Enumerating the Non-IsomorphicAssembly Con gurations of Modular Robotic

A modular robotic system consists of standardized joint and link units that can be assembled into a number of different kinematic configu rations to meet various task requirements. Owing to typical symme tries in module design, different assembly configurations may lead to robotic structures that are geometrically identical, or isomorphic. This paper focuses on the problem of enumerating the set ofkinemat ically distinct modular robot assembly configurations from a given set of modules. We first consider how to enumerate the nonisomor phic (or geometrically unique) assembly configurations of a modular robotic system. Our scheme is based on a novel representation of a modular robot assembly configuration as an assembly incidence matrix (AIM). Equivalence relations based on symmetries in mod ule geometry and graph isomorphisms are defined on the AIMs. An enumeration algorithm to generate nonisomorphic assembly config urations based on this equivalence relation is proposed. We then present an algorithm to identify the kinematically equivalent robots. The application of these two algorithms will result in the set of kine matically unique assembly configurations. Examples demonstrate that this method is a significant improvement over a brute-force enu meration process.

[1]  R. Cohen,et al.  Conceptual Design of a Modular Robot , 1992 .

[2]  John B. Fraleigh.  A first course in abstract algebra , 1967 .

[3]  Sandra Mitchell Hedetniemi,et al.  Constant Time Generation of Rooted Trees , 1980, SIAM J. Comput..

[4]  Roger W. Brockett,et al.  Robotic manipulators and the product of exponentials formula , 1984 .

[5]  L. S. Woo,et al.  Type Synthesis of Plane Linkages , 1967 .

[6]  S. Shankar Sastry,et al.  Optimal Kinematic Design of 6R Manipulators , 1988, Int. J. Robotics Res..

[7]  Frank Harary,et al.  Logical Foundations of Kinematic Chains: Graphs, Line Graphs, and Hypergraphs , 1990 .

[8]  A. V. Kozina.  Coding and generation of nonisomorphic trees , 1979 .

[9]  Brendan D. McKay,et al.  Constant Time Generation of Free Trees , 1986, SIAM J. Comput..

[10]  J. Denavit,et al.  A kinematic notation for lower pair mechanisms based on matrices , 1955 .

[11]  Pradeep K. Khosla,et al.  Integration of real-time software modules for reconfigurable sensor-based control systems , 1992, Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems.

[12]  Toshio Fukuda,et al.  Dynamically reconfigurable robotic system , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[13]  Ferdinand Freudenstein,et al.  Some Applications of Graph Theory to the Structural Analysis of Mechanisms , 1967 .

[14]  W. Miller.  Symmetry groups and their applications , 1972 .

[15]  Takeo Kanade,et al.  The CMU reconfigurable modular manipulator system , 1988 .

[16]  Steven Skiena,et al.  Implementing discrete mathematics - combinatorics and graph theory with Mathematica , 1990 .

[17]  Éva Tardos,et al.  Combinatorics in computer science , 1996 .

[18]  Richard M. Murray,et al.  A Mathematical Introduction to Robotic Manipulation , 1994 .

[19]  E. Beckenbach,et al.  Applied Combinatorial Mathematics , 1965 .

[20]  C. L. Liu,et al.  Introduction to Combinatorial Mathematics. , 1971 .

[21]  Erik K. Antonsson,et al.  Grammars for engineering design , 1994 .

[22]  Hong-Sen Yan,et al.  The specialization of mechanisms , 1991 .

[23]  Narsingh Deo,et al.  Graph Theory with Applications to Engineering and Computer Science , 1975, Networks.

[24]  I. Herstein,et al.  Topics in algebra , 1964 .

[25]  J. J. Uicker,et al.  A Generalized Symbolic Notation for Mechanisms , 1971 .

[26]  I. Chen.  Theory and applications of modular reconfigurable robotic systems , 1994 .