The configuration-function transition digraphs of metamorphic mechanisms or variable topology mechanisms
Chuanhe Liu · ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots · 2009
In this paper, the new concepts such as configuration set, topological function, topological function set, configuration-function, and configuration-function set etc. of the metamorphic mechanisms or variable topology mechanisms are put forward for the first time. Six topological properties which include the μ degree of the motion of basic input and output motions are revealed, and the Boolean motion screw categorical data matrix describing these six topological properties is also established, which provides a new scientific and practical tool for the description of the topological function. The mathematic definition of the configuration-function transition digraph is given for the first time, and the configuration-function transition directed figure set T is divided into three subsets T 1−1 , T 1−α and T α−1 . Two new types of the configuration-function transition digraph are given and the new concepts such as the reversible digraph, the irreversible digraph, the directed-edge-digraph, the forked digraph, the convergent digraph and the combined digraph are found out. The definitions and representation examples of the complete incidence matrix, adjacency matrix and its adjoint matrix of the configuration-function transition digraph are expressed. Six new properties of the configuration-function transition digraph are revealed. Two configuration-function transition principles are advanced. Ten new theorems are given and proved strictly: i.e. the equivalent theorem of theT 1−α -type and the T α−1 -type directed figure with the directed edges, the connection theorem, the simple-non-trivial theorem, the complete-reversible theorem, the complete-irreversible theorem, and the reversible-degree theorem about reversibility, the “OR” theorem and the “AND” theorem on second order relation, and two theorems reflecting the relation of condition control signal (weight). These theorems reveal the change laws of the topological structure and its topological function of the metamorphic mechanisms or variable topology mechanisms.