design space pruning heuristics and global optimization method for ...

Another example of applying a costate transformation is presented by Russell. 27
In his work, the unknown initial position and velocity co-states are replaced with more
physically meaningful quantities: α and β (the in- and out-of-plane orientation angles,
which represent the direction of the initial thrust), S (the switching function), and their
time derivatives. This transformation is applied to the initial guesses for the targeting
routine and then directly iterates on the co-states. This transformation is part of a larger
effort, which applies primer vector theory to a global low-thrust trade study algorithm.
This methodology is applied to two multiple-revolution problems in the restricted threebody
problem: a phase-free transfer between two distant retrograde orbits at Europa and a
phase-free transfer from a distant near circular orbit at Earth to a distant retrograde orbit
at the Moon.
1.1.2 Improvements to Direct Methods
Sims and Flanagan developed a new direct method, which is implemented in
MALTO, a tool intended for the preliminary design of low-thrust trajectories including
those with gravity assists. 12,13
As shown in Figure 2, the trajectory is divided into legs
that begin and end at control nodes. Typically, these control nodes represent planets or
other bodies, but could also represent free points in space. On each leg is a match point,
and the trajectory is propagated forward from the previous control node and backward
from the subsequent control node to the match point. Each leg is also subdivided into
numerous segments containing an impulsive ∆V at the middle of each segment. In the
limit, as the number of segments is increased, this approximates the continuous thrust
problem. The magnitude of the ∆V is limited by the total amount of ∆V that could be
accumulated over the entire segment for the continuous thrust case. Propagation of the
trajectory assumes two-body motion, and gravity assists are assumed to cause an
instantaneous change in the direction of the V ∞ vector.
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