Basic methods for the type synthesis of planar mechanisms are proposed and tested in this study. New idea for isomorphism test of basic kinematic chains is suggested and further extended to find the symmetry in a kinematic chain. Based upon these studies, a method is proposed to efficiently enumerate all the basic kinematic chains, given the number of links and degree-of-freedoms. Finally a procedure of type synthesis is illustrated by an application to a realistic mechanism, starting from a set of complete enumeration. More specific contributions of the present study are summarized as follows:
(1) An algorithm is proposed to find the representative edge list of the graph related to a kinematic chain. It is defined as an edge list coming first in the lexicographical order of edge lists of all possible labelings of vertices of the graph. This edge list of the graph is named as a minimal edge list. A heuristic algorithm of obtaining the minimal edge list is proposed. A minimal edge list thus obtained represents a set of isomorphic graphs, and hence can be used as a code for isomorphism test of two arbitrary graphs. During the labeling procedure, vertex symmetries of a given graph can also be identified.
(2) Two kinds of symmetries of kinematic chains are defined. The first is related to the symmetric links and joints, and the second to the type of symmetry of kinematic chains. An algorithm for finding the type symmetry is considered in terms of the vertex-induced group of graphs. To reduce the huge number of infeasible vertex permutations, that would occurs when the whole vertex is used, the set is partitioned into several classes by using the structural properties of vertices. Then the permutations of vertices in the same class need only be considered.
(3) By using the vertex-induced groups of graphs of kinematic chains, a new way of checking the isomorphism between two mechanisms is proposed. Because mechanisms are derived from the basic kinematic chains by assigning various types to links and joints, they can be represented by colored graphs. If there exists a vertex mapping between two colored graphs, which preserves the vertex and edge properties, then they are considered to be isomorphic to each other. The algorithm is also used to check isomorphism between two contracted graphs.
(4) A duality between the graph of a kinematic chain and the corresponding binary chain is found. Thus, by enumerating binary chains, the graphs of kinematic chains can be directly generated. For isomorphism test, the minimal edge list is introduced. Also used is the investigation of symmetry of vertices in a graph. This can reduce the number of redundant isomorphic graphs generated during the earlier stage of enumeration. The algorithm is applied to several practically important cases;1962 chains for the 12 link, 3 degree-of-freedom system, and 6856 for the 12 link, one degree-of freedom case are some of the results.
(5) Type synthesis of a variable -stroke engine mechanism is tried to show the procedure. Rules for designing the structure of the variable-stroke engine mechanism are gathered form literatures, and classified into several groups. Each group contains rules which are necessary in a corresponding step of type synthesis. From the 16 basic kinematic chains enumerated for a 8 link, one degree-of-freedom system, 20 mechanisms are obtained. The present results have brought out one new type of mechanism in addition to those obtained in the literature.
