High-Order State Transition Polynomial with Time Expansion Based on Differential Algebra

In this study, the problem of propagating an initial orbital state around its reference value to a variable final time is addressed. Based on the differential algebra (DA) technique, a high-order state transition polynomial with time expansion (STP-T) method is developed. The STP-T is a high-order Taylor approximation of the final orbital state expanded around the reference initial state and the propagation time. Thus, given the initial displaced orbital state, any state around the reference final time can be efficiently obtained by evaluating the corresponding polynomial. Furthermore, an asynchronous-order scheme for STP-T is presented, which enables the STP-T to have different orders with respect to the variables with different nonlinearity. Besides, the manual derivation and integration of the high-order variational equations is avoided in the DA framework, which makes the order of the polynomial flexible and the method versatile for various dynamics. The simulation results indicate that the STP-T supplies a good approximation of the final state in pure and J2 perturbed Keplerian dynamics, and in a nonlinear relative motion.