  Given a reconfigurable system X, such as a robot moving on a grid or a set of
particles traversing a graph without colliding, the possible positions of X
naturally form a cubical complex S(X). When S(X) is a CAT(0) space, we can
explicitly construct the shortest path between any two points, for any of the
four most natural metrics: distance, time, number of moves, and number of steps
of simultaneous moves.
  CAT(0) cubical complexes are in correspondence with posets with inconsistent
pairs (PIPs), so we can prove that a state complex S(X) is CAT(0) by
identifying the corresponding PIP. We illustrate this very general strategy
with one known and one new example: Abrams and Ghrist's positive robotic arm on
a square grid, and the robotic arm in a strip. We then use the PIP as a
combinatorial "remote control" to move these robots efficiently from one
position to another.
