Semi-Analytical Time Series Solutions to the Circular Restricted Three-Body Problem

Abstract This work develops and verifies a semi-analytical propagation framework for the circular restricted three-body problem that represents the state as an explicit polynomial in time. The coefficient expressions are derived symbolically from the equations of motion, and the state transition matrix is obtained by differentiating the same truncated series so that state and sensitivity remain consistent. An adaptive step size follows from Taylor’s remainder integral and the method applies on intervals where the required derivatives exist and remain bounded. Validation across planar, spatial, and non-periodic Earth–Moon trajectories shows the expected truncation-error behavior and preserves invariants such as the Jacobi constant. Application studies use the same time-series model for both orbit propagation and STM construction. In impulsive targeting the STM maps terminal-state corrections to small adjustments at the initial epoch. In relative navigation with range, range-rate, and optical-angle measurements, the Extended Kalman Filter advances the state with the time-series propagation and carries the covariance and linearized measurement sensitivities with the analytical STM, while the Unscented Kalman Filter advances sigma points and predicted measurements with the same time-series dynamics. The results demonstrate accurate guidance and estimation performance in nonlinear cislunar dynamics with a consistent closed-form treatment of state and sensitivity.