Trajectory Optimization of a Swarm Orbiting 67P/Churyumov-Gerasimenko Maximising Gravitational Signal
Authors:   Advanced Concepts Team, European Space Agency, European Space Research and Technology Centre (ESTEC), Noordwijk, Netherlands, Maråk Rasmus , Blazquez Emmanuel , Gómez Pablo 
Published: January 2024

Proper modelling of the gravitational fields of irregular asteroids and comets is an essential yet difficult part of any spacecraft visit and flyby to these bodies. Accurate density representations provide crucial information, e.g., for proximity operations of spacecraft near such bodies which rely heavily on it to design safe and efficient trajectories. [2] Recently, so-called neural density fields [1] have emerged as a versatile tool that can provide an accurate description of the density distribution of a body’s mass, internal and external shape with few prior requirements. This representation has several advantages as it requires no prior information on the body, converges even inside the Brillouin sphere, and is extensible even to heterogeneous density distributions of the celestial bodies. [1,3] However, it remains an open question whether there are feasible, achievable trajectories that provide sufficient information, i.e. gravitational signal, to model the gravity and density field of an irregular body with high fidelity using neural density fields. For instance, a previous study demonstrated that the planned trajectory of the OSIRIS-REx spacecraft around Bennu produced a gravitational signal that proved to be too sparse for the task [4]. This difficulty could be circumvented using a distributed data acquisition and learning approach, where a swarm of spacecraft instead of a single one would work to acquire the gravity signal and learn a body’s density field.
In this work, we explore maximising the gravitational signal in a hypothetical mission around the comet 67P/Churyumov-Gerasimenko by using a swarm of spacecraft. Sets of spacecraft trajectories are simultaneously optimised to maximise overall signal return while minimising propellant budget for orbital manoeuvres. This proves to be a challenging optimization problem due to the complex topology of 67P’s gravitational field and its sidereal rotation. [5] In contrast to a single spacecraft scenario, this mission context allows us to improve the acquisition of the gravitational signal through multiple, simultaneous relative observation angles. Orbit propagation is based on an open-source polyhedral gravity model [6] using a detailed mesh of 67P and takes the comet’s sidereal rotation into account.
Trajectory optimization routines rely on the open-source pygmo framework maintained by ESA’s Advanced Concepts Team to formulate the problem as a constrained, multi-objective optimization problem. The developed code is designed independently of the celestial body of interest and provided online to allow follow-up studies with related models on other bodies. Constraints considered for this application include partial line of sight with the rest of the swarm, absence of collision with the comet, spacecraft power generation and telemetry. A parameter of particular interest for this study is the number of spacecraft constituting the swarm, in order to investigate the potential benefits and signal return thresholds when using a distributed approach. We compare results on different formation flying scenarios with varying complexity of the imposed constraints. Further, we consider heterogeneous measures of the gravitational signal characterised by different view angles and altitudes with respect to the celestial body. Additionally, we can directly correlate richness of the obtained dataset of trajectories to the duration of the mission. Based on a dataset of points and accelerations for the swarm after varying amounts of time, we investigate the training of a geodesyNet to model P67’s mass density field. We compare the obtained fidelity in detail with the established synthetic training using randomly sampled points as in the original work on geodesyNet [1]. Thus, we can directly relate the signal obtainable in a real mission scenario with an ideal one.
Overall, this work takes the next step in bringing neural density fields to an onboard mission scenario, where they can be a useful and potent tool complementing existing approaches such as polyhedral or mascon models. In practice, the developed, open-source code can serve as a testbed to evaluate whether a hypothetical mission scenario can reasonably rely on geodesyNets and neural density fields. It also serves as a first step in investigating the potential of autonomously learning small-body gravitational fields in a distributed fashion. The tools used for this study are fully available online and designed to be extended to more general distributed learning applications with spacecraft swarms around small celestial bodies.
References
[1] Izzo, D. and Gómez, P., 2021. Geodesy of irregular small bodies via neural density fields: geodesyNets. arXiv preprint arXiv:2105.13031.
[2] Leonard, J.M., Geeraert, J.L., Page, B.R., French, A.S., Antreasian, P.G., Adam, C.D., Wibben, D.R., Moreau, M.C. and Lauretta, D.S., 2019, August. OSIRIS-REx orbit determination performance during the navigation campaign. In 2019 AAS/AIAA Astrodynamics Specialist Conference (pp. 1-20).
[3] Cui, Pingyuan, and Dong Qiao. "The present status and prospects in the research of orbital dynamics and control near small celestial bodies." Theoretical and Applied Mechanics Letters 4.1 (2014): 013013.
[4] von Looz, M., Gómez, P. and Izzo, D., 2021. Study of the asteroid Bennu using geodesyANNs and Osiris-Rex data. arXiv preprint arXiv:2109.14427.
[5] Keller, H.U., Mottola, S., Skorov, Y. and Jorda, L., 2015. The changing rotation period of comet 67P/Churyumov-Gerasimenko controlled by its activity. Astronomy & Astrophysics, 579, p.L5.
[6] Schuhmacher, J. (2022) Efficient Polyhedral Gravity Modeling in Modern C++. Technical University of Munich.