  We present a computational approach for the design of continuous low thrust
transfers around an asteroid. These transfers are computed through the use of a
reachability set generated on a lower dimensional Poincar\'e surface. Complex,
long duration transfer trajectories are highly sensitive to the initial guess
and generally have a small region of convergence. Computation of the reachable
set alleviates the need to generate an accurate initial guess for optimization.
From the reachable set, we chose a trajectory which minimizes a distance metric
towards the desired target. Successive computation of the reachable set allows
for the design of general transfer trajectories which iteratively approach the
target. We demonstrate this method by determining a transfer trajectory about
the asteroid 4769 Castalia.
