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Parallel Implicit Runge-Kutta Methods Applied to Coupled Orbit/Attitude Propagation

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Abstract

A variable-step Gauss-Legendre implicit Runge-Kutta (GLIRK) propagator is applied to coupled orbit/attitude propagation. Concepts previously shown to improve efficiency in 3DOF propagation are modified and extended to the 6DOF problem, including the use of variable-fidelity dynamics models. The impact of computing the stage dynamics of a single step in parallel is examined using up to 23 threads and 22 associated GLIRK stages; one thread is reserved for an extra dynamics function evaluation used in the estimation of the local truncation error. Efficiency is found to peak for typical examples when using approximately 8 to 12 stages for both serial and parallel implementations. Accuracy and efficiency compare favorably to explicit Runge-Kutta and linear-multistep solvers for representative scenarios. However, linear-multistep methods are found to be more efficient for some applications, particularly in a serial computing environment, or when parallelism can be applied across multiple trajectories.

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Notes

  1. The quaternion is defined here such that \(\bar {\boldsymbol {q}}_{4 \times 1} = \left [ \boldsymbol {e}^{T} \sin \left (\phi / 2 \right ), \quad \cos \left (\phi / 2 \right ) \right ]^{T}\), where e 3×1 is the rotation axis and ϕ is the rotation angle.

  2. A cylindrical Earth shadowing function is assumed in this paper.

  3. The area and orientation of each individual panel i is taken into account in the calculation of C a,i .

  4. The defining arrays of the GLIRK method are calculable for arbitrary s, and may be obtained to high precision using, for example, the Mathematica function NDSolve‘ImplicitRungeKuttaGaussCoefficients [3].

  5. The embedded solution requires a single additional high-fidelity dynamics model evaluation at the initial time and state of the step.

  6. It is noted that, for any ODE solver, the propagation of state transition tensors may be parallelized within a single propagation step over the dimension of the state.

  7. All input options for LSODE are set to default values (with method flag = 10) except for the relative and absolute tolerance parameters: The absolute tolerance is set uniformly to machine epsilon, and the relative tolerance is varied.

  8. Each subfigure displays results corresponding to a single identical value of relative LTE tolerance, rtol=10−15.

  9. Note that the spherical harmonics formulation of the geopotential is used because it is more efficient than the interpolation model for the low degree and order used for GEO propagation.

  10. Excepting possibly the effect of aerodynamic acceleration on an SO in an extremely low orbit.

  11. This value is problem- and integrator-dependent.

  12. It is noted that erratic behavior for an implicit method can also be caused by frequent divergence of the iterative procedure used to solve the RK update equations, but divergence does not occur over the range of tolerances displayed in the figures in this paper.

  13. Unless the SO is in the Earth’s shadow.

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Acknowledgments

This work was funded, in part, by a Phase II SBIR from the Air Force Research Laboratory, contract FA9453-14-C-0295, under a subcontract from Emergent Space Technologies, Inc.

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  1. Department of Aerospace Engineering and Engineering Mechanics, University of Texas at Austin, Austin, TX, USA

    Noble Hatten & Ryan P. Russell

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Correspondence to Noble Hatten.

Appendix: A: Local Truncation Error Estimation

Appendix: A: Local Truncation Error Estimation

Position accuracy as a function of high-fidelity dynamics function evaluations (a proxy for CPU time) for three methods of calculating the “comparison solution” for estimating the local trunction error (LTE) is shown in Fig. 13. The figure displays results for three-orbit propagations of the tumbling LEO scenario (Fig. 13a) and non-tumbling GEO scenario (Fig. 13b) introduced in this paper. The variable-fidelity dynamics strategy is used, and the GLIRK solution uses eight stages. The “Kouya” method, which is based on a nearly embedded solution of order s, is the strategy used to produce the results given in this paper [28]. The “Jay” method uses an internal tolerance parameter to generate a less conservative estimate of the LTE based on the order-s comparison solution [25]. The “Radau” method produces a comparison solution using a non-embedded s-stage Radau-IA propagation, which produces a solution of order (2s−1) [19]. For the results presented in Fig. 13, it is assumed that the initial guess for the Radau-IA solution is accurate enough that only one high-fidelity dynamics function evaluation per stage is required to achieve convergence.

Fig. 13
figure 13

Position accuracy as a function of high-fidelity dynamics function evaluations for three LTE calculation methods; relative LTE tolerance of propagators is varied from 10−5−10−15. The Kouya method is used to produce the results given in this paper

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The inexpensiveness of the nearly embedded Kouya method generally produces a more efficient propagation than the Radau method for a given accuracy when using variable-fidelity dynamics models, even though the Radau method generally uses larger step sizes. On the other hand, the Jay method uses very few function evaluations, but the internal tolerance tends to produce such large step sizes that global error control is poor compared to the other two methods.

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Hatten, N., Russell, R.P. Parallel Implicit Runge-Kutta Methods Applied to Coupled Orbit/Attitude Propagation. J of Astronaut Sci 64, 333–360 (2017). https://doi.org/10.1007/s40295-016-0103-3

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