Non-isomorphic Configurations Research Articles

Configuration Enumeration of Multi-Limb Reconfigurable Modular Robots

By introducing modular and reconfigurable designs, the Multi-limb Reconfigurable Modular Robots (MLRMR) have many different configurations instead of a single fixed configuration, which greatly expands the boundaries of the operation. However, the number of non-isomorphic configurations increases exponentially with the number of modules, making it necessary to explore the enumeration method theoretically. Based on the group theory, the permutation groups of the Multi-limb (including three-limb, four-limb, and five-limb) Reconfigurable Modular Robots are established in this paper, and the theoretical formulas of configuration enumeration are derived by applying the Pólya enumeration theorem. This work can be used as the basis of configuration analysis.

Enumerating the Nonisomorphic Configurations of a Modular Reconfigurable Robot

Abstract By introducing the modular and reconfigurable design, the limbs of a hexapod robot can be assembled into different configurations to meet various task requirements. Contrary to the fixed configuration system, the task capabilities of the modular reconfigurable robot vary with the configurations. This article addresses the problem of the configuration enumeration and configuration expression of a modular reconfigurable robot. First, by establishing the permutation group of the regular hexagon (the base shape of the modular reconfigurable robot) and adopting the Pólya enumeration theorem, the theoretical formula of the nonisomorphic configuration enumeration is derived. Then, considering the change in structural features, equivalence relations based on structural features are defined and the structural feature method (SFM) is proposed to enumerate the nonisomorphic configurations. In addition, to express configurations more intuitively, a senary vertex identification array with the same dimension as the number of modules is presented, which can be transformed from the decimal index of the configuration. Simulation analysis demonstrates that the result of the SFM is consistent with that of the theoretical calculation, which also confirms the effectiveness and accuracy of the two methods applied in the nonisomorphic configuration enumeration of the modular reconfigurable robot.

Colouring problems for symmetric configurations with block size 3

AbstractThe study of symmetric configurations with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in which no block is monochromatic; the second is strong colouring, in which every block is multichromatic. The former has been studied before in relation to blocking sets. Results are proved on the possible sizes of blocking sets and we begin the investigation of strong colourings. We also show that the known and configurations without a blocking set are unique and make a complete enumeration of all nonisomorphic configurations. We discuss the concept of connectivity in relation to symmetric configurations and complete the determination of the spectrum of 2‐connected symmetric configurations without a blocking set. A number of open problems are presented.

On some generalization of the Möbius configuration

The Mobius (8 4 ) configuration is generalized in a purely combinatorial approach. We consider (2 n n ) configurations M ( n , φ ) depending on a permutation φ in the symmetric group S n . Classes of non-isomorphic configurations of this type are determined. The parametric characterization of M ( n , φ ) is given. The uniqueness of the decomposition of M ( n , φ ) into two mutually inscribed n -simplices is discussed. The automorphisms of M ( n , φ ) are characterized for n ≥ 3 .

Module Design and Functionally Non-Isomorphic Configurations of the Hex-DMR II System

Modular robots have captured the interest of the robotics community over the past several years. In particular, many modular robotic systems are reconfigurable, robust against faults, and low-cost due to mass production of a small number of different homogeneous modules. Faults in these systems are normally tolerated through redundancy or corrected by discarding damaged modules, which reduces the operational capabilities of the robot. To overcome these difficulties, we previously developed and discussed the general design constraints of a heterogeneous modular robotic system (Hex-DMR II) capable of autonomous team repair and diagnosis. In this paper, we discuss the design of each module, in detail, and present a new, novel elevator module. Then, we introduce a forestlike structure that enumerates every non-isomorphic, functional agent configuration of our system. Finally, we present a case study contrasting the kinematics and power consumption of two particular configurations during a mapping task.

Enumeration of the Non-Isomorphic Configurations for a Reconfigurable Modular Robot with Square-Cubic-Cell Modules

Configuration of a reconfigurable modular system is a tough issue because the possible configurations or structures grow exponentially with the number of modules. A library of the non-isomorphic configurations should be set up as a database for configuration design and control. In this paper, we propose a matrix-based enumerating approach for the non-isomorphic configurations of a reconfigurable modular robot system with square-cubic-cell(SCC) modules. Each SCC module considered in this study includes a cubic module body and four connectors on its lateral surfaces. Since it has a square lattice like 2D projection, configuration matrices are proposed to represent the topological information of the modular robot. Thus reconfiguration and enumeration can be programmed by matrix computation in simulation. According to combinational principle, recursive algorithms are used to find out all non-isomorphic configurations. Enumerating result for a multi-module SCC system is provided as an example. Potential applications of this approach to other reconfigurable systems with lattice-form or cubic-form modules are discussed as well. As an application, enumeration of the multi-module CONRO robot also proves the validity of this method.

Network-based reconfiguration routes for a self-reconfigurable robot

This paper presents a network-based analysis approach for the reconfiguration problem of a self-reconfigurable robot. The self-reconfigurable modular robot named “AMOEBA-I” has nine kinds of non-isomorphic configurations that consist of a configuration network. Each configuration of the robot is defined to be a node in the weighted and directed configuration network. The transformation from one configuration to another is represented by a directed path with nonnegative weight. Graph theory is applied in the reconfiguration analysis, where reconfiguration route, reconfigurable matrix and route matrix are defined according to the topological information of these configurations. Algorithms in graph theory have been used in enumerating the available reconfiguration routes and deciding the best reconfiguration route. Numerical analysis and experimental simulation results prove the validity of the approach proposed in this paper. And it is potentially suitable for other self-reconfigurable robots’ configuration control and reconfiguration planning.

REPRESENTING AND ENUMERATING THE NON-ISOMORPHIC CONFIGURATIONS OF THE RECONFIGURATION MODULAR ROBOT

A novel link-type reconfigurable robot which can shift shape is proposed and a three-module robot platform is introduced. A single module mainly includes module body, the link arm and the offset joints. The number of the module can be decided by actual need. The link type has two kinds of linkage, which is regular linkage and irregular linkage. The relative position of the adjacent modules can be changed through rotation of the offset joints. Numerous non-isomorphic configurations of the robot come from restructuring and shape shifting. The configuration matrices are proposed to represent the topology configurations of the reconfigurable module robot. According to combination principle, recursive algorithm is proposed to enumerate and evaluate the non-isomorphic configurations and non-isomorphic symmetry configurations. Three-module reconfigurable shape-shifting robot's symmetry con- figurations are designed to testify the algorithm's validity.

Number synthesis of kinematic chains based on permutation groups

A systematic and precise approach is developed for enumerating non-isomorphic kinematic chains based on the theory of permutation groups. First, we define contracted link adjacency matrices of kinematic chains, and elements in the matrices are separated into four sets. We then propose an algorithm for assigning values to elements of these sets to generate non-isomorphic configurations according to their permutation groups. As a result, the numbers of simple kinematic chains with up to twelve links and seven degrees of freedom are listed.