Enumeration of Graphs of Kinematic Chains In Chapter 3 we have shown that the topological structures of kinematic chains can be represented by graphs. Several useful structural characteristics of graphs of kinematic chains were derived. In this chapter we show that graphs of kinematic chains can be enumerated systematically by using graph theory and combinatorial analysis. There are enormous graphs. Obviously, not all of them are suitable for construction of kinematic chains. Only those graphs that satisfy the structural characteristics described in Chapter 4 along with some other special conditions, if any, are said to be feasible solutions | Chapter 5 Enumeration of Graphs of Kinematic Chains Introduction In Chapter 3 we have shown that the topological structures of kinematic chains can be represented by graphs. Several useful structural characteristics of graphs of kinematic chains were derived. In this chapter we show that graphs of kinematic chains can be enumerated systematically by using graph theory and combinatorial analysis. There are enormous graphs. Obviously not all of them are suitable for construction of kinematic chains. Only those graphs that satisfy the structural characteristics described in Chapter 4 along with some other special conditions if any are said to be feasible solutions. The following guidelines are designed to further reduce the complexity of enumeration 1. Since we are primarily interested in closed-loop kinematic chains all graphs should be connected with a minimal vertex degree of 2. 2. All graphs should have no articulation points or bridges. A mechanism that is made up of two kinematic chains connected by a common link but no common joint or by a common joint but no common link is called a fractionated mechanism. Fractionated mechanisms are useful for some applications. However the analysis and synthesis of such mechanisms can be accomplished easily by considering each nonfractionated submechanism. Therefore this type of mechanism is excluded from the study. 3. Unless otherwise stated nonplanar graphs will be excluded. Although this is somewhat arbitrary in view of the complexity of such mechanisms it is reasonable to exclude them. It has been shown that for the graph of a planar one-dof linkage to be nonplanar it must have at least 10 links. Perhaps Cayley Redfield and Polya are among the first few pioneers in the development of graphical enumeration theory. Polya s enumeration theorem provides a powerful tool for counting the number of graphs with a given number of vertices 2001 by CRC Press LLC and edges 7 . A tutorial paper on Polya s theorem can be found in .
