Dario Izzo

Zhong Zhang, Giacomo Acciarini, Dario Izzo et al.

Low-thrust trajectory design relies heavily on repeated evaluations of fuel consumption and transfer feasibility, which require expensive optimal control solutions. In this work, we show these quantities can be accurately approximated by machine learning surrogates, enabling fast and scalable evaluation across a wide range of scenarios. By increasing both dataset size and model capacity, we observe that low-thrust trajectory optimization follows a scaling law, with performance improving linearly with the logarithm of training data and network parameters, and no evidence of saturation within the explored regime. Guided by this observation, we construct a large-scale dataset using the proposed homotopy-ray strategy tailored to mission design requirements. A key is the introduction of a self-similar transformation, which allows generalization across semi-major axes, inclinations, and central bodies avoiding retraining. As a result, the same neural approximator can be applied to diverse orbital environments and mission classes. The proposed models accurately predict optimal fuel consumption and minimum transfer time for single- and multi-revolution transfers. Their performance and generalization are demonstrated on a public dataset, a multi-asteroid flyby problem from the Global Trajectory Optimization Competition, and an asteroid rendezvous mission design. The models and datasets are released as open-source to support the space community.

Carlos Sánchez-Sánchez, Dario Izzo

Recent research on deep learning, a set of machine learning techniques able to learn deep architectures, has shown how robotic perception and action greatly benefits from these techniques. In terms of spacecraft navigation and control system, this suggests that deep architectures may be considered now to drive all or part of the on-board decision making system. In this paper this claim is investigated in more detail training deep artificial neural networks to represent the optimal control action during a pinpoint landing, assuming perfect state information. It is found to be possible to train deep networks for this purpose and that the resulting landings, driven by the trained networks, are close to simulated optimal ones. These results allow for the design of an on-board real time optimal control system able to cope with large sets of possible initial states while still producing an optimal response.

Dario Izzo, Alexander Hadjiivanov, Dominik Dold et al.

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).

Dario Izzo, Sebastien Origer

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 $ΔV$ 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.

Dario Izzo, Gabriele Meoni, Pablo Gómez et al.

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.

Sofia McLeod, Gabriele Meoni, Dario Izzo et al.

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.

Marcus Märtens, Dario Izzo

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.

Loïc J. Azzalini, Emmanuel Blazquez, Alexander Hadjiivanov et al.

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.

Dario Izzo, Daniel Hennes, Marcus Märtens et al.

We consider the interplanetary trajectory design problem posed by the 8th edition of the Global Trajectory Optimization Competition and present the end-to-end strategy developed by the team ACT-ISAS (a collaboration between the European Space Agency's Advanced Concepts Team and JAXA's Institute of Space and Astronautical Science). The resulting interplanetary trajectory won 1st place in the competition, achieving a final mission value of $J=146.33$ [Mkm]. Several new algorithms were developed in this context but have an interest that go beyond the particular problem considered, thus, they are discussed in some detail. These include the Moon-targeting technique, allowing one to target a Moon encounter from a low Earth orbit; the 1-$k$ and 2-$k$ fly-by targeting techniques, enabling one to design resonant fly-bys while ensuring a targeted future formation plane% is acquired at some point after the manoeuvre ; the distributed low-thrust targeting technique, admitting one to control the spacecraft formation plane at 1,000,000 [km]; and the low-thrust optimization technique, permitting one to enforce the formation plane's orientations as path constraints.

Marcus Märtens, Dario Izzo, Emmanuel Blazquez et al.

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.

Leone Costi, Dario Izzo

In this work, we propose a simulator, based on the open-source physics engine MuJoCo, for the design and control of soft robotic nets for the autonomous removal of space debris. The proposed simulator includes net dynamics, contact between the net and the debris, self-contact of the net, orbital mechanics, and a controller that can actuate thrusters on the four satellites at the corners of the net. It showcases the case of capturing Envisat, a large ESA satellite that remains in orbit as space debris following the end of its mission. This work investigates different mechanical models, which can be used to simulate the net dynamics, simulating various degrees of compliance, and different control strategies to achieve the capture of the debris, depending on the relative position of the net and the target. Unlike previous works on this topic, we do not assume that the net has been previously ballistically thrown toward the target, and we start from a relatively static configuration. The results show that a more compliant net achieves higher performance when attempting the capture of Envisat. Moreover, when paired with a sliding mode controller, soft nets are able to achieve successful capture in 100% of the tested cases, whilst also showcasing a higher effective area at contact and a higher number of contact points between net and Envisat.

Giacomo Acciarini, Dario Izzo, Zhong Zhang

Reachability analysis plays a central role in low-thrust spacecraft trajectory optimization by identifying which target states can be achieved under constraints on time, thrust, and propellant. Classical approaches construct reachable sets by solving many optimal control problems over grids of terminal states, requiring extensive forward simulations with fixed initial conditions. While effective, this approach is computationally expensive and becomes impractical for high-dimensional systems or strongly nonlinear dynamics, such as those encountered in cislunar environments or solar sail missions. This work introduces a dual formulation of the reachability problem. Instead of computing reachable sets directly, we determine, for fixed transfer time and boundary conditions, the maximum allowable initial mass (or, for solar sails, a scalar sail-strength parameter) that permits a successful transfer. A target is reachable if the spacecraft's initial mass does not exceed this threshold. This reformulation reduces reachability assessment to a scalar optimization problem for each target, producing a smooth scalar field that encodes equivalent feasibility information to classical reachable sets. We develop indirect maximum-initial-mass (MIM) formulations for both electric low-thrust and solar-sail dynamics and show how they can serve as efficient reachability oracles. Building on this formulation, we construct data-driven surrogate models to approximate the MIM-based reachability indicator. We investigate fully connected neural networks and demonstrate that residual networks provide the best trade-off between accuracy, training stability, and model complexity. The resulting surrogates enable rapid reachability evaluation while preserving the numerical advantages of the dual formulation, offering a practical tool for preliminary mission design and feasibility assessment.

Sebastien Origer, Dario Izzo, Giacomo Acciarini et al.

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.

Giacomo Acciarini, Francesco Biscani, Dario Izzo

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.

Jonas Schuhmacher, Fabio Gratl, Dario Izzo et al.

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

Loïc J. Azzalini, Dario Izzo

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.

Giacomo Acciarini, Atılım Güneş Baydin, Dario Izzo

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.

Dario Izzo, Marcus Märtens, Laurent Beauregard et al.

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.

Zacharia A. Rudge, Dario Izzo, Moritz Fieback et al.

In recent years, the space community has been exploring the possibilities of Artificial Intelligence (AI), specifically Artificial Neural Networks (ANNs), for a variety of on board applications. However, this development is limited by the restricted energy budget of smallsats and cubesats as well as radiation concerns plaguing modern chips. This necessitates research into neural network accelerators capable of meeting these requirements whilst satisfying the compute and performance needs of the application. This paper explores the use of Phase-Change Memory (PCM) and Resistive Random-Access Memory (RRAM) memristors for on-board in-memory computing AI acceleration in space applications. A guidance and control neural network (G\&CNET) accelerated using memristors is simulated in a variety of scenarios and with both device types to evaluate the performance of memristor-based accelerators, considering device non-idealities such as noise and conductance drift. We show that the memristive accelerator is able to learn the expert actions, though challenges remain with the impact of noise on accuracy. We also show that re-training after degradation is able to restore performance to nominal levels. This study provides a foundation for future research into memristor-based AI accelerators for space, highlighting their potential and the need for further investigation.

Dario Izzo, Sebastien Origer, Giacomo Acciarini et al.

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.

Dominik Dold, Amy Thomas, Nicole Rosi et al.

Recently, a class of mechanical lattices with reconfigurable, zero-stiffness structures has been proposed, called Totimorphic lattices. In this work, we introduce a computational framework that enables continuous reprogramming of a Totimorphic lattice's effective properties, such as mechanical and optical behaviour, through geometric changes alone, demonstrated using computer simulations. Our approach is differentiable and guarantees valid Totimorphic configurations throughout the optimisation process, providing not only target states with desired properties but also continuous trajectories in configuration space that connect them. This enables reprogrammable structures in which actuators are controlled via automatic differentiation on an objective-dependent cost function, continuously adapting the lattice to achieve a given goal. We focus on deep space applications, where harsh and resource-constrained environments demand solutions that combine flexibility, efficiency, and autonomy. As proof of concept, we present two scenarios: a reprogrammable disordered lattice material and a space telescope mirror with adjustable focal length. The introduced framework is adaptable to a wide range of Totimorphic designs and objectives, providing a lightweight model for endowing physical systems with autonomous self-configuration and self-repair capabilities.

Pietro Fanti, Dario Izzo

The geodesy of irregularly shaped small bodies presents fundamental challenges for gravitational field modeling, particularly as deep space exploration missions increasingly target asteroids and comets. Traditional approaches suffer from critical limitations: spherical harmonics diverge within the Brillouin sphere where spacecraft typically operate, polyhedral models assume unrealistic homogeneous density distributions, and existing machine learning methods like GeodesyNets and Physics-Informed Neural Networks (PINN-GM) require extensive computational resources and training time. This work introduces MasconCubes, a novel self-supervised learning approach that formulates gravity inversion as a direct optimization problem over a regular 3D grid of point masses (mascons). Unlike implicit neural representations, MasconCubes explicitly model mass distributions while leveraging known asteroid shape information to constrain the solution space. Comprehensive evaluation on diverse asteroid models including Bennu, Eros, Itokawa, and synthetic planetesimals demonstrates that MasconCubes achieve superior performance across multiple metrics. Most notably, MasconCubes demonstrate computational efficiency advantages with training times approximately 40 times faster than GeodesyNets while maintaining physical interpretability through explicit mass distributions. These results establish MasconCubes as a promising approach for mission-critical gravitational modeling applications requiring high accuracy, computational efficiency, and physical insight into internal mass distributions of irregular celestial bodies.

Nadezhda Dobreva, Emmanuel Blazquez, Jai Grover et al.

We present a decentralized model for autonomous reconfiguration of homogeneous pivoting cube modular robots in two dimensions. Each cube in the ensemble is controlled by a neural network that only gains information from other cubes in its local neighborhood, trained using reinforcement learning. Furthermore, using geometric deep learning, we include the grid symmetries of the cube ensemble in the neural network architecture. We find that even the most localized versions succeed in reconfiguring to the target shape, although reconfiguration happens faster the more information about the whole ensemble is available to individual cubes. Near-optimal reconfiguration is achieved with only nearest neighbor interactions by using multiple information passing between cubes, allowing them to accumulate more global information about the ensemble. Compared to standard neural network architectures, using geometric deep learning approaches provided only minor benefits. Overall, we successfully demonstrate mostly local control of a modular self-assembling system, which is transferable to other space-relevant systems with different action spaces, such as sliding cube modular robots and CubeSat swarms.

Zacharia A. Rudge, Dominik Dold, Moritz Fieback et al.

Memristors are an emerging technology that enables artificial intelligence (AI) accelerators with high energy efficiency and radiation robustness -- properties that are vital for the deployment of AI on-board spacecraft. However, space applications require reliable and precise computations, while memristive devices suffer from non-idealities, such as device variability, conductance drifts, and device faults. Thus, porting neural networks (NNs) to memristive devices often faces the challenge of severe performance degradation. In this work, we show in simulations that memristor-based NNs achieve competitive performance levels on on-board tasks, such as navigation \& control and geodesy of asteroids. Through bit-slicing, temporal averaging of NN layers, and periodic activation functions, we improve initial results from around $0.07$ to $0.01$ and $0.3$ to $0.007$ for both tasks using RRAM devices, coming close to state-of-the-art levels ($0.003-0.005$ and $0.003$, respectively). Our results demonstrate the potential of memristors for on-board space applications, and we are convinced that future technology and NN improvements will further close the performance gap to fully unlock the benefits of memristors.

Harry Holt, Sebastien Origer, Dario Izzo

Guidance & control networks (G&CNETs) provide a promising alternative to on-board guidance and control (G&C) architectures for spacecraft, offering a differentiable, end-to-end representation of the guidance and control architecture. When training G&CNETs, two predominant paradigms emerge: behavioural cloning (BC), which mimics optimal trajectories, and reinforcement learning (RL), which learns optimal behaviour through trials and errors. Although both approaches have been adopted in G&CNET related literature, direct comparisons are notably absent. To address this, we conduct a systematic evaluation of BC and RL specifically for training G&CNETs on continuous-thrust spacecraft trajectory optimisation tasks. We introduce a novel RL training framework tailored to G&CNETs, incorporating decoupled action and control frequencies alongside reward redistribution strategies to stabilise training and to provide a fair comparison. Our results show that BC-trained G&CNETs excel at closely replicating expert policy behaviour, and thus the optimal control structure of a deterministic environment, but can be negatively constrained by the quality and coverage of the training dataset. In contrast RL-trained G&CNETs, beyond demonstrating a superior adaptability to stochastic conditions, can also discover solutions that improve upon suboptimal expert demonstrations, sometimes revealing globally optimal strategies that eluded the generation of training samples.

Giacomo Acciarini, Dario Izzo, Francesco Biscani

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.

Sebastien Origer, Dario Izzo

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.

Dario Izzo, Emmanuel Blazquez, Robin Ferede et al.

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.

Tae Ha Park, Marcus Märtens, Gurvan Lecuyer et al.

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.

Dario Izzo, Pablo Gómez

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.

Anne Mergy, Gurvan Lecuyer, Dawa Derksen et al.

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.

Dawa Derksen, Dario Izzo

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.

Thomas Uriot, Dario Izzo, Luís F. Sim{õ}es et al.

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.

Thomas Uriot, Dario Izzo

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.

Dario Izzo, Ekin Öztürk

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.

Shuo Li, Ekin Ozturk, Christophe De Wagter et al.

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.

Mate Kisantal, Sumant Sharma, Tae Ha Park et al.

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.

Marcus Märtens, Dario Izzo

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.

Marcus Märtens, Dario Izzo, Andrej Krzic et al.

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.

Dario Izzo, Ekin Öztürk, Marcus Märtens

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.

Christopher Iliffe Sprague, Dario Izzo, Petter Ögren

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.

Dharmesh Tailor, Dario Izzo

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.

Dario Izzo, Marcus Märtens, Binfeng Pan

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.

Dario Izzo, Dharmesh Tailor, Thomas Vasileiou

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.

Dario Izzo, Christopher Sprague, Dharmesh Tailor

After providing a brief historical overview on the synergies between artificial intelligence research, in the areas of evolutionary computations and machine learning, and the optimal design of interplanetary trajectories, we propose and study the use of deep artificial neural networks to represent, on-board, the optimal guidance profile of an interplanetary mission. The results, limited to the chosen test case of an Earth-Mars orbital transfer, extend the findings made previously for landing scenarios and quadcopter dynamics, opening a new research area in interplanetary trajectory planning.

Luís F. Simões, Dario Izzo, Evert Haasdijk et al.

The design of spacecraft trajectories for missions visiting multiple celestial bodies is here framed as a multi-objective bilevel optimization problem. A comparative study is performed to assess the performance of different Beam Search algorithms at tackling the combinatorial problem of finding the ideal sequence of bodies. Special focus is placed on the development of a new hybridization between Beam Search and the Population-based Ant Colony Optimization algorithm. An experimental evaluation shows all algorithms achieving exceptional performance on a hard benchmark problem. It is found that a properly tuned deterministic Beam Search always outperforms the remaining variants. Beam P-ACO, however, demonstrates lower parameter sensitivity, while offering superior worst-case performance. Being an anytime algorithm, it is then found to be the preferable choice for certain practical applications.

Dario Izzo, Francesco Biscani, Alessio Mereta

We introduce the use of high order automatic differentiation, implemented via the algebra of truncated Taylor polynomials, in genetic programming. Using the Cartesian Genetic Programming encoding we obtain a high-order Taylor representation of the program output that is then used to back-propagate errors during learning. The resulting machine learning framework is called differentiable Cartesian Genetic Programming (dCGP). In the context of symbolic regression, dCGP offers a new approach to the long unsolved problem of constant representation in GP expressions. On several problems of increasing complexity we find that dCGP is able to find the exact form of the symbolic expression as well as the constants values. We also demonstrate the use of dCGP to solve a large class of differential equations and to find prime integrals of dynamical systems, presenting, in both cases, results that confirm the efficacy of our approach.

Kevin van Hecke, Guido de Croon, Laurens van der Maaten et al.

Self-Supervised Learning (SSL) is a reliable learning mechanism in which a robot uses an original, trusted sensor cue for training to recognize an additional, complementary sensor cue. We study for the first time in SSL how a robot's learning behavior should be organized, so that the robot can keep performing its task in the case that the original cue becomes unavailable. We study this persistent form of SSL in the context of a flying robot that has to avoid obstacles based on distance estimates from the visual cue of stereo vision. Over time it will learn to also estimate distances based on monocular appearance cues. A strategy is introduced that has the robot switch from stereo vision based flight to monocular flight, with stereo vision purely used as 'training wheels' to avoid imminent collisions. This strategy is shown to be an effective approach to the 'feedback-induced data bias' problem as also experienced in learning from demonstration. Both simulations and real-world experiments with a stereo vision equipped AR drone 2.0 show the feasibility of this approach, with the robot successfully using monocular vision to avoid obstacles in a 5 x 5 room. The experiments show the potential of persistent SSL as a robust learning approach to enhance the capabilities of robots. Moreover, the abundant training data coming from the own sensors allows to gather large data sets necessary for deep learning approaches.