Abstract A myriad of methods, including many optimization and estimation algorithms, require the sensitivities of a dynamic system governed by a set of ordinary differential equations (ODEs). In this paper, the decoupled direct method (DDM) is derived for the calculation of first- and second-order state transition matrices (STMs) using fixed-step-size Runge-Kutta (RK) ODE solvers. Implications for both explicit and implicit RK methods are discussed. Emphasis is placed on the implicit class because of recent interest in the astrodynamics community and because the decoupling of the state propagation and STM calculation eliminates the need to evaluate the Jacobian and Hessian iteratively at each propagation step. Thus, significant computational savings are achievable compared to the traditional propagation of the variational equations, particularly for systems with expensive dynamics. In the current work, two implementations of the DDM are extended both to second-order STMs and to the double-integrator form of the RK equations. One of the strategies presented in this paper, newly derived to include second-order sensitivities, takes advantage of the structure of the system, allowing for the STMs of nonlinear systems to be calculated using efficient linear-algebra techniques rather than iterative methods. This linear-algebra-based approach is shown to be more efficient than the second alternative, and is also more efficient than the propagation of the variational equations for computationally intensive dynamical systems. The effects of important implicit RK customizations, such as variable-fidelity dynamics models and parallelization, are also examined in two examples: a simple, proof-of-concept forced harmonic oscillator, and an Earth-orbiting space object subject to either a low- or high-computational-cost geopotential model.


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    Titel :

    Decoupled Direct State Transition Matrix Calculation with Runge-Kutta Methods


    Beteiligte:
    Hatten, Noble (Autor:in) / Russell, Ryan P. (Autor:in)


    Erscheinungsdatum :

    2018




    Medientyp :

    Aufsatz (Zeitschrift)


    Format :

    Print


    Sprache :

    Englisch



    Klassifikation :

    BKL:    55.60 Raumfahrttechnik
    Lokalklassifikation TIB:    770/7040



    Decoupled Direct State Transition Matrix Calculation with Runge-Kutta Methods

    Hatten, Noble / Russell, Ryan P. | Springer Verlag | 2018


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    J. S. Rosen | NTIS | 1968


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    Runge-Kutta Method

    Ying, Shuh-Jing (Benj | AIAA | 1997


    Runge-Kutta methods for linear problems

    Zingg, D / Chisholm, T | AIAA | 1995