## 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.

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## Notes

- 1.
In the Gaussian case, a proper characterization of uncertainty is sometimes called covariance realism or covariance consistency (Drummond et al. 2006). In the more general case, it is called uncertainty realism or uncertainty consistency.

- 2.
Uncertainty in a model parameter can be treated in many ways, including as part of the estimation process, assuming that the model parameter is observable.

- 3.
- 4.
The solution of the unperturbed Kepler problem, for example, requires solving only a single transcendental equation in the universal anomaly. Efficient methods for doing so are well known (Montenbruck and Gill 2005).

- 5.
Local error is incurred over each time step, and results from a combination of truncation error and round-off error. Accumulation and transport of local error leads to the global error. In the context of many applications including orbital propagation (Shampine 2005; Aristoff et al. 2014b), local error control gives rise to global error control and more efficient propagation compared to methods that employ fixed steps sizes.

- 6.
For super-convergent IRK methods such as Gauss–Legendre IRK, estimation of the local error can be expensive. Hence, the reuse of error estimates leads to a considerable speedup.

- 7.
- 8.
The equinoctial elements \((a,h,k,p,q,\ell )\) are defined in terms of the Keplerian elements \((a,e,i,\varOmega ,\omega ,M)\) by

$$\begin{aligned} \begin{aligned}&a = a, \quad h = e \sin (\omega + \varOmega ), \quad k = e \cos (\omega + \varOmega ), \\&p = \tan \tfrac{i}{2} \sin \varOmega , \quad q = \tan \tfrac{i}{2} \cos \varOmega , \quad \ell = M + \omega + \varOmega , \end{aligned} \end{aligned}$$(11)where \(a\) is the semi-major axis, \(e\) the eccentricity, \(i\) the inclination, \(\varOmega \) the longitude of the ascending node, \(\omega \) the argument of periapsis, and \(M\) the mean anomaly (Broucke and Cefola 1972). Note that unlike the Keplerian elements, the equinoctial elements are non-singular for zero eccentricity and inclination.

- 9.
- 10.
By neglecting all but the dominant term in the spherical harmonic expansion of the Earth’s gravitational field, the equations of motion permit an analytical solution.

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## 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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## Appendix: Convergence of the fixed-point iterations—proof and example

### Appendix: Convergence of the fixed-point iterations—proof and example

Fixed-point methods for solving (3) or (6) are sometimes used because they do not require taking derivatives of \(f\) or \(g\), which can be computationally prohibitive. However, unlike Newton or quasi-Newton methods, care must be taken when choosing the step size in order to ensure that the iterations converge on each (time) step.

Convergence of the nonlinear system (3) via fixed-point methods is guaranteed provided \(f\) fulfills the Lipshitz condition

and \(s\cdot h \cdot (\max |a_{ij}|) \cdot L < 1\), where \(L\) is the Lipshitz constant for the function \(f\), and where \(s\) is the number of stages in the IRK method (Shampine 1994). Similarly, one may apply the contraction mapping theorem to show that convergence of a second-order nonlinear system is guaranteed provided \(g\) fulfills the Lipshitz condition (12), \(s\cdot h \cdot (\max |a_{ij}|) \cdot L' < 1\) and \(s\cdot h^2 \cdot (\max |\bar{a}_{ij}|) \cdot L < 1\), where \(L\) and \(L'\) are the Lipshitz constants for the function \(g\) (corresponding to the position and velocity components, respectively). Hence, there exists a step size \(h\) below which an IRK method will converge (given a well-behaved \(f\) or \(g\)), regardless of \(L\) and \(L'\). Implementations of IRK can therefore detect if the iterative method used to solve (3) or (6) is converging, and if it is not, then the step size can be reduced.

For the two-body problem, we may estimate the magnitude of the step size \(h\) below which the iterations will converge by considering the case of unperturbed Keplerian dynamics, wherein \(g(t,r) = -(\mu /r^3) r\), \(\mu \) is the gravitational coefficient, and \(r\) is the distance from the orbiting object to the center of the Earth. Consider two nearby orbits, \(r_1\) and \(r_2=r_1+\varDelta r\). By retaining only the first two terms in the Taylor series expansion of \(g(t,r_2)\), the Lipschitz condition (12) becomes

Since \(\Vert \frac{\partial g}{\partial r}\Vert \le \mu /(27r^3)\), we find that \(L\le \mu /(27r^3)\). In addition, since \(\partial g / \partial r' = 0\), we have that \(L'=0\). Thus, the Lipshitz constant is small. Convergence of the fixed-point iterations require a step size \(h\) that satisfies

By making the observation that \(0.73<\sqrt{s\cdot \max _{i,j=1,\ldots , s}|\bar{a}_{ij}|}<1.00\) (for Gauss–Legendre IRK methods having 50 or fewer stages per step), a conservative estimate of \(h_{ max}\) can be made:

For nearly-circular orbits, \(r\approx a\), where \(a\) is the semi-major axis. Since the mean motion and the orbital period \(T\) are related via \(T=2\pi /n\) where \(n=\sqrt{\mu /a^3}\), we can rewrite (15) as

This result suggests that time steps up to \(4/5\) of an orbital period should converge, which is much greater than the time steps taken in the examples presented in Sect. 3.

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Aristoff, J.M., Horwood, J.T. & Poore, A.B. Implicit-Runge–Kutta-based methods for fast, precise, and scalable uncertainty propagation.
*Celest Mech Dyn Astr* **122, **169–182 (2015). https://doi.org/10.1007/s10569-015-9614-7

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### Keywords

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