Analytical approach to coasting trajectory prediction with higher zonal harmonics

Abstract

This paper concentrates on the motion of coasting bodies under the influence of zonal harmonics of arbitrary degree, deriving an approximate closed-form solution based on state space perturbation theory. The derived closed-form solution is articulated as a linear combination of a series of definite integrals. A recursive formula for these definite integrals, parameterized by the degree of the zonal harmonics, has been established to expedite the calculation process. Additionally, a hybrid algorithm that leverages the strengths of the classical Vinti algorithm and the presented analytical algorithm has been proposed, aiming to achieve higher prediction accuracy. Numerical results suggest that this proposed method outperforms existing analytical algorithms and numerical integration techniques in both computational accuracy and efficiency. The trajectory prediction errors for the hybrid algorithm are below 1 m within one revolution for various scenarios, and the error propagation rates along the x, y, and z axes are all less than 1.5 m per hour over numerous revolutions, which is significantly superior to other analytical methods. Moreover, the computational efficiency of this method is approximately 6–20 times greater than that of the classical adaptive-step integration method, while maintaining the same level of calculation accuracy.