continuation for Space Trajectory Design

This paper explores the application of stochastic continuation methods in the context of mission analysis for spacecraft trajectories around libration points in the planar circular restricted three-body problem. Traditional deterministic approaches have limitations in accounting for uncertainties, requiring a two-step process involving Monte Carlo techniques for assessing the robustness of the deterministic design. This might lead to suboptimal solutions and to a long and time-consuming design process. Stochastic continuation methods, which extend numerical continuation techniques to moments of probability density functions, offer a promising alternative. This paper aims to pioneer the application of stochastic continuation procedures in mission analysis, incorporating and acknowledging the stochastic nature of spacecraft missions from the early design phases. By extending existing frameworks to handle fixed points of stroboscopic or Poincaré mappings, the study focuses on robustifying and enhancing trajectory design by considering uncertainties in the determination of periodic orbits. The proposed approach has the potential to discover new solutions that may remain hidden in deterministic analyses, offering improved mission design outcomes. Specifically, this work concentrates on the planar circular restricted three-body problem, assuming uncertainties in both initial conditions and the mass ratio parameter. Stochastic continuation is employed to identify equilibrium points and periodic orbits in this uncertain dynamical system. The generalization of steady states and periodic orbits in uncertain environments is discussed, demonstrating the effectiveness of stochastic continuation in identifying safe operational regions in uncertain astrodynamics problems.

Giacomo Acciarini, Edward Brown, Chris Bridges, Atılım Günes Baydin, Thomas E Berger, Madhulika Guhathakurta (2023)Karman - a Machine Learning Software Package for Benchmarking Thermospheric Density Models

Recent events, such as the loss of 38 satellites by SpaceX due to a geomagnetic storm have highlighted the importance of having more accurate estimation and prediction of thermospheric density. Solar and geomagnetic activities wield significant influence over the behavior of the thermospheric density, exerting an important impact on spacecraft motion in low-Earth orbit (LEO). The impending Solar Cycle 25's peak arrives at a time when the number of operational satellites in LEO is surging, driven by the proliferation of mega-constellations. This escalating satellite presence, spanning sectors from defense to commercial applications, increases the intricacy of the operational environment. The accuracy of thermospheric neutral density models, which underpin crucial safety-oriented tasks like satellite collision avoidance and space traffic management, is therefore pivotal. While the importance of solar events on thermospheric density is apparent, currently, the influence of the Sun in thermospheric density models is only included in the form of solar proxies (such as F10.7). This can be underwhelming, leading to mispredictions of thermospheric density values. A shared framework that supports the ingestion of inputs from various sources to devise thermospheric density models, and where thermospheric density models can be compared, is currently lacking. Furthermore, the recent advancements in machine learning (ML) offer a unique opportunity to construct thermospheric density models that use these models to describe the relationship between the Sun and the Earth's thermosphere. For this reason, this study introduces an open-source software package, called Karman, to help solve this problem. Essential for this, are three steps: first, the preparation and ingestion of input data from several sources in an ML-readiness fashion. Then, the construction of ML models that can be trained on these datasets. Finally, the creation of a benchmarking platform to compare ML models against state-of-the-art empirical models, evaluating their performances under varying conditions, such as geomagnetic storm strength, altitude, and solar irradiance levels.The utility of this framework is demonstrated through various experiments, showcasing its effectiveness in both benchmarking density models and discerning factors driving thermospheric density variations. The study compares the performance of traditional empirical models (NRLMSISE-00 and JB-08) with machine learning models trained on identical inputs. The results reveal a consistent 20-40\% improvement in accuracy, highlighting the potential of machine learning techniques.One particularly significant area addressed by this research involves the incorporation of additional inputs to refine density estimations. Current approaches rely on solar proxies for estimating the Sun's impact on the thermosphere. However, it is suggested that direct Extreme Ultraviolet (EUV) irradiance data could enhance accuracy. The framework outlined in this paper enables the integration of such inputs, facilitating the validation of hypotheses and supporting the evolution of thermospheric density models.In conclusion, this study presents a comprehensive framework for advancing thermospheric density modeling in the context of LEO satellites. Through the development of neural network models, an extensive dataset, and a benchmarking platform, the paper contributes significantly to the improvement of satellite trajectory predictions. As the space environment becomes increasingly intricate, tools such as the presented framework are crucial for maintaining the safety and effectiveness of satellite operations in LEO.

Giacomo Acciarini, Nicola Baresi, Christopher Bridges, Leonard Felicetti, Stephen Hobbs, Atılım Günes Baydin (2023)Observation strategies and megaconstellations impact on current LEO population, In: Proceedings of the 2nd NEO and Debris Detection Conference (NEOSST2) ESA Space Debris Office

The risk of collisions in Earth’s orbit is growing markedly. In January 2021, SpaceX and OneWeb released an operator-to-operator fact sheet that highlights the critical reliance on conjunction data messages (CDMs) and observations, demonstrating the need for a diverse sensing environment for orbital objects. Recently, the University of Oxford and the University of Surrey developed, in collaboration with Trillium Technologies and the European Space Operations Center, an opensource Python package for modeling the spacecraft collision avoidance process, called Kessler. Such tools can be used for importing/exporting CDMs in their standard format, modeling the current low-Earth orbit (LEO) population and its short-term propagation from a given catalog file, as well as modeling the evolution of conjunction events based on the current population and observation scenarios, hence emulating the CDMs generation process of the Combined Space Operations Center (CSpOC). The model also provides probabilistic programming and ML tools to predict future collision events and to perform Bayesian inference (i.e., optimal use of all available observations). In the framework of a United Kingdom Space Agency-funded project, we analyze and study the impact of megaconstellations and observation models in the collision avoidance process. First, we monitor and report how the estimated collision risk and other quantities at the time of closest approach (e.g. miss distance, uncertainties, etc.) vary, according to different observation models, which emulate different radar observation accuracy. Then, we analyze the impact of future megaconstellations on the number of warnings generated from the increase in the number of conjunctions leading to an increased burden on space operators. FCC licenses were used to identify credible megaconstellation sources to understand how a potential consistent increase in active satellites will impact LEO situational safety. We finally present how our simulations help understand the impact of these future megaconstellations on the current population, and how we can devise better ground observation strategies to quantify future observation needs and reduce the burden on operators.