An efficient methodology for the structural synthesis of geared kinematic chains

~

Mech. Mach. Theory Vol. 32, No. 8, pp. 957-973. 1997

Pergamon

PII: S0094-114X(96)00051-X

© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 009,114x/97 $17.00 + 0.00

AN EFFICIENT METHODOLOGY FOR THE STRUCTURAL SYNTHESIS OF GEARED KINEMATIC CHAINS CHENG-HO HSU and JIN-JUH HSU Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, Republic of China

(Received 3 January 1995; in revised form 5 July 1996; received for publication 1996) Abstract--Through the use of acyclic graphs, an efficient methodology has been developed for the synthesis of the kinematic structure of geared kinematic chains with any number of links. First, a systematic approach is proposed for the enumeration of N-vertex acyclic graphs. For each N-vertex acyclic graph, all the N-vertex geared graphs are generated by adding (N - F - 1) geared edges to the acyclic graphs, eliminating geared graphs which violate the fundamental rules, and non-isomorphic geared graphs are obtained by comparing the structural codes. Finally, the catalog of the graphs of one-degree-of-freedom (DOF) geared kinematic chains with N links are synthesized from the catalog of acyclic graphs with N vertices. As a result, the structural synthesis of one-DOF geared kinematic chains with up to eight links has been successfully constructed. © 1997 Elsevier Science Ltd INTRODUCTION

The analysis and synthesis of kinematic structure of geared kinematic chains have been the subject of several studies for a number of years [1-14]. Basically, three different methods have been developed for the structural synthesis of geared kinematic chains. In the first method, all the one-DOF geared kinematic chains with N links are generated from the catalog of N-vertex admissible graphs by the assignment of N - 2 geared edges [1-3]. The second method is called the recursive generation method [4-6]. One-DOF geared kinematic chains with N links are generated from one-DOF geared kinematic chains with N - 1 links by the process of adding one link with one turning pair and one gear pair. The third method uses the parent-bar-linkage technique for the enumeration of geared kinematic chains [7]. Using these methods, one- and two-DOF geared kinematic chains with up to eight links have been studied. However, for geared kinematic chains, the problem of isomorphism is further convoluted by the fact that two mathematically non-isomorphic graphs can represent mechanisms that are kinematically equivalent. Such graphs are called pseudoisomorphic graphs [5, 8]. It also results in a difficulty in synthesizing the kinematic structure of geared kinematic chains. The total numbers of non-isomorphic geared kinematic chains with more than six links synthesized by the researchers [3, 6] are inconsistent. Therefore, Chatterjee and Tsai [9, 10] presented the canonical graph representation of geared kinematic chains to avert the problem of pseudoisomorphic graphs. They also proposed a method based on the concept of trees for the enumeration of planetary gear trains for automotive automatic transmissions. Recently, Hsu and Lam[ll, 12] presented a graph representation to clarify the kinematic structure of geared kinematic chains. Consequently, all the pseudoisomorphic graphs correspond to the same graph [11, 12]. Hsu [13] also defined the structural codes of geared kinematic chains to detect the structural isomorphism efficiently. Based on the graph representation and the concept of admissible graphs, Hsu [14] proposed a methodology for the structural synthesis of geared kinematic chains. However, the catalog of admissible graphs is not feasible for the synthesis of geared kinematic chains with more than six links. Therefore, based on the graph representation and the concept of acyclic graphs [15], our purpose here is to develop an efficient methodology for the structural synthesis of geared kinematic chains with any number of links and degrees of freedom. KINEMATIC

STRUCTURE

OF GEARED KINEMATIC

CHAINS

Generally, an F-DOF geared kinematic chain with N links refers to an assemblage of N links, N - 1 turning pairs and N - F - 1 gear pairs. It is a closed and connected kinematic chain. A geared chain with zero degrees of freedom is called a geared rigid chain. A geared kinematic chain is a 957

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Cheng-Ho Hsu and Jin-Juh Hsu

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Fig. 1. Functional representation of the Simpson gear system and its geared kinematic chain.

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(a) gearedkinematic graph

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Fig. 2. Graph representation of the Simpson gear system.

geared chain without any geared rigid subchains. Otherwise, it is a degenerate geared chain. Two kinematically equivalent geared kinematic chains are said to be isomorphic in kinematic structure. According to the researches of Hsu and Lam [11, 12], a geared kinematic chain is transformed to the corresponding geared kinematic graph by denoting the links, gear pairs, simple turning pairs, and multiple turning pairs of the gear train by vertices, dashed (geared) edges, solid (turning) edges, and solid (turning) polygons, respectively. If all the geared edges of an N-vertex geared kinematic graph are removed, then we will obtain an N-vertex acyclic graph, which is a connected graph without any circuits [15]. Conversely, if any geared edge is added to an acyclic graph, then a fundamental circuit (f-circuit) is generated. In a geared kinematic graph, an f-circuit consists of three vertices, one geared edge and two turning edges or polygons. The vertex, not incident to the geared edge, is the associated transfer vertex. Moreover, each turning edge or polygon is characterized by a level which identifies the location of its axis in space. For example, Figs 1(a) and 1(b) show the functional representation of the Simpson gear system and the geared kinematic chain obtained by releasing ground link 0. The geared kinematic chain shown in Fig. l(b) is transformed to be the corresponding graph as shown in Fig. 2(a). It contains six vertices, four geared edges and three levels. There are four f-circuits 1-2-3-1, 1-2--4-1, 4--5-3-4

lq

2

(a) Canonical labeling

(b) Adjacencymatrix

[4442214421/410/0210] (c) Structtnal code Fig. 3. Canonical labeling, adjacencymatrix, and structural code of the Fig. 2(a) graph.

Structural synthesisof geared kinematicchains

959

and 4-5-6-4, and vertex 1 is the transfer vertex of f-circuits 1-2-3-1 and 1-2-4-1, and vertex 4 is the transfer vertex of f-circuits 4-5-3-4 and 4-5-6-4. Removing all the geared edges 2-3, 2-4, 3-5 and 5-6 from the graph shown in Fig. 2(a), the corresponding acyclic graph is obtained as shown in Fig. 2(b). FUNDAMENTAL RULES OF THE GRAPHS OF GEARED KINEMATIC CHAINS Based on the basic characteristics of geared kinematic chains, the fundamental rules for the graphs of one-DOF geared kinematic chains with N links, which obey the general degree-of-freedom equation, can be induced as follows: F1. The graphs are closed, connected graphs without any rigid subgraphs, and contain N vertices and N - 2 geared edges. F2. The subgraph obtained by removing all the geared edges from each graph is an N-vertex connected acyclic graph. F3. Any geared edge added to the acyclic graph forms an f-circuit, which consists of three vertices, one geared edge and two turning edges or polygons. F4. The number of f-circuits is equal to the number of geared edges. F5. Each turning edge (polygon) is characterized by a level, so the number of levels is equal to the total number of turning edges and polygons. F6. Each vertex must have at least one incident turning edge or polygon. STRUCTURAL CODES OF GEARED GRAPHS AND ACYCLIC GRAPHS The identification of graph isomorphism is one of the important steps in the structural synthesis of geared kinematic chains. The concept of structural codes [13] have been shown to be an effective means for the identification of the isomorphism of geared kinematic chains. According to the research of Hsu [14], the adjacency matrix of an N-vertex geared graph (acyclic graph) is defined as an N x N symmetric matrix with the entry a~ = 1 if vertex i is adjacent to vertex j with a turning edge, ao = 2 if vertex i is adjacent to vertexj with a geared edge, ao = m if vertex i is adjacent to vertex j with a turning polygon of m vertices, and as/= 0 otherwise. Furthermore, a, = 0. In a graph vertex i is a neighbor of vertex j, if they are incident with a common edge or polygon. The neighborhood degree sequence of vertex i with p neighbors, NDS(i), is defined as a p-digit number formed by the number of neighbors of its p neighbors in descending order. The canonical adjacency matrix of an N-vertex graph is defined as the matrix obtained by labeling the vertices such that the NDSs of vertices are in descending order and the value of the adjacency code is maximum among all the labelings. The symmetric adjacency matrix can be further represented by its upper triangular portion and concatenates the rows in order to obtain the corresponding adjacency code of a graph. For reasons of convenience, the structural code of this graph is denoted as a set of N - 1 integers in which the ith integer, ci, is constructed by the ith row of the upper right triangle of the canonical adjacency matrix, and expressed as: C ~- [ C l / C 2 / . . . / c i / . . . /CN-I]

For example, Figs 3(a), (b), and (c) show the canonical labeling graph, canonical adjacency matrix and structural code of the geared graph shown in Fig. 2(a). Similarly, Figs 4(a), (b), and (c) show the canonical labeling graph, canonical adjacency matrix and structural code of the acyclic graph shown in Fig. 2(b). SYSTEMATIC SYNTHESIS METHODOLOGY In accordance with the fundamental rules, all the graphs of one-DOF geared kinematic chains with N links can be synthesized by adding N - 2 geared edges to all the N-vertex acyclic graphs. However, some graphs synthesized may be open geared graphs or degenerate geared graphs, which violate fundamental rules FI-F6. Such graphs are not geared kinematic graphs and should be screened out. Moreover, some geared graphs synthesized may be isomorphic. It is true that geared

960

Cheng-Ho Hsu and Jin-Juh Hsu

"6 (a) Canonical labeling

(b) Adjacency matrix

[4441014401140010010] (c) Structuralcode Fig. 4. Canonical labeling, adjacency matrix and structural code of the Fig. 2(b) acyclic graph.

graphs synthesized from different acyclic graphs are non-isomorphic. Based on the above facts and the concept of acyclic graphs, a systematic methodology can be summarized for the structural synthesis of the graphs of one-DOF geared kinematic chains with N links as follows: Step 1. Enumerate all the N-vertex acyclic graphs. Step 2. Enumerate all the possible N-vertex geared graphs for each acyclic graph by adding N - 2 geared edges. Step 3. Eliminate geared graphs that violate the fundamental rules. Step 4. Determine the structural codes of geared graphs and check the graph isomorphism by comparing the structural codes of geared graphs. Step 5. Repeat Steps 2-4 until all the N-vertex acyclic graphs have been used, then the catalog of the graphs of one-DOF non-isomorphic geared kinematic chains with N links will be constructed.

STRUCTURAL SYNTHESIS OF ACYCLIC GRAPHS The first step of the synthesis methodology is to enumerate all the N-vertex connected acyclic graphs. In what follows, a simple approach has been proposed to enumerate all the N-vertex acyclic graphs. In the acyclic graph of a geared kinematic chain with N links, each turning edge (polygon) is characterized by a level, so we conclude that if a new vertex and one incident edge are added to any vertex of an (N - 1)-vertex acyclic graph, then an N-vertex acyclic graph is generated, in which the new edge may have a new level or have the same level as one of existing edges or polygons. For example, Fig. 5 shows the steps of the enumeration of all the five-vertex acyclic graphs by adding vertex 5 and its incident turning edge to a four-vertex acyclic graph. Comparing their structural codes, five non-isomorphic acyclic graphs are obtained as shown in Fig. 5. Based on the above conclusion, a systematic approach for the enumeration of N-vertex acyclic graphs from all the ( N - 1)-vertex acyclic graphs can be summarized as follows: 1. List all the ( N - 1)-vertex acyclic graphs. 2. For each acyclic graph, add a new vertex and one incident edge to every vertex to obtain all possible N-vertex acyclic graphs. 3. Determine the structural codes of the N-vertex acyclic graphs enumerated, and check the graph isomorphism by comparing the structural codes of acyclic graphs. Using the proposed procedure, the catalog of acyclic graphs with any number of vertices can be synthesized in a systematic way. As a result, the catalog of acyclic graphs with up to eight vertices has been successfully synthesized. Table 1 lists the number of acyclic graphs with up to eight vertices. Figure 6 shows the atlas of acyclic graphs with up to seven vertices.

Structural synthesis of geared kinematic chains

Acyclic graph

Generated acyclic graphs 1

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Fig. 5. The synthesis of five-vertex acyclic graphs.

E N U M E R A T I O N OF GEARED G R A P H S

The second stgep of the synthesis methodology is to enumerate all the N-vertex geared graphs from the catalog of N-vertex acyclic graphs. Since fundamental circuits in geared kinematic graphs are three-vertex circuits, it is shown that for an acyclic graph geared edges should be added between any two vertices of different levels, called the feasible pairs of vertices, which are adjacent to a common vertex. Otherwise, it cannot form a fundamental circuit. For example, Fig. 7 shows a five-vertex acyclic graph, which can be used to synthesize one-DOF geared kinematic chains with five links. We can find that four feasible pairs of vertices (1, 4), (1, 5), (2, 5) and (3, 4) can be suitable to add three (5 - 1 - 1 = 3) geared edges. Consequently, four (C(4, 3) = 4) geared graphs can be enumerated as shown in Figs 7(a)-(d). Based on the above conclusion, the procedure for the

Table 1. The number of acyclic graphs with three to eight vertices Number of vertices 3 4 5 6 7 8 Number of acyclic graphs 1 3 8 21 58 164

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Acyclie graphs

4

~

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Fig. 6. Atlas of acyclic graphs with three to seven vertices.

enumeration of all possible geared graphs of one-DOF geared kinematic chains with N links from a given N-vertex acyclic graph can be summarized as follows: 1. Identify all the feasible pairs of vertices, say K feasible pairs, for the given acyclic graph. 2. Add N - 2 geared edges to any N - 2 feasible pairs of vertices, then C ( K , N - 2) geared graphs are enumerated.

IDENTIFICATION OF GRAPHS VIOLATING FUNDAMENTAL RULES The third step of the synthesis methodology is to eliminate graphs violating the fundamental rules of geared kinematic chains from the geared graphs enumerated in the second step. Using the

Structural synthesisof geared kinematicchains

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Fig. 7. The enumeration of geared graphs from an acyclic graph. aforementioned method to enumerate geared graphs, then fundamental rules F2-F6 are automatically satisfied. Hence, geared graphs enumerated that violate fundamental rule F1 are open geared graphs or degenerate geared graphs. Such graphs are not the graphs of geared kinematic chains and should be eliminated. In what follows, the concept of fundamental circuits is applied to derive a method for detecting graphs violating fundamental rule F1.

Fig. 8. A seven-vertex open geared graph.

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Fig. 9. A degenerategeared graph and a geared rigid graph.

964

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I. Identification of open graphs Since an open graph contains at least one pendant vertex, which does not belong to any f-circuits, the number of vertices in the vertex set of all the fundamental circuits is less than the number of vertices of this graph. For example, Fig. 8 shows a seven-vertex geared graph, which has five f-circuits (2, 3)1, (3, 4)1, (3, 6)1, (4, 5)1 and (5, 6)1. The vertex set of all the f-circuits is {1, 2, 3, 4, 5, 6}. Since vertex 7 does not belong to this vertex set, it is an open geared graph and vertex 7 is a pendant vertex. Based on this reasoning, open geared graphs can be detected by the following steps: (1) determine all the f-circuits of the given graph, (2) count the number of vertices in the vertex set of all the f-circuits, and (3) check whether it is an open graph or not.

2. Detection of degenerate geared graphs A geared rigid chain refers to a geared chain with zero degree of freedom. A geared kinematic chain (graph) containing geared rigid subchains is called a degenerate geared chain (graph) and should be replaced by a geared kinematic chain with fewer number of links. According to fundamental rule F2, it is shown that an M-vertex geared rigid graph contains M - 1 revolute edges and M - 1 geared edges. Based on the definition of degenerate geared graph, we conclude that if an N-vertex geared graph contains a rigid subgraph with M - 1 f-circuits (geared edges) and M vertices, then it is a degenerate geared graph. For example, Fig. 9(a) shows a seven-vertex graph, in which vertices 4, 5, 6 and 7 form a geared rigid graph with four vertices and three geared edges as shown in Fig. 9(b), it is a degenerate geared graph. Based on this fact, degenerate geared graphs can be identified by using the concept of fundamental circuits. During the synthesis of one-DOF geared kinematic graphs with seven vertices, some degenerate geared graphs have been found as shown in Fig. 10. IDENTIFICATION OF ISOMORPHISM The fourth step of the synthesis methodology is to detect the isomorphism of geared kinematic graphs obtained in the third step. Based on the concept of structural codes proposed by Hsu [13],

Structural synthesis of geared kinematic chains

965

the isomorphism of geared graphs can be identified by the following steps: (1) determine the structural codes of the graphs to be identified, and (2) check for graph isomorphism by comparing their structural codes. Based on the concept of structural code, Hsu and Lain [13] have successfully developed a computer program by using the notation of adjacency matrices such that the isomorphism of geared kinematic chains can be identified automatically. In this paper, the developed computer program is used to detect the isomorphism of geared kinematic graphs. For example, the structural codes of the geared graphs shown in Figs 7(a), (b), (c) and (d) are [3322/321 / 10/0], [3322/321 / 10/0], [3321/312/20/0], and [3321/312/20/0], respectively. Therefore, two non-isomorphic geared graphs are obtained as shown in Fig. 7(a) or (b) and Fig. 7(c) or (d).

Step I: Enumerate geared graphs

Acyclic graph

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Step 2: Delete graphs violating the fundamentalrules f-circuits: (2,4)1, (2,5)1, (3,4)1

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subgraphs Fig. I I. The synthesis procedure of geared kinematic graphs from an acyclic graph.

966

Cheng-Ho Hsu and Jin-Juh Hsu

Step I: Number of geared graphs

Acyclic graph

No.

~

'w

20

Step 3: Atlas of nonisomorphic geared kinematic graphs

geared kinematic graphs

(D~/

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Fig. 12. The synthesis of the graphs of one-DOF geared kinematic chains with five links.

RESULTS Using the proposed synthesis methodology, all the non-isomorphic geared kinematic graphs can be enumerated systematically. In what follows, the structural synthesis of one-DOF geared kinematic chains with five links is used to illustrate the utility of the proposed synthesis methodology. The steps of this procedure are described as follows. Step 1. Eight acyclic graphs with five vertices are listed from the catalog of acyclic graphs as shown in Fig. 6. Step 2. For the acyclic graph shown in Fig. 11, there are five feasible pairs of vertices (2, 4), (2, 5), Table 2. The number of one-DOF geared kinematic chains with three to eight links Number of links

3

4

5

6

7

8

Number of geared kinematic chains

1

3

13

81

647

6360

Structuralsynthesisof gearedkinematicchains

967

(3, 4), (3, 5) and (4, 5). Then three geared edges are added to this acyclic graph, we obtain 10 (C(5, 3) = 10) geared graphs as shown in Figs 1 l(a)-(j). Step 3. Since Figs 1l(i) and (j) are open and degenerate geared graphs, which violate fundamental rule F1, we obtain eight geared kinematic graphs. Step 4. The structural codes of the geared kinematic graphs shown in Figs 1 l(a)-(h) are determined. Compare these structural codes, three non-isomorphic geared kinematic graphs are obtained as shown in Figs l l(a), (d) and (e).

Nonisomorphicgraphsof gearedkinematicchains 1-2 1-3

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Nonisomorphie graphs of geared kinematic chains

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Fig. 13. Atlas of six-vertex, one-DOF geared kinematic graphs. Step 5. Repeat Steps 2-4 for the other seven acyclic graphs, Fig. 12 lists the consequence of the enumeration of 13 one-DOF non-isomorphic geared kinematic graphs with five vertices. Using the proposed synthesis methodology, we have successfully synthesized one-DOF geared kinematic chains with three to eight links from the catalog of three- to eight-vertex acyclic graphs shown in Fig. 6 and Table 1 in a systematic way. Table 2 lists the number of one-DOF geared kinematic chains with three to eight links. Figure 13 shows the enumeration result of 81 one-DOF geared kinematic graphs with six vertices, which are synthesized from 21 six-vertex acyclic graphs. Table 3 and the Appendix show the results of the enumeration of one-DOF geared kinematic graphs with seven vertices from 58 seven-vertex acyclic graphs shown in Fig. 6.

Structural synthesis of geared kinematic chains

969

Table 3. Number of one-DOF, seven-vertex geared kinematic graphs enumerated from 58 seven-vertex acyclic graphs Acyclic Number of non-isomorphic Acyclic Number of non-isomorphic Acyclic Number of non-isomorphic graph no. kinematic Acyclics graph no. kinematic Acyclics graph no. kinematic Acyclics 1 6 (No. 1-6) 2 9 (No. 7-15) 3 27 (No. 16-42) 4 8 (No. 43-50) 5 6 (No. 51-56) 6 4 (No. 57-60) 7 40 (No. 61-100) 8 23 (No. 101-123) 9 4 (No. 124-127) 10 16 (No. 128-143) I1 47 (No. 144-190) 12 6 (No. 191-196) 13 20 (No. 197-216) 14 10 (No. 217-226) 15 1 (No. 227) 16 3 (No. 228-230) 17 24 (No. 231-254) 18 3 (No. 255-257) 19 3 (No. 258-260) 20 3 (No. 261-263) Total number of nonisomorphic kinematic

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 graphs

2 (No. 264-265) 13 (No. 266--278) 5 (No. 279-283) 8 (No. 284-291) 2 (No. 292-293) 20 (No. 294-313) 42 (No. 314-355) 66 (No. 356~21) 22 (No. 422-443) 7 (No. 444-450) 8 (No. 451-458) 5 (No. 459-463) 2 (No. 464-465) 5 (No. 466-470) 27 (No. 471-497) 11 (No. 498-508) 1 (No. 509) 2 (No. 510-511) 4 (No. 512-515) 1 (No. 516)

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

7 (No. 8 (No. 21 (No. 7 (No. 4 (No. 1 (No. 1 (No. 17 (No. 10 (No. 6 (No. 25 (No. 10 (No. 6 (No. 1 (No. 1 (No. 1 (No. 2 (No. 3 (No.

517-523) 524-531) 532-552) 553-559) 560-563) 564) 565) 566-582) 583-592) 593-598) 599-623) 624-633) 634-639) 640) 641) 642) 643-644) 645-647)

647

DISCUSSION The acyclic graph method proposed in this investigation provides several advantages: (1) all pseudoisomorphic graphs are represented by the only geared kinematic graph, (2) there is no need to represent either the level associated with each turning edge or the transfer vertex associated with each f-circuit, (3) geared kinematic graphs generated from different acyclic graphs are non-isomorphic, and (4) geared graphs violating fundamental rules can be identified by the vertex set of fundamental circuits. Based on these advantages, the task of the structural synthesis of geared kinematic chains are largely simplified. The probability of human error is significantly reduced. The results of one-DOF geared kinematic chains with up to five links are in complete agreement with those of Freudenstein [8], Ravisankar and Mruthyunjaya [2] and Tsai [4]. The catalog of 81 one-DOF geared kinematic chains with six links is consistent with the result of Hsu [14]. For one-DOF geared kinematic chains with seven links, Kim and Kwak [6] enumerated 642 displacement graphs by using the recursive generation method. However, Hsu and Lam [16] have showed that some isomorphic geared graphs were mistaken to be non-isomorphic graphs. Hsu and Lin [17] show that the correct number of one-DOF displacement graphs with seven vertices is 636. Using the admissible graph method, Shin and Krishnamurty [3] enumerated 659 geared kinematic chains with seven links from 37,884 geared (colored) graphs. However, Figs 14 (a) and (b) show two geared kinematic graphs with different standard codes, but they are pseudoisomorphic graphs [4]. These graphs have identical kinematic structure and can be represented by the same graph as shown in Fig. 14(c). Based on the proposed synthesis methodology and the structural code technique, one-DOF geared kinematic chains are enumerated from the catalog of acyclic graphs in an effective way. Further, the numbers of one-DOF geared kinematic chains with seven and eight links are 647 and 6360, respectively. These results are believed to be correct. CONCLUSIONS The concept of acyclic graphs, as defined in this investigation, has been shown to be a powerful tool in the structural synthesis of the graphs of geared kinematic chains. In this study, through the use of acyclic graphs, the effort in the identification of graph isomorphism is largely reduced during the process of structural synthesis of geared kinematic chains. Using the proposed synthesis methodology, one-DOF geared kinematic chains with any number of links can be systematically synthesized. The synthesis methodology can be also applied for the structural synthesis of multi-DOF geared kinematic chains. Based on the proposed methodology, it is hoped that a fully computerized algorithm can be easily developed for automatic synthesis of geared kinematic chains by using the notation of adjacency matrix [18].

970

Cheng-Ho Hsu and Jin-Juh Hsu

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(e) Stmct~al code = [444211/44220/4100/002/00/2] Fig. 14. Two pseudoisomorphic graphs having different standard codes and their corresponding geared kinematic graph.

Acknowledgements--The authors are grateful to the National Science Council of the Republic of China for the support of this research under grant Nos. NSC81-0422-E110-532 and NSC83-0401-E110-032. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Buchsbaum, F. and Freudenstein, F., Journal of Mechanisms, 1970, 5, 357-392. Ravisankar, R. and Mruthyunjaya, T. S., Mechanism and Machine Theory, 1985, 20, 367-387. Shin, J. K. and Krishnamurty, S., Mechanism and Machine Theory, 1993, 28, 347-355. Tsai, L. W., ASME Journal of Mechanisms, Transmissions, and Automation in Design, 1987, 109, 329-337. Tsai, L. W. and Lin, C. C., ASME Journal of Mechanisms, Transmissionsand Automation in Design, 1989, 111,524-529. Kim, J. U. and Kwak, B. M., Mechanism and Machine Theory, 1990, 25, 563-574. Sohn, W., A computer aided approach to the creative design of mechanisms. Ph.D. dissertation, Columbia University, New York, 1987. Freudenstein, F., ASME Journal of Engineeringfor Industry, 1971, 931}, 176-182. Chatterjee, G. and Tsai, L. W., SAE Transactions, Paper No. 941012, 1994. Chatterjee, G. and Tsai, L. W., ASME 1994 Design Technical Conferences, 1994, DE-Vol. 71, 275-282. Hsu, C. H. and Lam, K. T., ASME Journal of Mechanical Design, 1992, 114, 196-200. Hsu, C. H. and Lain, K. T., ASME Journal of Mechanical Design, 1993, 115, 631-638. Hsu, C. H., Mechanism and Machine Theory, 1994, 29, 513-523. Hsu, C. H., Journal of the Franklin Institute, 1993, 330, 913-927. Harary, F., Graph Theory. Addison-Wesley, Massachusetts, 1969. Hsu, C. H. and Lain, K. T., ASME Flexible Mechanisms: Dynamics and Analysis, 1992, DE-47, 515-522. Hsu, C. H. and Lin, Y. L., Mathematical and Computer Modelling, 1994, 19, 67-81. Hsu, C. H., Kinematic structure of planetary gear trains. Technical Report NSC 83-0401-El10-032, Taiwan, 1994,

Structural synthesis of geared kinematic chains APPENDIX

Catalog of One-DOF Geared Kinematic Graphs with Seven Vertices

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Cheng-Ho Hsu and Jin-Juh Hsu APPENDIX

continued

Catalog of One-DOF Geared Kinematic Graphs with Seven Vertices

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Structural synthesis of geared kinematic chains A P P E N D I X continued

Catalog of One-DOF Geared Kinematic Graphs with Seven Vertices

973