Trajectory Design Combining Invariant Manifolds with Discrete Mechanics and Optimal Control

Abstract

A mission design technique that combines invariant manifold techniques, discrete mechanics, and optimal control produces locally optimal low-energy trajectories. Previously, invariant manifolds of the planar circular restricted three-body problem have been used to design trajectories with relatively small midcourse change in velocity V. A different method of using invariant manifolds is explored to design trajectories directly in the four-body problem. Then, using the local optimal control method DMOC (Discrete Mechanics and Optimal Control), it is possible to reduce the midcourse V to zero. The influence of different boundary conditions on the optimal trajectory is also demonstrated. These methods are tested on a trajectory that begins in Earth orbit and ends in ballistic capture at the moon. Impulsive DMOC trajectories require up to 19% less V than trajectories using a Hohmann transfer. When applied to low-thrust trajectories, DMOC produces an improvement of up to 59% in the mass fraction and 22% in travel time when compared with results from shooting methods. = mass consumption, kg P = spacecraft Q = configuration space q = configuration variable q k = discrete configuration variable r a , r p = radius of elliptical orbit at apoapsis and perigee, km r M H , a H = perigee radius, semimajor axis of Hohmann transfer ellipse, km T = low-thrust magnitude, N T k = discrete thrust magnitude applied at node k, N T max = maximum allowable thrust, N TQ = state space T t = normalized time denoting transition from low thrust to no thrust T x;k = discrete thrust at node k in x direction, N T y;k = discrete thrust at node k in y direction, N t k = discrete time variable u = continuous control parameter u k = discrete control variable u x = control force in x direction u y = control force in y direction v r = normalized radial velocity v x;k = normalized discrete velocity at node k in x direction v y;k = normalized discrete velocity at node k in y direction x i , y i = x, y position of body i for i E; S; M V = scales nondimensional velocity to meters per second V = magnitude of change in velocity, m=s or km=s V C = V necessary to enter elliptical capture orbit at moon, km=s V H = V necessary to leave Earth orbit using a Hohmann transfer, km=s V i = V for circular orbit insertion at body i, i E; M, m=s or km=s V traj = V applied throughout trajectory, m=s t = discrete time grid t = refined discrete time grid = variational operator q k = discrete configuration variation i = angle of body i with respect to x axis of coordinate system for i M; S, rad = mass parameter of planar circular restricted three-body problem k = discrete thrust optimization variable, N = angle of semimajor axis of elliptical orbit at the moon with respect to the Earth-moon x axis, deg = phase of Earth-moon frame with respect to sun-Earth frame, deg = effective potential in rotating frame ! i = normalized rotation rate of body i for i M; S

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