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Celestial Mechanics and Dynamical Astronomy

, Volume 122, Issue 2, pp 169–182 | Cite as

Implicit-Runge–Kutta-based methods for fast, precise, and scalable uncertainty propagation

  • Jeffrey M. Aristoff
  • Joshua T. Horwood
  • Aubrey B. Poore
Original Article

Abstract

A large class of methods for propagating the state of an object and its associated uncertainty require the propagation of an ensemble of particles or states through nonlinear dynamics. Existing sigma point- or particle-based methods for uncertainty propagation solve the ensemble of initial value problems one-by-one without exploiting the proximity of the initial conditions. In this paper, we demonstrate how implicit Runge–Kutta methods can be modified to solve initial-value-problem ensembles collectively in order to significantly reduce the computational cost of uncertainty propagation. Examples and variations of this approach are given in the context of the perturbed two-body problem of orbital mechanics that arises in space-object tracking, conjunction analysis, maneuver detection, and other Astrodynamics functions. Particular attention is given to the accuracy of the solution to the initial-value-problem ensemble when the particles are propagated collectively.

Keywords

Implicit Runge–Kutta (IRK) Initial value problem (IVP) Orbit and uncertainty propagation Satellites 

Notes

Acknowledgments

The authors thank Navraj Singh for fruitful discussions and the reviewers for helpful feedback. This work was funded, in part, by a Phase II STTR from the Air Force Office of Scientific Research (FA9550-12-C-0034) and a grant from the Air Force Office of Scientific Research (FA9550-11-1-0248).

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Jeffrey M. Aristoff
    • 1
  • Joshua T. Horwood
    • 1
  • Aubrey B. Poore
    • 1
  1. 1.Numerica CorporationFort CollinsUSA

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