Type synthesis of unified planar–spatial mechanisms by systematic linkage and topology matrix-graph technique

Using characteristic strings to derive valid contracted graphs with hexagonal links plus other links for type synthesis of closed mechanisms

“…Each of strings includes serial digits. An associated linkage (AL) is an acceptable group of links [5,6]. Generally, several different CGs corresponding to one AL can be constructed.…”

“…A TG is similar to a CG, except that each of the edges is replaced by some B connected in series by some joints (J) with one DoF. A vertex vs. H, P, Q and T is connected with (6,5,4,3) edges, respectively. Let n 6 , n 5 , n 4 and n 3 be the numbers of H, P, Q and T, respectively.…”

“…The topology graph (TG) is a simple and effective tool for type synthesis of closed mechanisms [1][2][3][4][5][6]. The contracted graph (CG) is a basic and effective tool for deriving various TGs [5,6].…”

“…Gogu [1] studied type synthesis of PMs by morphological and evolutionary approaches. Lu and Leinonen [6] derived some CGs from ALs by the adjacency matrices and derived some TGs from CGs. By changing types and motion orientations of joints, Yan et al [10,11] studied the configuration synthesis of mechanisms with variable topological representations and characteristics of variable kinematic joints.…”

“…However, to directly derive and identify the isomorphic kinematic chain from TGs before identifying isomorphic/invalid CGs may lead to great unnecessary mental work, and may even have different solutions when identifying the same kinematic chain. In fact, many different CGs can be derived from the same AL, many different TGs can be derived from the same CG, and many different mechanisms can be derived from the same TG [1][2][3][4][5][6][7][8][9]26]. Suppose there are two different CG x and CG y.…”