Pushing Hypercubes Around

We study collections of identical, connected modules which may relocate relative to each other; such a collection is called a modular metamorphic system (see [CPE], [MKK], [PEC], [RV], [YMK]). While such a system must always remain connected, it is possible for such a system to reconfigure its shape through successive motions of individual modules, either by rotation and sliding (e.g., [MKK]) or by expansion and contraction (e.g., [PEC]). The theory of modular metamorphic systems has important applications in the study and use of reconfigurable robots, small, modular robots with limited motion abilities (see [Yi], [Ch], [MKK]). Such robots reorganize themselves by changing shape locally, while maintaining connectivity. Reconfigurable robots are easily adaptable, as they are relatively inexpensive to produce and exhibit high fault tolerance. Given the desire to have reconfigurable robots take on specific configurations, it is natural to ask whether a collection of modules can achieve specified configurations. Formally, the motion planning problem for a modular metamorphic system asks for a sequence of motions which transform a given configuration of modules V into a specified configuration V ′. We denote the motion planning problem asking for a transformation taking V to V ′ by [V 7→ V ′]. When a solution to [V 7→ V ′] exists, we say that [V 7→ V ′] is feasible. For configurations V and V ′ of two-dimensional, hexagonal modules, the problem [V 7→ V ′] is feasible whenever the configurations have the same number of modules and do not contain a single three-module pattern, as shown by Nguyen, Guibas, and Yim [NGY]. Recently, Dumitrescu and Pach [DP] showed that the motion planning problem is even simpler for square modules in two dimensions. Indeed, for any two configurations V and V ′ of n square modules, the problem [V 7→ V ′] is feasible (see [DP]). We find a similar reconfiguration result for metamorphic systems of d-dimensional hypercubic modules. In particular, we will show in Section II-B that for any two n-module configurations V and V ′ of d-dimensional hypercubic modules, the problem [V 7→ V ′] is feasible. This result fully generalizes Dumitrescu and Pach’s [DP] result for squares. Furthermore, our result for d = 3 affirmatively answers the “Pushing Cubes Around” problem proposed by O’Rourke at CCCG 2007 [DO].