Abstract
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 $\mathcal{S}(X)$. When $\mathcal{S}(X)$ 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 $\mathcal{S}(X)$ 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.
Highlights
There are numerous contexts in mathematics, robotics, and other fields where a discrete system changes according to local, reversible moves
Perhaps the most natural and important question that arises is the motion-planning or shape-planning question: how does one efficiently get a reconfigurable system X from one position to another one? Abrams, Ghrist, and Peterson observed that the transition graph G(X) is the 1-skeleton of the state complex S(X): a cubical complex whose vertices are the states of X, whose edges correspond to allowable moves, and whose cubes correspond to collections of moves which can be performed simultaneously
The idea is simple: we identify a posets with inconsistent pairs (PIPs) whose corresponding CAT(0) cubical complex is X
Summary
There are numerous contexts in mathematics, robotics, and other fields where a discrete system changes according to local, reversible moves. [1, 8] the state complex of some reconfigurable systems is globally non-positively curved, or CAT(0). This stronger property implies that for any two points p and q there is a unique shortest path between them. Roller [13] and Sageev [14], and Ardila, Owen, and Sullivant [3] gave two completely combinatorial descriptions of CAT(0) cubical complexes. The idea is simple: we identify a PIP whose corresponding (rooted) CAT(0) cubical complex is X In principle, this method is completely general, though its implementation in a particular situation is not trivial. We close by showing how to find the shortest path between states in a CAT(0) state complex S(X) under four natural metrics
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