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A graph representation for the structural synthesis of geared kinematic chains
A Graph Representation for the Structural Synthesis of Geared Kinematic Chains by CHENG-HOHSU
Department of Mechanical Engineering, Kaohsiung, Taiwan 80424, R.O.C.
National Sun Yat-Sen
University,
This paper presents a systematic approach, which is based on a new graph ABSTRACT: representation of geared kinematic chains, for the structural synthesis of geared kinematic chains. First, the new graph representation of geared kinematic chains is introduced. Next, a simple method is proposed to generate all possible (N+ I)-link geared kinematic chains from N-link geared kinematic chains. Then, isomorphic geared kinematic chains are identified to obtain nonisomorphic geared kinematic chains by comparing their Structural Codes. Finally, an atlas of nonisomorphic geared kinematic chains for one deqree-of-freedom planetary gear trains with up to seven links is constructed.
I. Introduction
Graph theory has shown to be a useful tool for the structural analysis and synthesis of mechanisms. Buchsbaum and Freudenstein (1. 2) first applied graph theory to the structural synthesis and analysis of one degree-of-freedom (DOF) geared kinematic chains. Ravisankar and Mruthyunjaya (3) computerized Buchsbaum and Freudenstein’s method (1) to derive geared kinematic chains with up to six links. Tsai and Lin (4, 5) presented a systematic approach for the structural synthesis of one- and two-DOF planetary gear trains with up to four gear pairs. Kim and Kwak (6) computerized the method proposed by Tsai (4) to synthesize geared kinematic chains for one-DOF planetary gear trains with up to seven links. The synthesis procedure (4-6) includes the following steps : (1) enumerate all the permissible unlabeled graphs of Nf 1 vertices for all known unlabeled graphs of N vertices by the generation method. (2) Check for graph isomorphism to obtain a set of nonisomorphic unlabeled graphs. (3) Identify the transfer vertices by inspection or by using Boolean algebra technique. (4) Eliminate unlabeled graphs violating the fundamental rules of planetary gear trains. (5) Transform the remaining graphs into rotation graphs and check for rotational isomorphism. (6) Label the revolute edges of the rotationally nonisomorphic graphs to obtain all possible nonisomorphic displacement graphs. This synthesis procedure is systematic and works well for simple mechanisms, but it becomes quite complicated and tedious as the number of links increases. Furthermore, it has a major disadvantage in that a planetary gear train may correspond to several different graphs (7-10). Recently, Hsu and Lam (9, 10) have presented a new graph representation to
The Frankhn lnst1lute00l&Q032.'93 $5.00+0.00
131
Chenq- Ho Hsu clarify the kinematic structure of planetary gear trains effectively. Based on the new graph representation, the kinematic and structural analyses of geared kinematic chains for planetary gear trains can be performed in an unambiguous and systematic way. The purpose of this work is to extend the new graph representation to develop a systematic method for the structural synthesis of geared kinematic chains. In what follows, we shall first review the graph representation of a planetary gear train including the fundamental rules of the graph, and then introduce an efficient method for detecting isomorphic graphs. Finally, a systematic approach for the synthesis of the graphs of geared kinematic chains for planetary gear trains is developed. II. Graph Representation
of Geared Kinematic
Chains
Functional representation of a planetary gear train refers to the conventional schematic drawing of the mechanism. Figure l(a) shows the functional representation of the Simpson gear system, which contains six links, five turning pairs and four gear pairs. The links are numbered arbitrarily from 1 to 6, and the location (level) of each axis is labeled as a-a, b-b and c--c, respectively. The corresponding kinematic chain is obtained by releasing the ground link as shown in Fig. 1(b). Based on the researches of Hsu and Lam (9, lo), a geared kinematic chain can be transformed into its corresponding graph by denoting the links, gear pairs, simple and multiple revolute pairs of the kinematic chain by vertices, dashed (geared) edges, solid (revolute) edges and polygons, respectively. The degree of a vertex is defined as the total number of revolute edges and polygons incident to the vertex. A fundamental circuit (f-circuit) is a circuit consisting of three vertices, one geared edge and two revolute edges/polygons, and the vertex, not incident to the geared edge, is the associated transfer vertex. Each revolute edge or polygon represents a level, which indicates the location of its axis in space.
(a) FIG. I. The Simpson gear system and its corresponding 132
0) geared kinematic chain Journal althe Franklm Institute Pergamon Press Ltd
Graph Representation
I\
4 I
\
/6
\
,’
3’,
\
a
\
\
\
16
\
‘I
I I
Chains
\
1 ‘\ \
I’
of Geared Kinematic
\
\
4
C
5
FIG. 2. The graph representation
of the Simpson gear system.
In the graph of a geared kinematic chain, vertices i and,j are said to be adjacent and vertex i is a neighbor of vertex j, if they are incident with a common edge/ polygon. The neighborhood of vertex i is the set N(i) consisting of all vertices adjacent to vertex i. The number of neighbors of vertex i, n(i), is the number of vertices in N(i). The neighborhood degree sequence of vertex i with p neighbors, denoted as NDS(i), is defined as a p-digit number forming by the numbers of neighbors of its p neighbors in descending order. For example, Fig. 2 shows the graph of the Simpson gear system, shown in Fig. 1, which contains four vertices 2, 3,4 and 5 of degree three, two vertices 1 and 6 of degree two, four geared edges 2-3,224, 335 and 5-6, two revolute edges l-2 and 4-5, and one revolute polygon of four vertices 1, 3, 4 and 6. It has three different levels and four f-circuits l-2-31, l-241, 553-45 and 5-6645. Vertex 1 is the associated transfer vertex of f-circuits 1-2-3-1 and I-241, and vertex 4 is the associated transfer vertex of f-circuits 5-3345 and 5-6-45. Since vertices 1, 4 and 6 are adjacent to vertex 3 with a polygon, and vertices 2 and 5 are adjacent to vertex 3 with an edge, the neighborhood and the number of neighbors of vertex 3 are N(3) = {l, 2, 4, 5, 6) and n(3) = 5, respectively. Since the number of neighbors of vertices 1, 2,4, 5 and 6 are four, three, five, three and four, the NDS of vertex 3 is 54433. III. Fundamental Rules of the Graphs of Geared Kinematic Chains According to Hsu and Lam (9, lo), the following fundamental rules have been established for the graphs of geared kinematic chains, which obey the general degree-of-freedom equation : (Fl) For an N-link, F-DOF geared kinematic chain, let its corresponding graph have N vertices, e, geared edges, e2 revolute edges, e3 revolute polygons of three vertices, e4 revolute polygons of four vertices,. , and ek revolute polygons of k vertices, then it satisfies the following equations : e,=N-1-F e,=e2+
(1)
t(i-l)e,=N-1
(2)
i= 3
in which eK is the total number Vol 330. No. I, pp. 131-143.1993 Printed in Great Bnlam
of equivalent
revolute
edges in this graph.
133
Cheng- Ho Hsu (F2) The subgraph obtained by deleting all the geared edges is a connected graph with no circuits. (F3) Any geared edge added to the connected graph forms an f-circuit. An f-circuit has three vertices, one geared edge and two revolute edges/polygons, and the vertex, not incident to the geared edge, is the associated transfer vertex. (F4) The number of f-circuits equals the number of geared edges. (F5) Each revolute edge/polygon can be characterized a level, which identifies the location of its axis in space. Thus, the number of levels equals the total number of revolute edges and polygons. (F6) Each vertex must have at least one incident revolute edge/polygon.
IV. Identification
of Isomorphism
of Geared Kinematic
Chains
In the synthesis of the structure of geared kinematic chains, we need to provide a precise means for checking isomorphism. Hsu (11) has presented an efficient method, called the Structural Codes, to identify the isomorphism of the graphs of geared kinematic chains, hence we apply the concept of Structural Code for this purpose. The vertex-vertex adjacency matrix (A) of the graph of an N-link geared kinematic chain, which was first defined by Hsu and Lam (9), and defined as a square matrix of order N with the entry a, = 1 if vertex i is adjacent to vertexj with a revolute edge, a, = 2 if vertex i is adjacent to vertex j with a geared edge, ai, = rn if vertex i is adjacent to vertex j with a revolute polygon of rn vertices, and a, = 0 otherwise. Furthermore, arr = 0. For example, the adjacency matrix of the graph shown in Fig. 2 is given by -0
1
4
4
0
4-
102200 A=
420424 424014’ 002102 404420
For an N-vertex graph, there are N! possible labelings and N! associated adjacency matrices. Hsu (11) defines the canonical adjacency matrix of this graph as an adjacency matrix obtained by relabeling the vertices such that the NDSs of vertices in this graph are in descending order, and the value of the code produced by the upper right triangle of the adjacency matrix of the relabeled graph is the largest among all possible labelings of this graph. Hsu (11) also defines the Structural Code of this graph as a set of N- 1 numbers in which the ith number, c,, is constructed by the ith row of the upper right triangle of the canonical adjacency matrix of this graph. For reasons of simplicity and clarity, the general form of the Structural Code of an N-vertex graph is expressed as : c = [c-,/c-z/. . ./c,/. . ./c,_ ,] Journal
134
of the Franklin lnst~tute Pergamon Press Lrd
Graph Representation
of Geared Kinematic
Chains
6 FIG. 3. The canonical labeling of the Fig. 2 graph.
For every graph there is a unique Structural Code corresponding to it. If the graphs of geared kinematic chains have the same Structural Code, then they are isomorphic. Moreover, the Structural Code can describe the topological structure of graphs up to isomorphism. For example, since the NDSs of vertices 1, 2, 3, 4, 5 and 6 of the graph shown in Fig. 2 are 5543, 554, 54433, 54433, 554, 5543, there are eight (2!2!2! = 8) different possible relabelings. It is shown that the canonical labeling of this graph can be achieved by the relabeling (3, 4, 1, 6, 5, 2), which means vertex 3 is relabeled as the new vertex 1, vertex 4 as the new vertex 2, and so on. Consequently, the graph with canonical labeling of the graph is shown in Fig. 3, and the Structural Code is [44422/4421/410/02/O]. Similarly, Fig. 4 shows the functional representation and the graph with canonical labeling of the HydraMatic 440-T4 gear system (9), and its Structural Code is [44421/4412/420/02/O]. Since the Structural Codes of the graphs of Figs 3 and 4(b) are different, they are nonisomorphic.
CL
1 aT cTc
1
I4
.a
T
(a) FIG. 4. Functional Vol. 330, No. 1. pp. 131-143, 1993 Printed in Great Britain
and graph representations
of the Hydra-Matic
gear system.
135
Cheng-Ho
Hsu
V. Stvuctuval Synthesis Procedure In view of the fundamental
rules Fl and F3, we conclude
that :
If one vertex with one incident geared edge and one incident revolute edge are added between any two vertices, which are adjacent with one revolute edge/polygon, of the graph of a known N-link geared kinematic chain, then we can obtain a new (N+ I)-link geared kinematic chain. Moreover, the added revolute edge may be characterized by a new level or by the level of one of its adjacent edges/polygons. Based on the above conclusion, (N+ I)-link geared kinematic chains can be systematically enumerated from the generic N-link geared kinematic chains and described as follows : (1) A set of one vertex with one incident revolute edge and one incident geared edge is added to two vertices, which are adjacent with a revolute edge/polygon. For example, in Fig. 5, if vertex 5 and its incident revolute edge d and geared edge y3 are added between the vertices 1 and 2 of a four-link geared kinematic chain, then we can obtain two new graphs of five-link geared kinematic chains. (2) A set of one vertex with one incident revolute edge and one incident geared edge is added to two vertices, which are adjacent with a revolute edge/polygon, and the level of the added revolute edge is the same as that of its adjacent revolute edge/polygon. For example, in Fig. 6, if vertex 5 and its incident revolute edge d and geared edge g3 are added between vertices 1 and 2, and the incident revolute edges d and h have the same level, then we obtain a new graph of a five-link geared kinematic chain. Using the method for the enumeration of the graphs of geared kinematic chains, the fundamental rules Fl to F6 are automatically satisfied. Hence, the procedure for the structural synthesis of the nonisomorphic graphs of (Nf 1)-link geared kinematic chains is developed and summarized as follows :
1
3
5
1 2 3
4+ ,43
1
2 3
FIG. 5. 136
Journal of the Frankhn lnstitutc Pergamon Press Ltd
Graph Representation
of Geared Kinematic
Chains
2 3
4+
5-w / /‘g3
d
2
FE. 6.
(1) List all the graphs of N-link geared kinematic chains as generic graphs. (2) Enumerate all possible graphs of (N+ I)-link geared kinematic chains for each generic graph of N-link geared kinematic chains. (3) Check for graph isomorphism by comparing the Structural Codes to obtain all nonisomorphic graphs of (N+ 1)-link geared kinematic chains.
VI. Structural Synthesis of Geared Kinematic Chains Using the systematic synthesis procedure, the graphs of geared kinematic chains with any number of links can be synthesized systematically. The following describes the structural synthesis of the graphs of geared kinematic chains for one-DOF planetary gear trains. Three-link geared kinematic chains The only possible three-link geared kinematic chain is shown in Fig. 7 where the three vertices are connected together by two revolute edges and one geared edge, and form an f-circuit.
1
_____ 3n FIG.
7. The graph of a three-link,
one-DOF
2 geared kinematic
chaih.
137
Cheng-Ho
Hsu 1
1
4
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1
4
4\
2
3
W
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I \ 11 I I
/’
/’
/’ ‘2
(0
FIG. 8. The graphs of four-link, one-DOF geared kinematic chains.
Four-link geared kinematic chains (1) Figure 7 shows the three-vertex graph, which is a generic graph. (2) Using the proposed method, we obtain six possible graphs as Fig. 8. (3) Since the Structural Codes of Fig. 8(a) and (d) graphs are [122/l of Fig. 8(b) and (e) graphs are [121/12/O], and those of Fig. 8(c) and are [332/32/l], there are three nonisomorphic graphs of four-link geared chains as shown in Fig. S(a-c).
shown
in
l/O], those (f) graphs kinematic
Five-link geared kinematic chains (1) Figure S(a), (b) and (e) are all the four-vertex generic graphs. (2) Using the proposed method, 12, 10 and 11 graphs with five vertices are enumerated from Fig. 8(a), (b) and (e) generic graphs, respectively, and shown in Fig. 9(a<). 138
Journalofthe
Frankhn Institute Pergallon Press Lid
Graph Representation of Geared Kinematic Chains
(al)
WY
(a@
(bl)
W
(a9
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WV
WI
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(a3
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(c7)
(~8)
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(ClO)
(cl11
FIG. 9. The graphs of five-link, one-DOF geared kinematic chains.
(3) The Structural Codes of those 33 derived graphs are given in Table I. Comparing the Structural Codes, 13 nonisomorphic graphs of five-link geared kinematic chains are obtained and shown in Fig. 10. Based on the developed synthesis procedure, the graphs of geared kinematic chains with any number of links can be synthesized systematically. Table II lists the numbers of nonisomorphic graphs of geared kinematic chains for one-DOF planetary gear trains with up to seven links. Figure 11 lists the graphs of 80 geared kinematic chains for one-DOF, seven-link planetary gear trains. Vol. 3x, NO. I. pp. 131-143.1993 Printed III Great Britam
139
Cheng- Ho Hsu
(a)
(4
(4
6)
6)
(k)
FIG. IO. The graphs of nonisomorphic
(4 0) five-link, one-DOF geared kinematic chains
VII. Conclusions Based on the new graph representation of geared kinematic chains (9, 10) and the concept of Structural Codes (ll), a systematic approach for the structural synthesis of geared kinematic chains has been successfully developed. This method is simpler and more efficient than the previous methods (l-6). An atlas of the graphs of geared kinematic chains of planetary gear trains with up to seven links has been developed. The results are in complete agreement with those of Freudenstein (2), Ravisankar and Mruthyunjaya (3) and Tsai (4). It is shown that there are 636 nonisomorphic graphs of geared kinematic chains from which all the oneDOF, seven-link planetary gear trains can be generated. Moreover, we also show that the result of Kim and Kwak (6) is incorrect, because there are some isomorphic geared kinematic chains in the atlas of 642 nonisomorphic seven-link geared kinematic chains.
Graph Representation
‘.
Chains
: \1:
A ‘\
of Geared Kinematic
‘.
#’
.’
’
FIG. Il. Atlas of nonisomorphic
Vol. 330, No. I. pp. 131-143. Printed 10 Great Bntain
1993
six-link, one-DOF
geared kinematic
chains.
ChenpHo
Hsu
The Structural al a2 a3 a4 a5 a6 a7 a8 a9 a10 all al2
1222/l 1 l/00/0* 1221/l 12/00/o* 133 1/222/30/O* 133 1/222/30/O = 1111/220/02/0* 1121/210/02/0* 332213 11/20/O* 33 1 l/320/02/2* 1111/220/02/O = 1121/210/02/O = 332213 11/20/O = 33 1 l/320/02/2 =
a3
a5 a6 a7 a8
bl b2 b3 b4 b5 b6 b7 b8 b9 b10
TABLE I Codes of the graphs shown in Fig. 9 1221/l 12/00/O 1221/l 12/00/O 1332/221/30/O* 1332/221/30/O 1221/210/02/O 1122/210/01/0* 3321/312/20/O* 1121/210/02/O 1122/210/01/O 3321/312/20/O
= a2 = a2 = b3 = a6
= a6 = b6 = b7
cl c2 c3 c4 C5 c6 c7 C8 c9 cl0 cl1
1331/222/30/O = 4442/442/42/ 1 * 1332/221/30/O = 3322/311/20/O = 3321/312/20/O = 3333/232/23/O* 3322/311/20/O = 3321/312/20/O = 3333/232/23/O = 3322/321/10/O* 3322/321/10/O =
a3 b3 a7 b7 a7 b7 c6 cl0
TABLE II The numbers of the graphs of geared kinematic chains for one-DOF planetary gear trains with up to seven links
Number of links
Number of generated graphs
3 4 5 6 7
1 6 33 256 2348
Number of nonisomorphic graphs 1 3 13 80 636
Acknowledgement The author is grateful to the National Science financial support under Grant Nos NSC79-0401-El
Council of the Republic of China lo-13 and NSCSl-0422-EllO-01.
for
References (1). F. Buchsbaum and F. Freudenstein, “Synthesis of kinematic structure of gear kinematic chains and other mechanisms”, J. Mech., Vol. 5, pp. 357-392, 1970. (2) F. Freudenstein, “An application of Boolean algebra to the motion of epicyclic drives”, ASMEJ. Engng Ind., Vol. 938, pp. 176182, 1971. “Computerized synthesis of the structures of (3) R. Ravisankar and T. S. Mruthyunjaya, geared kinematic chains”, Mech. Mach. Theory, Vol. 20, pp. 367-387, 1984. (4) L. W. Tsai, “An application of the linkage characteristic polynomial to the topological synthesis of epicyclic gear trains”, ASME J. Mech. Trans. Autom. Des., Vol. 109, pp. 3299337, 1987. (5) L. W. Tsai and C. C. Lin, “The creation of nonfractionated, two-degree-of-freedom Journal
142
of the Franklin Institute Pergamon Press Ltd
Graph Representation
(6) (7)
(8)
(9)
(10) (11)
of Geared Kinematic
Chains
epicyclic gear trains”, ASME J. Mech. Trans. Autom. Des., Vol. 111, pp. 524-529, 1989. J. U. Kim and B. M. Kwak, “Application of edge permutation group to structural synthesis of epicyclic gear trains”, Mech. Mach. Theory, Vol. 25, pp. 563-574, 1990. D. G. Olson, A. G. Erdman and D. R. Riley, “A new graph theory representation for the topological analysis of planetary gear trains”, Proc. 7th World Congr. Theory Mach. Mech., Sevilla, Spain, Vol. 3, pp. 1421-1425, 1987. D. G. Olson, A. G. Erdman and D. R. Riley, “Topological analysis of single-degreeof-freedom planetary gear trains”, Trends Dev. Mech. Mach. Robotics, DE-Vol. 151, pp. 1255131, 1988. C. H. Hsu and K. T. Lam, “A new graph representation for the automatic kinematic analysis of planetary spur-gear trains”, Mech. Syst. Anal. Des Simul., De-Vol. 193, pp. 403408, 1989. C. H. Hsu and K. T. Lam, “Automatic analysis of the kinematic structure of planetary gear trains”, Cam. Gear. Mech. Syn., DE-Vol. 26-3, pp. 81-86, 1990. C. H. Hsu, “The identification of displacement isomorphism of planetary gear trains”. Presentation on the 1992 ASME Mechanism Conference. submitted
Vol. 330. No. I, pp. 131-143, 1993 Punted in Great Brmm
143
Department of Mechanical Engineering, Kaohsiung, Taiwan 80424, R.O.C.
National Sun Yat-Sen
University,
This paper presents a systematic approach, which is based on a new graph ABSTRACT: representation of geared kinematic chains, for the structural synthesis of geared kinematic chains. First, the new graph representation of geared kinematic chains is introduced. Next, a simple method is proposed to generate all possible (N+ I)-link geared kinematic chains from N-link geared kinematic chains. Then, isomorphic geared kinematic chains are identified to obtain nonisomorphic geared kinematic chains by comparing their Structural Codes. Finally, an atlas of nonisomorphic geared kinematic chains for one deqree-of-freedom planetary gear trains with up to seven links is constructed.
I. Introduction
Graph theory has shown to be a useful tool for the structural analysis and synthesis of mechanisms. Buchsbaum and Freudenstein (1. 2) first applied graph theory to the structural synthesis and analysis of one degree-of-freedom (DOF) geared kinematic chains. Ravisankar and Mruthyunjaya (3) computerized Buchsbaum and Freudenstein’s method (1) to derive geared kinematic chains with up to six links. Tsai and Lin (4, 5) presented a systematic approach for the structural synthesis of one- and two-DOF planetary gear trains with up to four gear pairs. Kim and Kwak (6) computerized the method proposed by Tsai (4) to synthesize geared kinematic chains for one-DOF planetary gear trains with up to seven links. The synthesis procedure (4-6) includes the following steps : (1) enumerate all the permissible unlabeled graphs of Nf 1 vertices for all known unlabeled graphs of N vertices by the generation method. (2) Check for graph isomorphism to obtain a set of nonisomorphic unlabeled graphs. (3) Identify the transfer vertices by inspection or by using Boolean algebra technique. (4) Eliminate unlabeled graphs violating the fundamental rules of planetary gear trains. (5) Transform the remaining graphs into rotation graphs and check for rotational isomorphism. (6) Label the revolute edges of the rotationally nonisomorphic graphs to obtain all possible nonisomorphic displacement graphs. This synthesis procedure is systematic and works well for simple mechanisms, but it becomes quite complicated and tedious as the number of links increases. Furthermore, it has a major disadvantage in that a planetary gear train may correspond to several different graphs (7-10). Recently, Hsu and Lam (9, 10) have presented a new graph representation to
The Frankhn lnst1lute00l&Q032.'93 $5.00+0.00
131
Chenq- Ho Hsu clarify the kinematic structure of planetary gear trains effectively. Based on the new graph representation, the kinematic and structural analyses of geared kinematic chains for planetary gear trains can be performed in an unambiguous and systematic way. The purpose of this work is to extend the new graph representation to develop a systematic method for the structural synthesis of geared kinematic chains. In what follows, we shall first review the graph representation of a planetary gear train including the fundamental rules of the graph, and then introduce an efficient method for detecting isomorphic graphs. Finally, a systematic approach for the synthesis of the graphs of geared kinematic chains for planetary gear trains is developed. II. Graph Representation
of Geared Kinematic
Chains
Functional representation of a planetary gear train refers to the conventional schematic drawing of the mechanism. Figure l(a) shows the functional representation of the Simpson gear system, which contains six links, five turning pairs and four gear pairs. The links are numbered arbitrarily from 1 to 6, and the location (level) of each axis is labeled as a-a, b-b and c--c, respectively. The corresponding kinematic chain is obtained by releasing the ground link as shown in Fig. 1(b). Based on the researches of Hsu and Lam (9, lo), a geared kinematic chain can be transformed into its corresponding graph by denoting the links, gear pairs, simple and multiple revolute pairs of the kinematic chain by vertices, dashed (geared) edges, solid (revolute) edges and polygons, respectively. The degree of a vertex is defined as the total number of revolute edges and polygons incident to the vertex. A fundamental circuit (f-circuit) is a circuit consisting of three vertices, one geared edge and two revolute edges/polygons, and the vertex, not incident to the geared edge, is the associated transfer vertex. Each revolute edge or polygon represents a level, which indicates the location of its axis in space.
(a) FIG. I. The Simpson gear system and its corresponding 132
0) geared kinematic chain Journal althe Franklm Institute Pergamon Press Ltd
Graph Representation
I\
4 I
\
/6
\
,’
3’,
\
a
\
\
\
16
\
‘I
I I
Chains
\
1 ‘\ \
I’
of Geared Kinematic
\
\
4
C
5
FIG. 2. The graph representation
of the Simpson gear system.
In the graph of a geared kinematic chain, vertices i and,j are said to be adjacent and vertex i is a neighbor of vertex j, if they are incident with a common edge/ polygon. The neighborhood of vertex i is the set N(i) consisting of all vertices adjacent to vertex i. The number of neighbors of vertex i, n(i), is the number of vertices in N(i). The neighborhood degree sequence of vertex i with p neighbors, denoted as NDS(i), is defined as a p-digit number forming by the numbers of neighbors of its p neighbors in descending order. For example, Fig. 2 shows the graph of the Simpson gear system, shown in Fig. 1, which contains four vertices 2, 3,4 and 5 of degree three, two vertices 1 and 6 of degree two, four geared edges 2-3,224, 335 and 5-6, two revolute edges l-2 and 4-5, and one revolute polygon of four vertices 1, 3, 4 and 6. It has three different levels and four f-circuits l-2-31, l-241, 553-45 and 5-6645. Vertex 1 is the associated transfer vertex of f-circuits 1-2-3-1 and I-241, and vertex 4 is the associated transfer vertex of f-circuits 5-3345 and 5-6-45. Since vertices 1, 4 and 6 are adjacent to vertex 3 with a polygon, and vertices 2 and 5 are adjacent to vertex 3 with an edge, the neighborhood and the number of neighbors of vertex 3 are N(3) = {l, 2, 4, 5, 6) and n(3) = 5, respectively. Since the number of neighbors of vertices 1, 2,4, 5 and 6 are four, three, five, three and four, the NDS of vertex 3 is 54433. III. Fundamental Rules of the Graphs of Geared Kinematic Chains According to Hsu and Lam (9, lo), the following fundamental rules have been established for the graphs of geared kinematic chains, which obey the general degree-of-freedom equation : (Fl) For an N-link, F-DOF geared kinematic chain, let its corresponding graph have N vertices, e, geared edges, e2 revolute edges, e3 revolute polygons of three vertices, e4 revolute polygons of four vertices,. , and ek revolute polygons of k vertices, then it satisfies the following equations : e,=N-1-F e,=e2+
(1)
t(i-l)e,=N-1
(2)
i= 3
in which eK is the total number Vol 330. No. I, pp. 131-143.1993 Printed in Great Bnlam
of equivalent
revolute
edges in this graph.
133
Cheng- Ho Hsu (F2) The subgraph obtained by deleting all the geared edges is a connected graph with no circuits. (F3) Any geared edge added to the connected graph forms an f-circuit. An f-circuit has three vertices, one geared edge and two revolute edges/polygons, and the vertex, not incident to the geared edge, is the associated transfer vertex. (F4) The number of f-circuits equals the number of geared edges. (F5) Each revolute edge/polygon can be characterized a level, which identifies the location of its axis in space. Thus, the number of levels equals the total number of revolute edges and polygons. (F6) Each vertex must have at least one incident revolute edge/polygon.
IV. Identification
of Isomorphism
of Geared Kinematic
Chains
In the synthesis of the structure of geared kinematic chains, we need to provide a precise means for checking isomorphism. Hsu (11) has presented an efficient method, called the Structural Codes, to identify the isomorphism of the graphs of geared kinematic chains, hence we apply the concept of Structural Code for this purpose. The vertex-vertex adjacency matrix (A) of the graph of an N-link geared kinematic chain, which was first defined by Hsu and Lam (9), and defined as a square matrix of order N with the entry a, = 1 if vertex i is adjacent to vertexj with a revolute edge, a, = 2 if vertex i is adjacent to vertex j with a geared edge, ai, = rn if vertex i is adjacent to vertex j with a revolute polygon of rn vertices, and a, = 0 otherwise. Furthermore, arr = 0. For example, the adjacency matrix of the graph shown in Fig. 2 is given by -0
1
4
4
0
4-
102200 A=
420424 424014’ 002102 404420
For an N-vertex graph, there are N! possible labelings and N! associated adjacency matrices. Hsu (11) defines the canonical adjacency matrix of this graph as an adjacency matrix obtained by relabeling the vertices such that the NDSs of vertices in this graph are in descending order, and the value of the code produced by the upper right triangle of the adjacency matrix of the relabeled graph is the largest among all possible labelings of this graph. Hsu (11) also defines the Structural Code of this graph as a set of N- 1 numbers in which the ith number, c,, is constructed by the ith row of the upper right triangle of the canonical adjacency matrix of this graph. For reasons of simplicity and clarity, the general form of the Structural Code of an N-vertex graph is expressed as : c = [c-,/c-z/. . ./c,/. . ./c,_ ,] Journal
134
of the Franklin lnst~tute Pergamon Press Lrd
Graph Representation
of Geared Kinematic
Chains
6 FIG. 3. The canonical labeling of the Fig. 2 graph.
For every graph there is a unique Structural Code corresponding to it. If the graphs of geared kinematic chains have the same Structural Code, then they are isomorphic. Moreover, the Structural Code can describe the topological structure of graphs up to isomorphism. For example, since the NDSs of vertices 1, 2, 3, 4, 5 and 6 of the graph shown in Fig. 2 are 5543, 554, 54433, 54433, 554, 5543, there are eight (2!2!2! = 8) different possible relabelings. It is shown that the canonical labeling of this graph can be achieved by the relabeling (3, 4, 1, 6, 5, 2), which means vertex 3 is relabeled as the new vertex 1, vertex 4 as the new vertex 2, and so on. Consequently, the graph with canonical labeling of the graph is shown in Fig. 3, and the Structural Code is [44422/4421/410/02/O]. Similarly, Fig. 4 shows the functional representation and the graph with canonical labeling of the HydraMatic 440-T4 gear system (9), and its Structural Code is [44421/4412/420/02/O]. Since the Structural Codes of the graphs of Figs 3 and 4(b) are different, they are nonisomorphic.
CL
1 aT cTc
1
I4
.a
T
(a) FIG. 4. Functional Vol. 330, No. 1. pp. 131-143, 1993 Printed in Great Britain
and graph representations
of the Hydra-Matic
gear system.
135
Cheng-Ho
Hsu
V. Stvuctuval Synthesis Procedure In view of the fundamental
rules Fl and F3, we conclude
that :
If one vertex with one incident geared edge and one incident revolute edge are added between any two vertices, which are adjacent with one revolute edge/polygon, of the graph of a known N-link geared kinematic chain, then we can obtain a new (N+ I)-link geared kinematic chain. Moreover, the added revolute edge may be characterized by a new level or by the level of one of its adjacent edges/polygons. Based on the above conclusion, (N+ I)-link geared kinematic chains can be systematically enumerated from the generic N-link geared kinematic chains and described as follows : (1) A set of one vertex with one incident revolute edge and one incident geared edge is added to two vertices, which are adjacent with a revolute edge/polygon. For example, in Fig. 5, if vertex 5 and its incident revolute edge d and geared edge y3 are added between the vertices 1 and 2 of a four-link geared kinematic chain, then we can obtain two new graphs of five-link geared kinematic chains. (2) A set of one vertex with one incident revolute edge and one incident geared edge is added to two vertices, which are adjacent with a revolute edge/polygon, and the level of the added revolute edge is the same as that of its adjacent revolute edge/polygon. For example, in Fig. 6, if vertex 5 and its incident revolute edge d and geared edge g3 are added between vertices 1 and 2, and the incident revolute edges d and h have the same level, then we obtain a new graph of a five-link geared kinematic chain. Using the method for the enumeration of the graphs of geared kinematic chains, the fundamental rules Fl to F6 are automatically satisfied. Hence, the procedure for the structural synthesis of the nonisomorphic graphs of (Nf 1)-link geared kinematic chains is developed and summarized as follows :
1
3
5
1 2 3
4+ ,43
1
2 3
FIG. 5. 136
Journal of the Frankhn lnstitutc Pergamon Press Ltd
Graph Representation
of Geared Kinematic
Chains
2 3
4+
5-w / /‘g3
d
2
FE. 6.
(1) List all the graphs of N-link geared kinematic chains as generic graphs. (2) Enumerate all possible graphs of (N+ I)-link geared kinematic chains for each generic graph of N-link geared kinematic chains. (3) Check for graph isomorphism by comparing the Structural Codes to obtain all nonisomorphic graphs of (N+ 1)-link geared kinematic chains.
VI. Structural Synthesis of Geared Kinematic Chains Using the systematic synthesis procedure, the graphs of geared kinematic chains with any number of links can be synthesized systematically. The following describes the structural synthesis of the graphs of geared kinematic chains for one-DOF planetary gear trains. Three-link geared kinematic chains The only possible three-link geared kinematic chain is shown in Fig. 7 where the three vertices are connected together by two revolute edges and one geared edge, and form an f-circuit.
1
_____ 3n FIG.
7. The graph of a three-link,
one-DOF
2 geared kinematic
chaih.
137
Cheng-Ho
Hsu 1
1
4
I 1’ 3 K __-_--
4
__----
I 1’ __-_-3 /v
2
(a)
2
0) 1 4 \
-n
\
\
\
\
\
\ ____--
2
__---vi__---3
Cc)
(4
1
4
4\
2
3
W
P 3
I \ 11 I I
/’
/’
/’ ‘2
(0
FIG. 8. The graphs of four-link, one-DOF geared kinematic chains.
Four-link geared kinematic chains (1) Figure 7 shows the three-vertex graph, which is a generic graph. (2) Using the proposed method, we obtain six possible graphs as Fig. 8. (3) Since the Structural Codes of Fig. 8(a) and (d) graphs are [122/l of Fig. 8(b) and (e) graphs are [121/12/O], and those of Fig. 8(c) and are [332/32/l], there are three nonisomorphic graphs of four-link geared chains as shown in Fig. S(a-c).
shown
in
l/O], those (f) graphs kinematic
Five-link geared kinematic chains (1) Figure S(a), (b) and (e) are all the four-vertex generic graphs. (2) Using the proposed method, 12, 10 and 11 graphs with five vertices are enumerated from Fig. 8(a), (b) and (e) generic graphs, respectively, and shown in Fig. 9(a<). 138
Journalofthe
Frankhn Institute Pergallon Press Lid
Graph Representation of Geared Kinematic Chains
(al)
WY
(a@
(bl)
W
(a9
(W
(all)
WV
WI
W
635)
(a3
W’)
(c7)
(~8)
WI
w
@lo)
(ClO)
(cl11
FIG. 9. The graphs of five-link, one-DOF geared kinematic chains.
(3) The Structural Codes of those 33 derived graphs are given in Table I. Comparing the Structural Codes, 13 nonisomorphic graphs of five-link geared kinematic chains are obtained and shown in Fig. 10. Based on the developed synthesis procedure, the graphs of geared kinematic chains with any number of links can be synthesized systematically. Table II lists the numbers of nonisomorphic graphs of geared kinematic chains for one-DOF planetary gear trains with up to seven links. Figure 11 lists the graphs of 80 geared kinematic chains for one-DOF, seven-link planetary gear trains. Vol. 3x, NO. I. pp. 131-143.1993 Printed III Great Britam
139
Cheng- Ho Hsu
(a)
(4
(4
6)
6)
(k)
FIG. IO. The graphs of nonisomorphic
(4 0) five-link, one-DOF geared kinematic chains
VII. Conclusions Based on the new graph representation of geared kinematic chains (9, 10) and the concept of Structural Codes (ll), a systematic approach for the structural synthesis of geared kinematic chains has been successfully developed. This method is simpler and more efficient than the previous methods (l-6). An atlas of the graphs of geared kinematic chains of planetary gear trains with up to seven links has been developed. The results are in complete agreement with those of Freudenstein (2), Ravisankar and Mruthyunjaya (3) and Tsai (4). It is shown that there are 636 nonisomorphic graphs of geared kinematic chains from which all the oneDOF, seven-link planetary gear trains can be generated. Moreover, we also show that the result of Kim and Kwak (6) is incorrect, because there are some isomorphic geared kinematic chains in the atlas of 642 nonisomorphic seven-link geared kinematic chains.
Graph Representation
‘.
Chains
: \1:
A ‘\
of Geared Kinematic
‘.
#’
.’
’
FIG. Il. Atlas of nonisomorphic
Vol. 330, No. I. pp. 131-143. Printed 10 Great Bntain
1993
six-link, one-DOF
geared kinematic
chains.
ChenpHo
Hsu
The Structural al a2 a3 a4 a5 a6 a7 a8 a9 a10 all al2
1222/l 1 l/00/0* 1221/l 12/00/o* 133 1/222/30/O* 133 1/222/30/O = 1111/220/02/0* 1121/210/02/0* 332213 11/20/O* 33 1 l/320/02/2* 1111/220/02/O = 1121/210/02/O = 332213 11/20/O = 33 1 l/320/02/2 =
a3
a5 a6 a7 a8
bl b2 b3 b4 b5 b6 b7 b8 b9 b10
TABLE I Codes of the graphs shown in Fig. 9 1221/l 12/00/O 1221/l 12/00/O 1332/221/30/O* 1332/221/30/O 1221/210/02/O 1122/210/01/0* 3321/312/20/O* 1121/210/02/O 1122/210/01/O 3321/312/20/O
= a2 = a2 = b3 = a6
= a6 = b6 = b7
cl c2 c3 c4 C5 c6 c7 C8 c9 cl0 cl1
1331/222/30/O = 4442/442/42/ 1 * 1332/221/30/O = 3322/311/20/O = 3321/312/20/O = 3333/232/23/O* 3322/311/20/O = 3321/312/20/O = 3333/232/23/O = 3322/321/10/O* 3322/321/10/O =
a3 b3 a7 b7 a7 b7 c6 cl0
TABLE II The numbers of the graphs of geared kinematic chains for one-DOF planetary gear trains with up to seven links
Number of links
Number of generated graphs
3 4 5 6 7
1 6 33 256 2348
Number of nonisomorphic graphs 1 3 13 80 636
Acknowledgement The author is grateful to the National Science financial support under Grant Nos NSC79-0401-El
Council of the Republic of China lo-13 and NSCSl-0422-EllO-01.
for
References (1). F. Buchsbaum and F. Freudenstein, “Synthesis of kinematic structure of gear kinematic chains and other mechanisms”, J. Mech., Vol. 5, pp. 357-392, 1970. (2) F. Freudenstein, “An application of Boolean algebra to the motion of epicyclic drives”, ASMEJ. Engng Ind., Vol. 938, pp. 176182, 1971. “Computerized synthesis of the structures of (3) R. Ravisankar and T. S. Mruthyunjaya, geared kinematic chains”, Mech. Mach. Theory, Vol. 20, pp. 367-387, 1984. (4) L. W. Tsai, “An application of the linkage characteristic polynomial to the topological synthesis of epicyclic gear trains”, ASME J. Mech. Trans. Autom. Des., Vol. 109, pp. 3299337, 1987. (5) L. W. Tsai and C. C. Lin, “The creation of nonfractionated, two-degree-of-freedom Journal
142
of the Franklin Institute Pergamon Press Ltd
Graph Representation
(6) (7)
(8)
(9)
(10) (11)
of Geared Kinematic
Chains
epicyclic gear trains”, ASME J. Mech. Trans. Autom. Des., Vol. 111, pp. 524-529, 1989. J. U. Kim and B. M. Kwak, “Application of edge permutation group to structural synthesis of epicyclic gear trains”, Mech. Mach. Theory, Vol. 25, pp. 563-574, 1990. D. G. Olson, A. G. Erdman and D. R. Riley, “A new graph theory representation for the topological analysis of planetary gear trains”, Proc. 7th World Congr. Theory Mach. Mech., Sevilla, Spain, Vol. 3, pp. 1421-1425, 1987. D. G. Olson, A. G. Erdman and D. R. Riley, “Topological analysis of single-degreeof-freedom planetary gear trains”, Trends Dev. Mech. Mach. Robotics, DE-Vol. 151, pp. 1255131, 1988. C. H. Hsu and K. T. Lam, “A new graph representation for the automatic kinematic analysis of planetary spur-gear trains”, Mech. Syst. Anal. Des Simul., De-Vol. 193, pp. 403408, 1989. C. H. Hsu and K. T. Lam, “Automatic analysis of the kinematic structure of planetary gear trains”, Cam. Gear. Mech. Syn., DE-Vol. 26-3, pp. 81-86, 1990. C. H. Hsu, “The identification of displacement isomorphism of planetary gear trains”. Presentation on the 1992 ASME Mechanism Conference. submitted
Vol. 330. No. I, pp. 131-143, 1993 Punted in Great Brmm
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