Distributed Linear Heterogeneous Reconfiguration of Cubic Modular Robots via Simultaneous Tunneling and Permutation

Two types of reconfiguration—heterogeneous reconfiguration, in which all the modules are considered to be nonidentical, and homogeneous reconfiguration, in which they are considered to be identical—are widely studied in modular robotics research. Homogeneous reconfiguration can manage the transformation of the whole configuration of the robot structure, whereas heterogeneous reconfiguration can manage not only this transformation, but also the permutation process for navigating all modules to their exact target positions in the transformed configuration. However, the transformation process and time-consuming permutation process cannot always be executed simultaneously in heterogeneous reconfiguration. As a result, it takes much more time to reconfigure heterogeneous modular robots than it does to reconfigure homogeneous ones. In the previous research, linear homogeneous transformation algorithms for modular robots have been proposed. However, only quadratic permutation algorithms are available for heterogeneous lattice modular robots. This article studies a reconfiguration algorithm for heterogeneous lattice modular robots with linear operation time cost. The algorithm is based on simultaneous tunneling and permutation; a robot transforms its configuration via homogeneous tunneling motion, but permutation of each module's position is carried out simultaneously during the tunneling transformation. The approach also makes the time cost for the whole permutation process linear. To achieve this, the idea of a transparent meta-module is introduced, which allows the modules belonging to a meta-module to pass through the spaces occupied by other meta-modules. The algorithm is provided in both centralized and distributed form. We prove the correctness and completeness of the proposed algorithm. We also show some results of the reconfiguration simulation of heterogeneous modular robots with three-dimensional 2 × 2 × 2 cubic meta-modules by the proposed algorithm.