High-order state transition polynomial with time expansion based on differential algebra

In this study, a high-order state transition polynomial with time expansion (STP-T) method is developed to propagate an initial orbital state around its reference value to a variable final time based on the differential algebra (DA) technique. STP-T is a high-order Taylor polynomial of the final orbital state expanded around the reference initial state and reference propagation time. Since the final state usually shows different nonlinearity with respect to different components of initial state and propagation time, a weighted-order scheme is combined with STP-T, which enables the STP-T to have higher orders on the components with higher nonlinearity and lower orders on the components with lower nonlinearity. Then, an error estimation method is presented, which can a priori provide the error profiles of a STP-T and is useful for selecting a proper order and determining the corresponding valid ranges of displacements. Finally, the STP-T method is tested for orbit propagation under three typical orbital dynamics: the unperturbed Keplerian dynamics, the J, perturbed two-body dynamics, and the nonlinear relative dynamics. The numerical simulation results indicate that the STP-T supplies a good approximation of the final state within certain valid ranges of initial state and propagation time, the a priori estimated error is close to the exact error in sense of trend and magnitude, and the computational cost can be significantly saved by the weighted-order scheme without loss of accuracy.