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Applying data mining techniques to higher-dimensional Poincaré maps in the circular restricted three-body problem

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Abstract

Poincaré maps are regularly used to facilitate rapid and informed trajectory design within multi-body systems. However, maps that capture a general set of spatial trajectories are often higher-dimensional and, as a result, challenging for a human to analyze. This paper addresses this challenge by employing techniques from data mining. Specifically, distributed clustering, dimension reduction and classification are used in combination to construct a data-driven approach to autonomously group higher-dimensional crossings on a Poincaré map according to the geometry of the associated trajectories generated over a short time interval. This procedure is demonstrated using a periapsis map that captures spatial trajectories at a single energy level in the Sun-Earth circular restricted three-body problem. Arcs along hyperbolic invariant manifolds associated with families of tori in the \(L_{1}\) and \(L_{2}\) gateways are also projected onto this clustering result to rapidly extract their fundamental geometries. Together, these examples demonstrate the potential for the presented data-driven approach to facilitate analysis of a complex solution space reflected on a higher-dimensional Poincaré map.

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Acknowledgements

This work was completed at the University of Colorado Boulder, partially funded under NASA Grant 80NSSC18K1536. The authors thank the anonymous reviewers for their feedback.

Funding

This work was completed at the University of Colorado Boulder, partially funded under NASA Grant 80NSSC18K1536.

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Authors and Affiliations

  1. Smead Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO, 80303, USA

    Stefano Bonasera & Natasha Bosanac

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  1. Stefano Bonasera
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  2. Natasha Bosanac
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Correspondence to Natasha Bosanac.

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An earlier version of this paper was presented in January 2020 as Paper AIAA 2020-2178 at the 30th AIAA/AAS Space Flight Mechanics Meeting in Orlando, FL.

Appendices

11 Appendix 1

See Figs. 13 and 14.

Fig. 13
figure 13

Subset of representatives for global clustering result summarizing trajectories associated with prograde perigees and generated for up to three returns to a perigee map in the Sun-Earth CR3BP at \(C_{J}\) = 3.00088, with the libration points displayed as magenta diamonds

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Fig. 14
figure 14

Subset of representatives for global clustering result summarizing trajectories associated with prograde perigees and generated for up to three returns to a perigee map in the Sun-Earth CR3BP at \(C_{J}\) = 3.00088, with the libration points displayed as magenta diamonds

Full size image

12 Appendix 2

See Table 1.

Table 1 Truncated period, initial state and stable and unstable eigenvalues, \(\lambda _{S}\) and \(\lambda _{U}\), respectively, of the monodromy matrix for the Lyapunov and vertical orbits at \(L_1\) and \(L_2\) in the Sun-Earth CR3BP at \(C_J = 3.00088\)
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Bonasera, S., Bosanac, N. Applying data mining techniques to higher-dimensional Poincaré maps in the circular restricted three-body problem. Celest Mech Dyn Astr 133, 51 (2021). https://doi.org/10.1007/s10569-021-10047-3

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