Modular pipeline for small bodies gravity field modeling: an efficient representation of variable density spherical harmonics coefficients

TL;DR

The paper addresses the challenge of modeling small-body gravity fields with non-uniform density for autonomous GNC. It introduces a modular pipeline that computes variable-density spherical-harmonics coefficients from a polyhedral shape via radial discretization, yielding a coherent SH-based gravity-field model. The method provides an exact semi-analytic evaluation of the normalized coefficients $\overline{C}_{n,m}$ and $\overline{S}_{n,m}$ for arbitrary density distributions by subdividing the body into tetrahedra and using Beta-function integrals. Validation across spheres, Arrokoth, and Eros demonstrates high accuracy (CoM errors < 1 m; acceleration errors < 1 mGal) and strong efficiency, with density-variation-driven trajectory differences of about 10 km over 24 h. The results support onboard GNC testing and hardware-in-the-loop simulations, enabling robust planning under density-structure uncertainties.

Abstract

Proximity operations to small bodies, such as asteroids and comets, demand high levels of autonomy to achieve cost-effective, safe, and reliable Guidance, Navigation and Control (GNC) solutions. Enabling autonomous GNC capabilities in the vicinity of these targets is thus vital for future space applications. However, the highly non-linear and uncertain environment characterizing their vicinity poses unique challenges that need to be assessed to grant robustness against unknown shapes and gravity fields. In this paper, a pipeline designed to generate variable density gravity field models is proposed, allowing the generation of a coherent set of scenarios that can be used for design, validation, and testing of GNC algorithms. The proposed approach consists in processing a polyhedral shape model of the body with a given density distribution to compute the coefficients of the spherical harmonics expansion associated with the gravity field. To validate the approach, several comparison are conducted against analytical solutions, literature results, and higher fidelity models, across a diverse set of targets with varying morphological and physical properties. Simulation results demonstrate the effectiveness of the methodology, showing good performances in terms of modeling accuracy and computational efficiency. This research presents a faster and more robust framework for generating environmental models to be used in simulation and hardware-in-the-loop testing of onboard GNC algorithms.

Figures (8)

• Figure 1: Representation of the single tetrehedron of the polyhedral shape model. (a) Represents the tetrahedron in the asteroid fixed frame $\mathcal{B}$, while (b) shows the standard simplex obtained after the coordinates change.
• Figure 2: Radial discretization of the tetrahedron
• Figure 3: CoM estimation comparison, the cross indicates the one computed with Eq.\ref{['eq: CoM']} while the circle the reference solution. (a) Uniform density sphere, (b) Variable density sphere, (c) Uniform density model of (486958) Arrokoth asteroid.
• Figure 4: Error in the acceleration obtained comparing the variable density spherical harmonics model discussed in this section with a Mascon model applied to a constant density model of Eros.
• Figure 5: Assumed density distribution for Eros with an heavier inner core.