Topological Structure Representation And Configuration Transformation Ananlysis On Rubik's Cube

With the continuous upgrading of the demand for engineering machinery products,higher requirements have been put forward for robot technology and advanced mechanical equipment.Mechanism is an important part of the machine equipment,and the innovation of its principle is the source and foundation of the independent innovation of machinery.Rubik's cube mechanism is different from traditional mechanism with the characteristics of variable topology and variable degree of freedom because of special connection mode among components.In this paper,the classical three order cub was taken as the research object,the method to desribe the topological characteristics of structure has been put forward and the mathematical model of configuration transformation for Rubik's cube has been created by analyZing its topology and topological structure transformation.The research contents of this paper are as follows.The origin and development of Rubik's cube and the character application of Rubik's cube are systematically introduced.The component types are classificatied according to component characteristics.Analyse the connections between components of Rubik's cube,summariZes the change law of topological structure and the degree of freedom in the process of movement.Then put forward the concepts definition of operation in the process of rotation.The topological graph of all different topological structures of mechanism which appears under the conditions of specific rotation are drawn based on the theory of topology and graph,This paper mainly studies all the topological structures of the upper and lower layers of Rubik's cube mechanism under the conditionsof single operation around the given axis,then establishes the adjacency matrix of topological structure and analyZes the transformation between different configurations.According to the topology graph,the corresponding adjacency matrix is established.But the traditional adjacency matrix is too cumbersome,the traditional matrix combined was optimized with the feature of topological graph encoding and the characteristics of Rubik's cube mechanism.The unit elements of the upper and lower are extracted from all the topological structure,and the unit adjacency matrix between the layer components is established.The elements that are not changed in the traditional adjacency matrix of Rubik's cube are removed,and two unit adjacency matrixes are combined into one configuration adjacency matrix,so as to propose a set of mathematical description for the topological structure of Rubik's cube mechanism.The topology structure of the mechanism will change during the rotation of the upper and lower layers in the forward single operation around a given axis.In this paper,the path and rule of Rubik's cube mechanism transformation are analyzed,combined with the optimized adjacency matrix,the transformation equations among the different configuration are established and solved.Thus,a set of mathematical description models suitable for the configuration change of Rubik's cube is formed,which lays a theoretical foundation for the study of the degree of freedom,kinematics and dynamics of Rubik's cube mechanism.