  Motion planning for underwater vehicles must consider the effect of ocean
currents. We present an efficient method to compute reachability and cost
between sample points in sampling-based motion planning that supports
long-range planning over hundreds of kilometres in complicated flows. The idea
is to search a reduced space of control inputs that consists of stream
functions whose level sets, or streamlines, optimally connect two given points.
Such stream functions are generated by superimposing a control input onto the
underlying current flow. A streamline represents the resulting path that a
vehicle would follow as it is carried along by the current given that control
input. We provide rigorous analysis that shows how our method avoids exhaustive
search of the control space, and demonstrate simulated examples in complicated
flows including a traversal along the east coast of Australia, using actual
current predictions, between Sydney and Brisbane.
