Low-Thrust Transfers in the Earth-Moon System, Including Applications to Libration Point Orbits

Abstract

Preliminary designs of low-thrust transfer trajectories are developed in the Earth-moon three-body problem with variable specific impulse engines and fixed engine power. The solution for a complete time history of the thrust magnitude and direction is initially approached as a calculus of variations problem to locally maximize the final spacecraft mass. The problem is then solved directly by sequential quadratic programming, using either single or multiple shooting. The coasting phase along the transfer exploits invariant manifolds and, when possible, considers locations along the entire manifold surface for insertion. Such an approach allows for a nearly propellant-free final coasting phase along an arc selected from a family of known trajectories that contract to the periodic libration point orbit. This investigation includes transfer trajectories from an Earth parking orbit to some sample libration point trajectories, including L 1 halo orbits, L 1 and L 2 vertical orbits, and L 2 butterfly orbits. Given the availability of variable specific impulse engines in the future, this study indicates that fuel-efficient transfer trajectories could be used in future lunar missions, such as south pole communications satellite architectures. Nomenclature c = constraint vector G = endpoint function H = Hamiltonian I sp = engine specific impulse i = instantaneous inclination angle m = total spacecraft mass P = engine power magnitude R = rotation matrix r, v = position and velocity vectors, Earth-moon rotating frame S = design variable vector T = engine thrust magnitude t = time U = pseudopotential function u c = control vector u T = thrust direction unit vector X = uncontrolled state vector, Earth-moon rotating framê = eigenvector at a fixed point on a periodic orbit = control constraint Lagrange multiplier vector = instantaneous argument of latitude M = anglelike manifold parameter = costate vector M = timelike manifold parameter = state transition matrix = kinematic boundary condition vector = instantaneous right ascension angle $, = kinematic boundary condition Lagrange multiplier vectors Subscripts c = control variable f = final condition FP = fixed point condition MS = multiple shooting phase p = powered phase PO = parking orbit condition s = stable manifold SS = single shooting phase u = unstable manifold 0 = initial condition

Multi-step optimization strategy for fuel-optimal orbital transfer of low-thrust spacecraft

“…A large literature is available on CRTBP, on the possibility of designing low-energy transfers within this dynamical framework, and the combination with low-thrust propulsion. The reader may refer to Mingotti et al (2011), Ozimek andHowell (2010) and the references therein for more details. The following subsections show how the combination of indirect methods and parametric optimization can be exploited to design low-thrust and lowenergy transfers in the CRTBP without making explicit use of invariant manifolds.…”

“…The same approach is adopted to avoid dealing with inequality constraints when gravity assists are included in the transfer: fixed pericenter radii are considered and the optimization of their values is achieved by including them in the optimization vector of the parametric optimization problem. The proposed approach falls in the category of the so-called hybrid methods (Kluver and Pierson 1995, Gao and Kluver 2004, Russel 2007, Ozimek and Howell 2010, but here the problem simplification is obtained by decomposition of multi-phase trajectories into a sequence of state-to-state transfers.…”

A Study on Earth-Moon Transfer Orbit Design

“…The system's governing equations for the optimal transfer trajectory design are the previously stated three-body orbital motion equations which are described by Eqs. (1) and (2) and they are considered in this study. The cost function or the performance index that is used to minimize the use of mixed impulsive and continuous thrust is defined as follows,…”

“…(1) and (2). Here, the purpose is to find out the control acceleration (u ξ , u ς ) that not only minimizes the performance index defined in Eq.…”

“…The solid rockets that use impulsive thrust are not able to control the thrust magnitude or the combustion time whereas liquid rockets that use continuous thrust can. The most general methods for transfer trajectory design using multiple impulsive maneuvers [2,3] are the Hohmann transfer, Bi-elliptic Transfer, Weak Stability Boundary(WSB) [4,5] that consider the possibility of the reignition of the rocket. Recently, research has progressed in the transfer trajectory design which actively uses electronic and ion thrusters.…”