Abstract
We present the first universal reconfiguration algorithm for transforming a modular robot between any two facet-connected square-grid configurations using pivot moves. More precisely, we show that five extra “helper” modules (“musketeers”) suffice to reconfigure the remaining n modules between any two given configurations. Our algorithm uses $$O(n^2)$$ pivot moves, which is worst-case optimal. Previous reconfiguration algorithms either require less restrictive “sliding” moves, do not preserve facet-connectivity, or for the setting we consider, could only handle a small subset of configurations defined by a local forbidden pattern. Configurations with the forbidden pattern do have disconnected reconfiguration graphs (discrete configuration spaces), and indeed we show that they can have an exponential number of connected components. But forbidding the local pattern throughout the configuration is far from necessary, as we show that just a constant number of added modules (placed to be freely reconfigurable) suffice for universal reconfigurability. We also classify three different models of natural pivot moves that preserve facet-connectivity, and show separations between these models.
Highlights
Shape shifting is a powerful idea in science fiction: T-1000 robots, Changelings, Symbiotes, Mystique, and Metamorphagi all have the ability to transform their shape nearly arbitrarily
We show that there exist robot configurations with many instances of the three forbidden patterns that are still reconfigurable, so the local separation condition is not necessary
We show that as soon as the local separation condition is relaxed, the reconfiguration graph breaks into an exponential number of connected components of exponential size
Summary
Shape shifting is a powerful idea in science fiction: T-1000 robots (from Terminator 2: Judgement Day), Changelings (from Star Trek: Deep Space 9 ), Symbiotes (from Venom), Mystique (from X-Men), and Metamorphagi (from Harry Potter) all have the ability to transform their shape nearly arbitrarily. We focus here on a more challenging model, pivoting squares/cubes [21, 20, 4], illustrated in Figure 1 right In this case, modules live in a square or cube lattice, move by rotating relative to each other, and require facet-connectivity. The number of pivots it makes is O(n2), which is optimal in the worst case by an earth-moving lower bound: each robot may need to move a distance of Θ(n) This result can be seen as proving connectivity of the reconfiguration graph Gn,k, where vertices represent facet-connected configurations of n modules and edges represent valid pivot moves, with the addition of k ≥ 5 musketeer modules.
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