## COMPUTATIONAL ASTRODYNAMICS RESEARCH

## Dynamics in near-Earth space

As near-Earth space becomes more congested, novel computational methods are needed to coordinate space traffic and avoid potentially catastrophic collisions.

### Regularized orbit propagation

I have developed high-accuracy numerical orbit propagation algorithms based on **regularized formulations of orbital mechanics**. The basic idea consists in performing a change of variables that makes the integration of the equations of motion more amenable to numerical schemes. The equations of motion are rewritten such that singularities are eliminated, and a so-called *analytical step-size regulation* is introduced to slow down the integration at periapsis, when the solution varies fastest. This is in fact a linearization of the dynamical system without introducing approximations, unlike Taylor and asymptotic expansions.

Regularized formulations are significantly more efficient than state-of-art orbit propagation methods, such as semi-analytical techniques, especially for MEOs and GTOs. Because they are non-averaged methods, they can easily be used for conjunction analysis across all orbital regimes.

### Dynamics in the Earth-Moon-Sun system

Regularized formulations implemented in the THALASSA orbit propagator have been used in several studies on the dynamics of resident space objects in the Earth-Moon-Sun system. With this approach, I showed that **resident space objects do not follow classical Lidov-Kozai cycles**, i.e., solutions of the circular restricted three-body problem that can lead to eccentricity growth and Earth collision, due to the presence of the Moon. Thus, Lidov-Kozai cycles cannot be used as disposal solutions without an in-depth numerical analysis.

I performed a numerical cartography of the LEO regime close to the Starlink mega-constellation region, which excluded the possibility of de-orbiting constellation elements purely through passive dynamical means (as proposed for the Galileo constellation in MEO). Furthermore, regularized formulations enabled a pioneering brute-force **conjunction analysis study for Starlink** that evidenced the need for careful orbit design to prevent collisions among constellation elements.

### Current developments

Alternative representations of a spacecraftâ€™s state can also improve the realism in propagating uncertainty, which is of relevance for stochastic control. I am currently exploring the application of regularized formulations to conjunction analysis and robust spacecraft guidance.

## Robust guidance for Entry, Descent, and Landing

Planetary orbit insertion and Entry, Descent, and Landing (EDL) present significant challenges for future spacecraft, which will carry payloads at least ten times larger than the current generation. In the case of planetary bodies with an atmosphere, spacecraft and entry vehicles must fly through atmospheres which are often unknown, with deceleration systems whose performance is difficult to characterize on-ground, face vexing aerothermal loads, and land with sufficient reliability and accuracy.

I am developing novel guidance algorithms that leverage **machine learning algorithms** to improve the on-board knowledge of the dynamics, and employ **uncertainty quantification methods** to make the guidance algorithm more robust through heuristic retargeting.

In particular, I devised a deep learning method for the reconstruction of density and wind during Martian atmospheric entry that is more general than other parametric methods, such as Kalman and fading memory filters.

## Scientific ML for Initial and Boundary Value Problems

In contrast to classical, time-marching methods, scientific machine learning algorithms assume that a PDE solution can be represented through a parametric functional. During training, the parameters of the functional are tuned such that the output function satisfies the PDE both within the solution domain and at its boundary.

Although ODE solutions are generally less computationally demanding than PDEs, they present their own challenges. In current approaches, the neural network functional must be re-trained (i.e., a new solution must be produced through training) if the boundary conditions change. In addition, although the SciML approach has proven effective for toy problems on relatively short time scales, its accuracy in reproducing long-term solutions is unexplored (especially for ODEs with oscillatory solutions).

I am investigating the generalization in time (long-term accuracy) and space (sets of initial conditions) of SciML ODE solutions.

## Software

**THALASSA (Tool for High-Accuracy, Long-term Analyses for SSA)**propagates orbits for bodies in the Earth-Moon-Sun system. Written in Fortran, it integrates either Newtonian equations in Cartesian coordinates or regularized equations of motion with the LSODAR (Livermore Solver for Ordinary Differential equations with Automatic Root-finding). THALASSA is a command-line tool; the repository also includes some Python3 scripts to perform batch propagations. Available on GitLab.**NAPLES (Numerical Analysis of PLanetary Encounters)**performs batch propagations of close encounters in the three-body problem and computes the numerical error with respect to reference trajectories computed in quadruple precision. It uses the LSODAR integrator from ODEPACK (ascl:1905.021) and the equations of motion correspond to several regularized formulations. Available on GitHub.