Reconfiguration planning for pivoting cube modular robots

Abstract

Abstract-In this paper, we present algorithms for selfreconfiguration of modular robots that move by pivoting. The modules are cubes that can pivot about their edges along thê x,ŷ, orẑ axes to move on a 3-dimensional substrate. This is a different model from prior work, which usually considers modules that slide along their faces. We analyze the pivoting cube model and give sufficient conditions for reconfiguration to be feasible. In particular, we show that if an initial configuration does not contain any of three subconfigurations, which we call rules, then it can reconfigure into a line. We provide provably correct algorithms for reconfiguration for both 2-D and 3-D systems, and we verify our algorithms via simulation on randomly generated 2-D and 3-D configurations.I. INTRODUCTION Modular robots consist of multiple connected modules, each with limited capabilities, that can be reconfigured to produce complex functionality as required by a task. Among the self-reconfigurable modular systems that have been developed [1]- [7], pivoting has emerged as a simple but powerful motion predicate [7]. In this paper, we describe a model for pivoting cubes in 3-D. We consider reconfiguration in both 2-D and 3-D settings and demonstrate that barring certain subconfigurations allows us to guarantee reconfiguration in O(n 2 ) moves. We provide provably correct algorithms for performing such reconfiguration. These are not optimal but are the first correct algorithms for the 3-D pivoting cube model. We perform simulations on random 2-D and 3-D configurations and show that in many cases, reconfiguration does not require the upper bound of n 2 moves. Pivoting modules, although prevalent in hardware [5], [7], are not well-studied. Pivoting modules sweep out a volume that extends past their initial and final positions, and any reconfiguration algorithm must take this motion constraint into account. An O(n 2 ) algorithm for 2-D pivoting modules was given in [8] but the model relaxed connectivity constraints compared to what many modular robots require and allowed movements that are often not physically realizable. Nguyen et al. [9] considered 2-D pivoting hexagons while requiring strong connectivity (connectivity through faces) and provided sufficient conditions for reconfiguration in O(n 5/2 ) moves. In [10], pivoting squares were reconfigured using a stochastic approach, and work in [5] analyzed the same system using meta-modules. A model for pivoting cubes in 3-D was introduced in [7], but the planning problem was not addressed. To our knowledge, there have been no studies of reconfiguration for pivoting cubes in 3-D.

Universal Reconfiguration of Facet-Connected Modular Robots by Pivots: The O(1) Musketeers

“…For this model, universal reconfiguration is possible between any two facet-connected configurations, in any dimension [7,1]. 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.…”

“…One algorithm follows some heuristics without a termination guarantee [3] (see also [11] for heuristics for hexagons). A recent algorithm guarantees reconfiguration by forbidding one or more local patterns in both the start and goal configurations [20], essentially preventing narrow holes in the shape. (A similar result was obtained for hexagons [15].)…”

“…Most modular robots follow a space-filling lattice structure (e.g., squares or hexagons in 2D, or cubes in 3D), to simplify both reconfiguration and the characterization of possible shapes. Pure lattice modular robots [13,6,10,17,2,20] have one robot per lattice element and always remain on the lattice, while hybrid modular robots [14,18,16,23] also allow units move out of the lattice. We focus here on the well-studied square lattice, though we suspect our results can be generalized to cube lattices.…”

Deterministic scaffold assembly by self-reconfiguring micro-robotic swarms

“…This is most relevant as these RD modules have motions constraints very similar to the ones of our own model, introduced in the next section. While it is usually assumed that all the modules composing a modular robot must remain connected at all times during reconfiguration, dropping this constraint can lead to interesting systems such as the rotating M-Blocks (Sung et al, 2015).…”

Correct-by-Construction Approach for Self-Evolvable Robots

“…Modules are represented as vertices while connections between the modules are represented as edges. Then tools from graph theory can be used to solve problems related to reconfigurable robots, such as configuration recognition [9] and motion planning [8]. Optimization-based approaches cast the design of a reconfigurable robot as an optimization problem with a objective function over the vector of design variables [10,11].…”