Semi-Analytical Method for Calculating the Elliptic Restricted Three-Body Problem Monodromy Matrix

R ECENT years have seen a rising interest in launching unmanned spacecraft into libration-point orbits for scientific missions. Past missions such as ISEE-3, SOHO, and Genesis successfully used quasiperiodic orbits about the L1 and L2 collinear libration points of the sun–Earth system. With the embarkation of future NASA and ESA missions such as Darwin, Terrestrial Planet Finder (TPF), and Single Aperture Far Infrared Observatory (SAFIR), to be launched into libration-point orbits, there is an opportunity to design and evaluate novel trajectory planning, simulation, and control schemes. In practical analysis and design of missions around libration points, the circular restricted three-body problem (CR3BP) model is usually adopted [1–4]. Although this model has proven fruitful, it possesses an inherent approximation, assuming that the orbits of the primaries are circular. However, both the motion of the Earth around the sun and the motion of moon around the Earth are eccentric. Incorporating the eccentricity term into the equations of motion renders a more general model, known as the elliptic restricted threebody problem (ER3BP). The ER3BP has significant topological differences compared with the CR3BP. For example, the position of the libration points is not constant, but rather pulsatingwith respect to Earth. Moreover, the Jacobi integral is time(true-anomaly-) dependent. Several works have addressed the problem of finding natural periodic orbits in the planar ER3BP based on specialized regularizations [5,6]. Derivation of such orbits through numerical searches was accomplished in [7,8]. Almost all methods for calculating orbits about the collinear libration points require the monodromy matrix, which is the state-transition matrix evaluated at the orbital period. Moreover, design of stationkeeping maneuvers, whether impulsive [9] or continuous [10], also requires the monodromy matrix. Although the evaluation of the monodromy matrix in the CR3BP is trivial, the problem is more involved in the more general setup of the ER3BP, because the linearized system is nonautonomous. In fact, the linearized system is linear parameter-varying (LPV) and periodic, exhibiting implicit dependence upon time through the true anomaly. It is therefore important to develop reliable algorithms for computing the monodromy matrix. In this work, we develop a new, semianalytical method for calculation of the monodromy matrix. We propose to expand the state-transition matrix into orthogonal Chebyshev polynomials of the first and second kind, shifted to fit the time interval in use. The Chebyshev approximation transforms the nonautonomous differential equations required to calculate the state-transitionmatrix into a set of algebraic equations. This approach can be used as a computationally efficient scheme for computing the state-transition matrix and as a simple check for verifying the accuracy of monodromymatrix calculations using a direct numerical integration.