Modelling Spacecraft-Surface Interactions in Low-Gravity Environments for Inferring Surface Mechanical Properties

Interactions between spacecrafts and the surfaces of small solar-system bodies (SSB), e.g. through landing or sampling, are incredibly valuable in terms of scientific return as the spacecraft-surface interaction provides direct information on the internal structure and material properties of the SSB while their instruments can do some in-situ measurements to characterize the SSB in more depth. Many missions have attempted, or are planning to attempt, interactions with the surface of a SSB. For example, the upcoming Hera mission will attempt to land its two CubeSats on the surface of asteroids Didymos and Dimorphos, while JAXA's Martian Moons eXploration (MMX) will deploy a rover on the surface of the Martian moon Phobos. One of the main difficulties for these missions is the uncertainty related to the surface properties of SSBs. A large number of SSBs are thought to have a large amount of granular material, also known as regolith, and boulders on their surface [1]. The interactions between the individual regolith particles, the boulders, and the spacecraft itself are governed by complex contact forces and are hard to model on Earth due to the low-gravity environment on these SSBs. There are two main methods for numerical modelling the surface of SSBs: the hard and soft surface method. First, the hard surface methods use a small amount of parameters that abstract away the different effects of the internal interactions happening within the regolith when in contact with the spacecraft. On the other hand, soft surface methods model the regolith using a large number of discrete bodies and simulate fully their interactions. This increases the fidelity of the model greatly and works well when trying to estimate surface properties from observing the landing, but drastically increases the numerical complexity. In this work, a novel approach is proposed which aims to generate an efficient model of the spacecraft-surface interaction like the hard surface models, but which retain the accuracy of the soft surface model. This is achieved using a data-driven approach where a small number of soft surface simulations are used to fit a model that can be more efficiently used. This model is created by a non-intrusive polynomial chaos expansion (PCE) method, which samples the initial conditions and system parameters in an optimal way, and takes the input-output relationship of these sample simulations to create a fit based on a family of orthogonal polynomials that represent the prior distribution of the input variables.  A discrete element modelling (DEM) technique is used to simulate the motion of the surface particles, the spacecraft, and their interaction. In this work, specifically the GRAINS software developed in [2] is used. A smooth contact model is chosen as this works best for cases where continuous contacts are present, as is the case for the settled surface. An example simulation of a spacecraft lander interacting with the surface is presented in Figure (1), where a constant acceleration of 0.5e-4 m/s^2 is used, representing roughly the surface acceleration of bodies like Didymos and Apophis.  Figure 1: Example landing simulation.   Using the DEM simulation environment, the PCE model can be generated as mentioned before. There are various properties of the PCE model that can be utilised. First, the PCE model allows for an analytical formulation of the moments of the output distribution through the orthogonal property of the polynomial basis. Using the analytical description of the variance, the first order Sobol indices can be calculated as well. These indices measure the expected reduction in the variance of the output  when fixing the input parameter. Thus, they also measure the contribution of the variance of an input parameter to the output parameter, allowing for a sensitivity analysis to determine which parameters are critical for the interaction. The other important property of the PCE model is that it is efficient to sample. This allows for the use of methods like Markov Chain Monte Carlo (MCMC) to perform inverse modelling, which requires sampling the simulation many times to estimate the probability distribution of the input parameters given a set of output observations. Where the DEM simulation would take hours to perform a single simulation, the PCE surrogate model is much more efficient. An example results is given in Figure (2), where a landing was simulated similar to Figure (1), and the PCE-MCMC method was used to estimate the probability distribution of the value of the surface properties given a certain observed outgoing velocity of the lander.   Figure 2: Estimated distribution of surface properties from an observed landing using the PCE-MCMC method. The solid lines are the true values.   Concluding, this work will show how to create an efficient and accurate model for the interaction between the surface of an SSB and a spacecraft. This model can be used to interpret the sensitivity of the post-contact dynamics of the spacecraft to uncertainties in the surface properties. It can then also be used to perform reliability analyses where the chance of failure after interaction is calculated as a function of the uncertainty in the surface properties. Finally it allows for estimating the actual surface properties from the interaction itself. This methodology will thus enable mission designers to perform robust and efficient interactions between the spacecraft and the surface of SSBs, creating novel science and exploration opportunities.   Acknowledgements: Funded by the European Union (ERC, TRACES, 101077758). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.   References: [1] N. Murdoch, M. Drilleau, C. Sunday, F. Thuillet, A. Wilhelm, G. Nguyen, and Y. Gourinat, “Low-velocity impacts into granular material: application to small-body landing,” Monthly Notices of the Royal Astronomical Society, Vol. 503, 4 2021, pp. 3460–3471, 10.1093/MN-RAS/STAB624.[2] F. Ferrari, A. Tasora, P. Masarati, and M. Lavagna, “N-body gravitational and contact dynamics for asteroid aggregation,” Multibody System Dynamics, Vol. 39, 1 2017, pp. 3–20, 10.1007/S11044-016-9547-2/FIGURES/9.