Efficient constant-velocity reconfiguration of crystalline robots

Abstract

Abstract:In this paper we propose novel algorithms for reconfiguring modular robots that are composed of n atoms. Each atom has the shape of a unit cube and can expand/contract each face by half a unit, as well as attach to or detach from faces of neighboring atoms. For universal reconfiguration, atoms must be arranged in 2×2×2 modules. We respect certain physical constraints: each atom reaches at most constant velocity and can displace at most a constant number of other atoms. We assume that one of the atoms has access to the coordinates of atoms in the target configuration.Our algorithms involve a total of O(n 2 ) atom operations, which are performed in O(n) parallel steps.This improves on previous reconfiguration algorithms, which either use O(n 2 ) parallel steps [Rus and Vona, 2001, Vassilvitskii et al., 2002, Butler and Rus, 2003 or do not respect the constraints mentioned above [Aloupis et al., 2009b]. In fact, in the setting considered, our algorithms are optimal. A further advantage of our algorithms is that reconfiguration can take place within the union of the source and target configuration space, and only requires local communication.

A new meta-module for efficient reconfiguration of hinged-units modular robots

“…AVOIDING META-META-MODULES By Theorem 4, we can apply the tunneling reconfiguration algorithms in [2], [5], [18], [22] for Crystalline and Telecube units to our meta-module. These algorithms, in turn, use meta-modules of Crystalline or Telecube units that are able to perform the following operations:…”

“…Maintaining the assumptions of constant velocity and strength, under which a module can pull or push only a constant number of other modules at constant speed, 2×2(×2)-unit meta-modules can be used to reconfigure in-place. This can be achieved by both centralized [2] and distributed [16] algorithms, and the overall number of unit moves needed is Θ(n 2 ), which is optimal in this setting. If modules have linear strength, the total number of unit moves can be reduced to O(n) [3].…”

Active self-assembly of algorithmic shapes and patterns in polylogarithmic time

“…For this model, Aloupis et al [5] give an algorithm for universal reconfiguration between a pair of connected shapes that works in time merely logarithmic in shape size. As pointed out by Aloupis et al in a subsequent paper [4], this high-speed reconfiguration can lead to strain on individual components, but they show that if each robot can displace at most a constant number of other robots, and reach at most constant velocity, then there is an optimal Θ(n) parallel time reconfiguration algorithm (which also has the desired property of working "in-place"). Reif and Slee [60] also consider physicaly constrained movement, giving an optimal Θ( √ n) reconfiguration algorithm where at most constant acceleration, but up to linear speed, is permitted.…”

Parallel Computation Using Active Self-assembly

“…One configuration transitions to another via the application of a single rule, r = (s1, s2, b, − → u ) → (s1 , s2 , b , − → u ) that acts on one or two monomers. 3 The left and right sides of the arrow respectively represent the contents of two monomer positions before and after the application of rule r. Here s1, s2 ∈ S ∪ {empty} are monomer states where at most one of s1, s2 is empty (denotes lack of a monomer), b ∈ {flexible, rigid, null} is the bond type between them, and − → u ∈ D is the relative position of the s2 monomer to the s1 monomer. If either of s1 or s2 (respectively s1 or s2 ) is empty then b (respectively b ) is null (monomers can not be bonded to empty space).…”