A Complete, Local and Parallel Reconfiguration Algorithm for Cube Style Modular Robots

We present a complete, local, and parallel reconfiguration algorithm for metamorphic robots made up of Telecubes, six degree of freedom cube shaped modules currently being developed at PARC. We show that by using 2x2x2 meta-modules we can achieve completeness of reconfiguration space using only local rules. Furthermore, this reconfiguration can be done in place and massively in parallel with many simultaneous module movements. Finally we present a loose quadratic upper bound on the total number of module movements required by the algorithm. 1 I n t r o d u c t i o n Modular Self Reconfigurable Systems consist of many identical robots that are very limited in their actions. As the number of modules in a system increases, the range of behaviors of the group of robots grows exponentially. The task of self-reconfiguration is important for developing self-sufficient systems. The overall system can reconfigure itself to help accomplish certain tasks such as locomotion, object manipulation and sorting, or interaction with other systems, especially when there is a need to adapt to the environment. Previous research has established that by grouping single modules into groups or meta-modules each unit in the system increases its number of degrees of freedom and the reconfiguration tasks are simplified [4, 5, 7, 10]. However using meta-modules limits the granularity of the possible configurations. Rus and Vona require 4x4 meta-modules for complete 2D reconfiguration for expanding cube style modules, while Nguyen et al. explore the possibility of 36 membered meta-modules for 2D reconfiguration with hexagonal modules. We propose meta-modules composed of 8 modules for 3D reconfiguration and guarantee completeness in the parallel reconfiguration. We proceed by motivating the need for a new reconfiguration algorithm. In section 4 we describe the hardware platform currently being developed at PARC (formerly Xerox PARC) that inspired our work. We then describe the new locomotion primitives for the 2x2x2 meta-modules. In section 7 we present the selfreconfiguration algorithm along with its analysis and correctness results. 2 R e l a t e d W o r k The problem of reconfiguration for modular selfreconfigurable robotic systems has received increased interest. This work includes [[1]-[8], [12]-[15]]. In [14] Walter et al. focus on limiting communication between the individual modules. Pamecha and Chirikjian explore probabilistic techniques such as Simulated Annealing in [6]. Rus and Vona have proposed the use of meta-modules to guarantee completeness of reconfiguration spaces (the space of all possible configurations) in [7, 8]. Their melt grow algorithm uses 4x4 metamodules to solve the general reconfiguration problem in two dimensions. More recently as the focus has shifted on decentralized control and parallel actuation, Butler et al. introduced Cellular Automata for distributed control[2], along with the PacMan algorithm for concurrent actuation by several modules [1]. 3 M o t i v a t i o n The ideal reconfiguration algorithm would be complete for all possible shapes, allow for more than one set of concurrent module movements and be completely autonomous. Each of the above algorithms lacks one of the above properties: The PacMan algorithm[I] along with the work by Walter et. al [14], is not complete, while Melt-Grow [8] is not distributed and does not allow for parallel actuation. To incorporate all of the desirable features into one algorithm, we begin by using 2x2x2 meta-modules. Vassilviskii et. al [12] prove completeness for reconfiguration using the 8 membered meta-modules and the algorithm presented here follows this work. We simplify the planning portion of the algorithm while retaining completeness for the meta-module configuration space. We provide an algorithm which performs in place parallel distributed reconfiguration in worst case quadratic time.