Detection of isomorphism in planar and geared kinematic chains (GKCs) is an interesting area since many years. Enumeration of planar and geared kinematic chains becomes easy only when isomorphism problem is resolved effectively. Many researchers proposed algorithms based on topological characteristics or some coding which need lot of computations and comparisons. In this paper, a novel and simple algorithm is proposed based on graph theory by which elimination of isomorphic chains can be done very easily without any tedious calculations or comparisons. A new concept “Net distance” is proposed based on the graph theory to be a quantitative measure to assess isomorphism in planar kinematic chains (PKCs) as well as GKCs. The proposed algorithm is applied on nine-link two-degrees-of-freedom (DOF) distinct kinematic chains completely and the results are presented. Algorithm is tested on examples from eight-link 1-DOF, ten-link 1-DOF, 12-link 1-DOF, and 15link 4-DOF PKCs. The algorithm is also tested on four-, six-link 1-DOF GKCs to detect isomorphism. All the results are in agreement with the existing literature.

References

References
1.
Woo
,
L. S.
,
1967
, “
Type Synthesis of Planar Linkages
,”
ASME J. Eng. Ind.
,
89
(
1
), pp.
159
172
.
2.
Mruthyunjaya
,
T. S.
,
1984
, “
A Computerized Methodology for Structural Synthesis of Kinematic Chains: Part 1—Formulation
,”
Mech. Mach. Theory
,
19
(
6
), pp.
487
495
.
3.
Mruthyunjaya
,
T. S.
,
1984
, “
A Computerized Methodology for Structural Synthesis of Kinematic Chains: Part 2—Application to Several Fully or Partially Known Cases
,”
Mech. Mach. Theory
,
19
(
6
), pp.
497
505
.
4.
Mruthyunjaya
,
T. S.
,
1984
, “
A Computerized Methodology for Structural Synthesis of Kinematic Chains: Part 3—Application to the New Case of 10-Link, Three-Freedom Chains
,”
Mech. Mach. Theory
,
19
(
6
), pp.
507
530
.
5.
Mruthyunjaya
,
T. S.
, and
Balasubramanian
,
H. R.
,
1987
, “
In Quest for Reliable and Efficient Test for Detection of Isomorphism in Kinematic Chains
,”
Mech. Mach. Theory
,
22
(
2
), pp.
131
139
.
6.
Mruthyunjaya
,
T. S.
,
2003
, “
Kinematic Structure of Mechanisms Revisited
,”
Mech. Mach. Theory
,
38
(
4
), pp.
279
320
.
7.
Tuttle
,
E. R.
,
Peterson
,
S. W.
, and
Titus
,
J. E.
,
1989
, “
Enumeration of Basic Kinematic Chains Using the Theory of Finite Groups
,”
ASME J. Mech. Transm. Autom. Des.
,
111
(
4
), pp.
498
503
.
8.
Kui
,
C. J.
, and
Qing
,
C. W.
,
1994
, “
Identification of Isomorphism Among Kinematic Chains and Inversions Using Link's Adjacent Chain Table
,”
Mech. Mach. Theory
,
29
(
1
), pp.
53
58
.
9.
Ambekar
,
A. G.
, and
Agrawal
,
V. P.
,
1986
, “
On Canonical Numbering of Kinematic Chains and Isomorphism Problem: Max Code
,”
ASME
Paper No. 86-DET-169.
10.
Ambekar
,
A. G.
, and
Agrawal
,
V. P.
,
1987
, “
Canonical Numbering of Kinematic Chains and Isomorphism Problem: Min Code
,”
Mech. Mach. Theory
,
22
(
5
), pp.
453
461
.
11.
Yadav
,
J. N.
,
Pratap
,
C. R.
, and
Agrawal
,
V. P.
,
1995
, “
Detection of Isomorphism Among Kinematic Chains Using the Distance Concept
,”
ASME J. Mech. Des.
,
117
(
4
), pp.
607
611
.
12.
Yadav
,
J. N.
,
Pratap
,
C. R.
, and
Agrawal
,
V. P.
,
1996
, “
Computer Aided Detection of Isomorphism Among Kinematic Chains and Mechanisms Using the Concept of Modified Distance
,”
Mech. Mach. Theory
,
31
(
4
), pp.
439
444
.
13.
Rao
,
A. C.
, and
Varada Raju
,
D.
,
1991
, “
Application of the Hamming Number Technique to Deduct Isomorphism Among Kinematic Chains and Inversions
,”
Mech. Mach. Theory
,
26
(
1
), pp.
55
75
.
14.
Rao
,
A. C.
, and
Rao
,
C. N.
,
1993
, “
Loop Based Pseudo Hamming Values-1: Testing Isomorphism and Rating Kinematic Chains
,”
Mech. Mach. Theory
,
28
(
1
), pp.
113
127
.
15.
Rao
,
A. C.
,
2000
, “
Application of Fuzzy Logic for the Study of Isomorphism, Inversions, Symmetry, Parallelism and Mobility in Kinematic Chains
,”
Mech. Mach. Theory
,
35
(
8
), pp.
1103
1116
.
16.
Rao
,
A. C.
, and
Prasad Raju
,
V. V. N. R.
,
2000
, “
Loop Based Detection of Isomorphism Among Kinematic Chains, Inversions and Type of Freedom in Multi Degree of Freedom Chain
,”
ASME J. Mech. Des.
,
122
(
1
), pp.
31
42
.
17.
Sanyal
,
S.
,
2009
, “
Detection of Isomorphism Amongst Planar Kinematic Chains Using Link Joint Connectivity Table
,” World Congress on Science, Engineering and Technology (
WCSET
), Bangkok, Thailand, Dec. 25–27, pp. 908–911.
18.
Sanyal
,
S.
, and
Bedi
,
G. S.
,
2010
, “
Joint Connectivity: A New Approach for Detection of Isomorphism and Inversions of Planar Kinematic Chains
,”
J. Inst. Eng. (India)
,
90
, pp.
23
26
.
19.
Sanyal
,
S.
, and
Bedi
,
G. S.
,
2011
, “
Modified Joint Connectivity Approach for Identification of Topological Characteristics of Planar Kinematic Chains
,”
Proc. Inst. Mech. Eng. Part C
,
225
(
11
), pp.
2700
2717
.
20.
Xiao
,
R.
,
Tao
,
Z.
, and
Liu
,
Y.
,
2005
, “
Isomorphism Identification of Kinematic Chains Using Novel Evolutionary Approaches
,”
ASME J. Comput. Inf. Sci. Eng.
,
5
(
1
), pp.
18
24
.
21.
Dargar
,
A.
,
Hasan
,
A.
, and
Khan
,
R. A.
,
2009
, “
Identification of Isomorphism Among Kinematic Chains and Inversions Using Link Adjacency Values
,”
IJMME
,
4
(
3
), pp.
309
315
.
22.
Bal
,
J. S.
,
Deshmukh
,
P. B.
, and
Jagadeesh
,
A.
,
2013
, “
Link Invariant Functions an Detection of Isomorphism and Inversions of Kinematic Chains
,” 1st International and 16th National Conference on Machines and Mechanisms (
iNaCoMM
), Roorkee, India, Dec. 18–20, pp. 554–561.
23.
Zou
,
Y.
,
He
,
P.
,
Xu
,
D.
, and
Li
,
J.
,
2017
, “
Automatic Synthesis of Planar Simple Joint Kinematic Chains by Single Kinematic Chain Adding Method
,”
Mechanism and Machine Science, Part V, ASIAN MMS & CCMMS
, Guangzhou, China, Dec. 15–17, pp.
901
909
.
24.
Ding
,
H.
,
Zi
,
B.
,
Huang
,
P.
, and
Kecskeméthy
,
A.
,
2013
, “
The Whole Family of Kinematic Structures for Planar 2- and 3-DOF Fractionated Kinematic Chains
,”
Mech. Mach. Theory
,
70
, pp.
74
90
.
25.
Lohumi
,
M. K.
,
Mohammad
,
A.
, and
Khan
,
I. A.
,
2012
, “
Hierarchical Clustering Approach for Determination of Isomorphism Among Planar Kinematic Chains and Their Derived Mechanisms
,”
J. Mech. Sci. Technol. (Springer)
,
26
(
12
), pp.
4041
4046
.
26.
Lohumi
,
M. K.
,
Mohammad
,
A.
, and
Khan
,
I. A.
,
2015
, “
A Computerized Loop Based Approach for Identification of Isomorphism and Type of Mobility in Planar Kinematic Chains
,”
Sadhana (Indian Acad. Sci.)
,
40
(
2
), pp.
335
350
.
27.
Xue
,
H.-L.
,
Liu
,
G.
, and
Yang
,
X.-H.
,
2015
, “
A Review of Graph Theory Application Research in Gears
,”
Proc. Inst. Mech. Eng. Part C
,
230
(10), pp. 1697–1714.
28.
Buchsbaum
,
F.
, and
Frudenstein
,
F.
,
1970
, “
Synthesis of Kinematic Structure of Geared Kinematic Chains and Other Mechanisms
,”
J. Mech.
,
5
(
3
), pp.
357
392
.
29.
Ravisankar
,
R.
, and
Mruthyunjaya
,
T. S.
,
1985
, “
Computerized Synthesis of the Structure of Geared Kinematic Chains
,”
Mech. Mach. Theory
,
20
(
5
), pp.
367
387
.
30.
Tsai
,
L. W.
,
1987
, “
An Application of the Linkage Characteristic Polynomial to the Topological Synthesis of Epicyclic Gear Trains
,”
ASME J. Mech. Trans. Autom. Des.
,
109
(
3
), pp.
329
336
.
31.
Kim
,
J. U.
, and
Kwak
,
B. M.
,
1990
, “
Application of Edge Permutation Group to Structural Synthesis of Epicyclic Gear Trains
,”
Mech. Mach. Theory
,
25
(
5
), pp.
563
574
.
32.
Olson
,
D. G.
,
Erdman
,
A. G.
, and
Riley
,
D. R.
,
1991
, “
Topological Analysis of Single Degree of Freedom Planetary Gear Trains
,”
ASME J. Mech. Des.
,
113
(
1
), pp.
10
16
.
33.
Hsu
,
C.-H.
,
1994
, “
Displacement Isomorphism of Planetary Gear Trains
,”
Mech. Mach. Theory
,
29
(
4
), pp.
513
523
.
34.
Rao
,
A. C.
, and
Prasad Raju Pathapati
,
V. V. N. R.
,
2002
, “
A New Technique Based on Loops to Investigate Displacement Isomorphism in Planetary Gear Trains
,”
ASME J. Mech. Des.
,
124
(
4
), pp.
662
675
.
35.
Rao
,
Y. V. D.
, and
Rao
,
A. C.
, 2008, “
Generation of Epicyclic Gear Trains of One Degree of Freedom
,”
ASME J. Mech. Des.
,
130
(5), p. 052604.
36.
Shin
,
J. K.
, and
Krishnamurty
,
S.
,
1993
, “
Standard Code Technique in the Enumeration of Epicyclic Gear Trains
,”
Mech. Mach. Theory
,
28
(
3
), pp.
347
355
.
37.
Del Castillo
,
J. M.
,
2002
, “
Enumeration of 1-DOF Planetary Gear Train Graphs Based on Functional Constraints
,”
ASME J. Mech. Des.
,
124
(
4
), pp.
723
732
.
38.
Xue
,
L.
,
Wang
,
Y.
,
Wang
,
H.
, and
Liu
,
R.
,
2005
, “
Classification and Synthesis of Planetary Gear Trains
,”
ASME
Paper No. DETC2005-84231.
39.
El-Gayyar
,
M. S.
,
El-Eashy
,
H. M.
, and
Zaki
,
M.
,
2006
, “
Structural Synthesis and Enumeration of Epicyclic Gear Mechanisms Up to 12 Links Using Acyclic Graph Method
,”
ASME
Paper No. GT2006-91136.
40.
Kong
,
F. G.
,
Li
,
Q.
, and
Zhang
,
W. J.
,
1999
, “
An Artificial Neural Network Approach to Mechanism Kinematic Chain Isomorphism Identification
,”
Mech. Mach. Theory
,
34
(
2
), pp.
271
283
.
41.
Marin
,
G. G.
,
Casermeiro
,
E. M.
, and
Rodriguez
,
D. L.
,
2007
, “
Improving Neural Networks for Mechanism Kinematic Chain Isomorphism Identification
,”
J. Neural Process Lett.
,
26
(
2
), pp.
133
143
.
42.
Galan-Marin
,
G.
,
Lopez-Rodríguez
,
D.
, and
Mérida-Casermeiro
,
E.
,
2010
, “
A New Multivalued Neural Network for Isomorphism Identification of Kinematic Chains
,”
ASME J. Comput. Inf. Sci. Eng.
,
10
(
1
), p.
011009
.
43.
Pennestri
,
E.
, and
Belfiore
,
N. P.
,
2015
, “
On Crossley's Contribution to the Development of Graph Based Algorithms for the Analysis of Mechanisms and Gear Trains
,”
Mech. Mach. Theory
,
89
, pp.
92
106
.
44.
Belfiore
,
N. P.
,
2000
, “
A Brief Note on the Concept of Planarity for Kinematic Chains
,”
Mech. Mach. Theory
,
35
(
12
), pp.
1745
1750
.
45.
Tischler
,
C. R.
,
Samuel
,
A. E.
, and
Hunt
,
K. H.
,
1995
, “
Kinematic Chains for Robot Hands-II: Kinematic Constraints, Classification, Connectivity, and Actuation
,”
Mech. Mach. Theory
,
30
(
8
), pp.
1217
1239
.
46.
Shoham
,
M.
, and
Roth
,
B.
,
1997
, “
Connectivity in Open and Closed Loop Robotic Mechanisms
,”
Mech. Mach. Theory
,
32
(
3
), pp.
279
293
.
47.
Belfiore
,
N. P.
, and
Benedetto
,
A. D.
,
2000
, “
Connectivity and Redundancy in Spatial Robots
,”
Int. J. Rob. Res.
,
19
(
12
), pp.
1245
1261
.
48.
Belfiore
,
N. P.
,
2000
, “
Distributed Databases for the Development of Mechanisms Topology
,”
Mech. Mach. Theory
,
35
(
12
), pp.
1727
1744
.
49.
Liberati
,
A.
, and
Belfiore
,
N. P.
,
2006
, “
A Method for the Identification of the Connectivity in Multi-Loop Kinematic Chains: Analysis of Chains With Total and Partial Mobility
,”
Mech. Mach. Theory
,
41
(
12
), pp.
1443
1466
.
50.
Yan
,
H. S.
, and
Chiu
,
Y. T.
,
2013
, “
An Algorithm for the Construction of Generalized Kinematic Chains
,”
Mech. Mach. Theory
,
62
, pp.
75
98
.
51.
Yan
,
H. S.
, and
Chiu
,
Y. T.
,
2014
, “
An Improved Algorithm for the Construction of Generalized Kinematic Chains
,”
Mech. Mach. Theory
,
78
, pp.
229
247
.
52.
Yan
,
H. S.
, and
Chiu
,
Y. T.
,
2015
, “
On the Number Synthesis of Kinematic Chains
,”
Mech. Mach. Theory
,
89
, pp.
128
144
.
53.
Chiu
,
Y. T.
, and
Yan
,
H. S.
,
2015
, “
An Algorithm for the Automatic Sketching of Generalized Kinematic Chains
,”
14th IFToMM World Congress
, Taipei, Taiwan, Oct. 25–30, Paper No.
OS2.048
.
54.
Ding
,
H. F.
,
Hou
,
F. M.
,
Kecskemethy
,
A.
, and
Huang
,
Z.
,
2012
, “
Synthesis of the Whole Family of Planar 1-DOF Kinematic Chains and Creation of Their Atlas Database
,”
Mech. Mach. Theory
,
47
(
1
), pp.
1
15
.
55.
Ding
,
H. F.
,
Cao
,
W. A.
,
Kecskemethy
,
A.
, and
Huang
,
Z.
,
2012
, “
Complete Atlas Database of 2-DOF Kinematic Chains and Creative Design of Mechanisms
,”
ASME J. Mech. Des.
,
134
(
3
), p.
031006
.
56.
Ding
,
H. F.
,
Huang
,
P.
,
Liu
,
J. F.
, and
Kecskemethy
,
A.
,
2013
, “
Automatic Structural Synthesis of the Whole Family of Planar 3-DOF Closed Loop Mechanisms
,”
ASME J. Mech. Rob.
,
5
(
4
), p.
041006
.
57.
Ding
,
H. F.
,
Huang
,
P.
,
Yang
,
W. J.
, and
Kecskemethy
,
A.
,
2016
, “
Automatic Generation of the Complete Set of Planar Kinematic Chains With Up to Six Independent Loops and Up to 19 Links
,”
Mech. Mach. Theory
,
96
(
Pt. 1
), pp.
75
93
.
58.
Huang
,
P.
,
Ding
,
H. F.
,
Yang
,
W.
, and
Kecskemethy
,
A.
,
2017
, “
An Automatic Method for the Connectivity Calculation in Planar Closed Kinematic Chains
,”
Mech. Mach. Theory
,
109
, pp.
195
219
.
59.
Rizvi
,
S. S. H.
,
Hasan
,
A.
, and
Khan
,
R. A.
,
2016
, “
An Efficient Algorithm for Distinct Inversions and Isomorphism Detection in Kinematic Chains
,”
Perspect. Sci.
,
8
, pp.
251
253
.
60.
Zeng
,
K.
,
Fan
,
X.
,
Dong
,
M.
, and
Yang
,
P.
,
2014
, “
A Fast Algorithm for Kinematic Chain Isomorphism Identification Based on Dividing and Matching Vertices
,”
Mech. Mach. Theory
,
72
, pp.
25
38
.
61.
Varadaraju
,
D.
, and
Mohankumar
,
Ch.
,
2016
, “
Split Hamming String as an Isomorphism Test for One Degree-of-Freedom Planar Simple-Jointed Kinematic Chains Containing Sliders
,”
ASME J. Mech. Des.
,
138
(
8
), p.
082301
.
62.
Pozhbelko
,
V.
,
2016
, “
A Unified Structure Theory of Multibody Open-, Closed-, and Mixed-Loop Mechanical Systems With Simple and Multiple Joint Kinematic Chains
,”
Mech. Mach. Theory
,
100
, pp.
1
16
.
63.
Kamesh
,
V. V.
,
Mallikarjuna Rao
,
K.
, and
Srinivasa Rao
,
A. B.
,
2016
, “
A Novel Method to Detect Isomorphism in Epicyclic Gear Trains
,”
Imanager's J. Future Eng. Technol.
,
12
(
1
), pp.
28
35
.
64.
Kamesh
,
V. V.
,
Mallikarjuna Rao
,
K.
, and
Srinivasa Rao
,
A. B.
,
2017
, “
Topological Synthesis of Epicyclic Gear Trains Using Vertex Incidence Polynomial
,”
ASME J. Mech. Des.
,
139
(
6
), p.
062304
.
65.
Kamesh
,
V. V.
,
Mallikarjuna Rao
,
K.
, and
Srinivasa Rao
,
A. B.
,
2017
, “
Detection of Degenerate Structure in Single Degree-of-Freedom Planetary Gear Trains
,”
ASME J. Mech. Des.
,
139
(
8
), p.
083302
.
66.
Ding
,
H.
, and
Huang
,
Z.
,
2007
, “
A Unique Representation of the Kinematic Chain and the Atlas Database
,”
Mech. Mach. Theory
,
42
(
6
), pp.
637
651
.
67.
Ding
,
H.
, and
Huang
,
Z.
,
2006
, “
The Establishment of the Canonical Perimeter Topological Graph of Kinematic Chains and Isomorphism Identification
,”
ASME J. Mech. Des.
,
129
(
9
), pp.
915
923
.
68.
Ding
,
H.
, and
Huang
,
Z.
,
2009
, “
Isomorphism Identification of Graphs: Especially for the Graphs of Kinematic Chains
,”
Mech. Mach. Theory
,
44
(
1
), pp.
122
139
.
69.
Robon Wilson
,
J.
,
1996
,
Introduction to Graph Theory
,
4th ed.
,
Addison Wesley Longman
,
London
.
70.
Balakrishnan
,
R.
, and
Ranganathan
,
K.
,
2012
,
A Textbook of Graph Theory
,
2ed.
,
Springer
,
New York
.
71.
Henley
,
E. J.
, and
Williams
,
R. A.
,
1973
,
Graph Theory in Modern Engineering
, Vol.
98
,
Academic Press
,
New York
.
72.
Wallis
,
W. D.
,
2000
,
A Beginner's Guide to Graph Theory
,
Springer Science + Business Media LLC
,
New York
.
You do not currently have access to this content.