  We introduce GeoSACS, a geometric framework for shared autonomy (SA). In
variable environments, SA methods can be used to combine robotic capabilities
with real-time human input in a way that offloads the physical task from the
human. To remain intuitive, it can be helpful to simplify requirements for
human input (i.e., reduce the dimensionality), which create challenges for to
map low-dimensional human inputs to the higher dimensional control space of
robots without requiring large amounts of data. We built GeoSACS on canal
surfaces, a geometric framework that represents potential robot trajectories as
a canal from as few as two demonstrations. GeoSACS maps user corrections on the
cross-sections of this canal to provide an efficient SA framework. We extend
canal surfaces to consider orientation and update the control frames to support
intuitive mapping from user input to robot motions. Finally, we demonstrate
GeoSACS in two preliminary studies, including a complex manipulation task where
a robot loads laundry into a washer.
