Abstract

The identification of Baranov chains is associated with the rigid subchain identification problem, which is a crucial step in several methods of structural synthesis of kinematic chains. In this article, a systematic approach for the detection of rigid subchains, based on matroid theory, is presented and proved. Based on this approach, a novel method for the enumeration of Baranov chains is proposed. A novel algorithm is applied to a database of nonisomorphic graphs of nonfractionated zero-mobility kinematic chains. By means of the proposed algorithm, the previous results for Baranov chains presented in the literature with up to 11 links are compared and validated. Furthermore, discrepancies in the number of Baranov chains with up to 13 links, presented in the literature, are pointed out, discussed, and the proven results are presented. Finally, the complete family of Baranov chains with up to 15 links is obtained. Examples of application of the proposed method are provided.

References

References
1.
Baranov
,
G.
,
1952
, “
Classification, Formation, Kinematics, and Kinetostatics of Mechanisms With Pairs of the First Kind
,”
Proceedings of Seminar on the Theory of Machines and Mechanisms
, Vol.
2
,
Moscow
, pp.
15
39
.
2.
Manolescu
,
N.
,
1973
, “
A Method Based on Baranov Trusses, and Using Graph Theory to Find the Set of Planar Jointed Kinematic Chains and Mechanisms
,”
Mech. Mach. Theory.
,
8
(
1
), pp.
3
22
. 10.1016/0094-114X(73)90003-7
3.
Tuttle
,
E.
,
1996
, “
Generation of Planar Kinematic Chains
,”
Mech. Mach. Theory.
,
31
(
6
), pp.
729
748
. 10.1016/0094-114X(95)00083-B
4.
Sunkari
,
R. P.
,
2006
,
Structural Synthesis and Analysis of Planar and Spatial Mechanisms Satisfying Gruebler’s Degrees of Freedom Equation, Ph.D thesis, University of Maryland, College Park, FL
.
5.
Galletti
,
C. U.
,
1986
, “
A Note on Modular Approaches to Planar Linkage Kinematic Analysis
,”
Mech. Mach. Theory.
,
21
(
5
), pp.
385
391
. 10.1016/0094-114X(86)90086-8
6.
Innocenti
,
C.
,
1993
, “Analytical Determination of the Intersections of Two Coupler-Point Curves Generated by Two Four-Bar Linkages,”
Comput. Kinematics
,
Springer
,
New York
, pp.
251
262
.
7.
Huang
,
P.
, and
Ding
,
H.
,
2019
, “
Structural Synthesis of Baranov Trusses With Up to 13 Links
,”
ASME J. Mech. Des.
,
141
(
7
), p.
1
.
8.
Tischler
,
C.
,
Samuel
,
A.
, and
Hunt
,
K.
,
1995
, “
Kinematic Chains for Robot Hands—ii. Kinematic Constraints, Classification, Connectivity, and Actuation
,”
Mech. Mach. Theory.
,
30
(
8
), pp.
1217
1239
. 10.1016/0094-114X(95)00044-Y
9.
Mruthyunjaya
,
T.
,
2003
, “
Kinematic Structure of Mechanisms Revisited
,”
Mech. Mach. Theory.
,
38
(
4
), pp.
279
320
. 10.1016/S0094-114X(02)00120-9
10.
Yan
,
H.-S.
, and
Chiu
,
Y.-T.
,
2015
, “
On the Number Synthesis of Kinematic Chains
,”
Mech. Mach. Theory.
,
89
, pp.
128
144
. 10.1016/j.mechmachtheory.2014.08.012
11.
Comanescu
,
A.
,
Comanescu
,
D.
,
Dugaesescu
,
I.
, and
Ungureanu
,
L.
,
2013
, “
Optimal Inverse Models for Bi-Mobile Mechanisms of Walking Robot Legs
,”
24th DAAAM International Symposium on Intelligent Manufacturing and Automation
,
Vienna, Austria
, pp.
417
430
.
12.
Lee
,
C.-C.
, and
Lo
,
C.-Y.
,
2013
, “
Movable Focal-Type 7-Bar Baranov-Truss Linkages
,”
ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Portland, OR
,
Aug. 4–7
.
13.
Kim
,
S.-H.
, and
Cho
,
C.
,
2016
, “
Transformation of Static Balancer From Truss to Linkage
,”
J. Mech. Sci. Technol.
,
30
(
5
), pp.
2093
2104
. 10.1007/s12206-016-0416-y
14.
Rojas
,
N.
,
Ma
,
R. R.
, and
Dollar
,
A. M.
,
2016
, “
The Gr2 Gripper: An Underactuated Hand for Open-Loop In-Hand Planar Manipulation
,”
IEEE Trans. Rob.
,
32
(
3
), pp.
763
770
. 10.1109/TRO.2016.2562122
15.
Huang
,
P.
, and
Ding
,
H.
,
2020
, “
Structural Synthesis of Assur Groups With Up to 12 Links and Creation of Their Classified Databases
,”
Mech. Mach. Theory.
,
145
, p.
103668
. 10.1016/j.mechmachtheory.2019.103668
16.
Pennock
,
G.
, and
Kamthe
,
G.
,
2006
, “
Study of Dead-Centre Positions of Single-Degree-of-Freedom Planar Linkages Using Assur Kinematic Chains
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
220
(
7
), pp.
1057
1074
. 10.1243/09544062JMES122
17.
Chung
,
W.-Y.
,
2007
, “
Double Configurations of Five-Link Assur Kinematic Chain and Stationary Configurations of Stephenson Six-Bar
,”
Mech. Mach. Theory.
,
42
(
12
), pp.
1653
1662
. 10.1016/j.mechmachtheory.2006.11.008
18.
Rojas
,
N.
, and
Thomas
,
F.
,
2011
, “
Closed-Form Solution to the Position Analysis of Watt–Baranov Trusses Using the Bilateration Method
,”
ASME J. Mech. Rob.
,
3
(
3
), p.
031001
. 10.1115/1.4004031
19.
Hahn
,
E.
, and
Shai
,
O.
,
2016
, “
Construction of Baranov Trusses Using a Single Universal Construction Rule
,”
ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Charlotte, NC
,
Aug. 21–24
.
20.
Sunkari
,
R. P.
, and
Schmidt
,
L. C.
,
2005
, “
Critical Review of Existing Degeneracy Testing and Mobility Type Identification Algorithms for Kinematic Chains
,”
ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Long Beach, CA
,
Sept. 24–28
, pp.
255
263
.
21.
Lee
,
H.-J.
, and
Yoon
,
Y.-S.
,
1992
, “
Detection of Rigid Structure in Enumerating Basic Kinematic Chain by Sequential Removal of Binary Link String
,”
JSME Int. J. Ser. 3, Vib., Control Eng., Eng. Indus.
,
35
(
4
), pp.
647
651
.
22.
Sunkari
,
R. P.
, and
Schmidt
,
L. C.
,
2006
, “
Structural Synthesis of Planar Kinematic Chains by Adapting a Mckay-Type Algorithm
,”
Mech. Mach. Theory.
,
41
(
9
), pp.
1021
1030
. 10.1016/j.mechmachtheory.2005.11.007
23.
Simoni
,
R.
,
Carboni
,
A. P.
, and
Martins
,
D.
,
2009
, “
Enumeration of Kinematic Chains and Mechanisms
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
223
(
4
), pp.
1017
1024
. 10.1243/09544062JMES1071
24.
Martins
,
D.
, and
Carboni
,
A. P.
,
2008
, “
Variety and Connectivity in Kinematic Chains
,”
Mech. Mach. Theory.
,
43
(
10
), pp.
1236
1252
. 10.1016/j.mechmachtheory.2007.10.011
25.
Ding
,
H.
,
Huang
,
Z.
, and
Mu
,
D.
,
2008
, “
Computer-Aided Structure Decomposition Theory of Kinematic Chains and Its Applications
,”
Mech. Mach. Theory.
,
43
(
12
), pp.
1596
1609
. 10.1016/j.mechmachtheory.2007.12.011
26.
Ding
,
H.
,
Zhao
,
J.
, and
Huang
,
Z.
,
2010
, “
The Establishment of Edge-Based Loop Algebra Theory of Kinematic Chains and Its Applications
,”
Eng. Comput.
,
26
(
2
), pp.
119
127
. 10.1007/s00366-009-0141-6
27.
Huang
,
P.
,
Ding
,
H.
,
Yang
,
W.
, and
Kecskeméthy
,
A.
,
2017
, “
An Automatic Method for the Connectivity Calculation in Planar Closed Kinematic Chains
,”
Mech. Mach. Theory.
,
109
, pp.
195
219
. 10.1016/j.mechmachtheory.2016.10.004
28.
Ding
,
H.
,
Hou
,
F.
,
Kecskeméthy
,
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
, pp.
1
15
. 10.1016/j.mechmachtheory.2011.08.011
29.
Ding
,
H.
,
Yang
,
W.
,
Huang
,
P.
, and
Kecskeméthy
,
A.
,
2013
, “
Automatic Structural Synthesis of Planar Multiple Joint Kinematic Chains
,”
ASME J. Mech. Des.
,
135
(
9
), p.
091007
. 10.1115/1.4024733
30.
Ding
,
H.
,
Huang
,
P.
,
Yang
,
W.
, and
Kecskeméthy
,
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
, pp.
75
93
. 10.1016/j.mechmachtheory.2015.09.006
31.
Morlin
,
F. V.
,
Barreto
,
R. L. P.
,
Carboni
,
A. P.
, and
Martins
,
D.
,
2020
, “
A Method for Generating Gripper Mechanisms Using Baranov Chains
,”
International Symposium on Multibody Systems and Mechatronics
,
Córdoba, Argentina
.
32.
Barreto
,
R. L. P.
,
Morlin
,
F. V.
,
Carboni
,
A. P.
, and
Martins
,
D.
,
2020
, “
Type Synthesis of Gripper Mechanisms Using Baranov Chains and Davies’ Method
,”
International Symposium on Multibody Systems and Mechatronics
,
Córdoba, Argentina
.
33.
Morlin
,
F. V.
,
Carboni
,
A. P.
,
de Souza
,
M. B.
, and
Martins
,
D.
,
2020
, “
Degeneracy Detection in Epicyclic Gear Trains Using a Matroid-Based Algorithm
,”
International Symposium on Multibody Systems and Mechatronics
,
Córdoba, Argentina
.
34.
Tsai
,
L.-W.
,
2000
,
Mechanism Design: Enumeration of Kinematic Structures According to Function
,
CRC Press
,
Boca Raton, FL
.
35.
Oxley
,
J.
,
2001
, “On the Interplay Between Graphs and Matroids,”
Surveys in Combinatorics
,
J. W. P
Hirschfeld
, ed.,
Cambridge University Press
,
Cambridge
, pp.
199
239
.
36.
Gordon
,
G.
, and
McNulty
,
J.
,
2012
,
Matroids: A Geometric Introduction
,
Cambridge University Press
,
Cambridge, UK
.
37.
Brualdi
,
R. A.
,
2009
,
Introductory Combinatorics
,
Pearson
,
New York
.
38.
Belfiore
,
N. P.
,
2000
, “
A Brief Note on the Concept of Planarity for Kinematic Chains
,”
Mech. Mach. Theory.
,
35
(
12
), pp.
1745
1750
. 10.1016/S0094-114X(00)00021-5
39.
Whitney
,
H.
,
1992
,
On the Abstract Properties of Linear Dependence
,
Birkhäuser Boston
,
Boston, MA
, pp.
147
171
.
40.
Oxley
,
J.
,
2003
, “
What Is a Matroid
,”
Cubo Matemática Educacional
,
5
(
3
), pp.
179
218
.
41.
Seymour
,
P. D.
,
1994
, “
A Note on Hyperplane Generation
,”
J. Combinat. Theory, Ser. B
,
61
(
1
), pp.
88
91
. 10.1006/jctb.1994.1033
42.
Khachiyan
,
L.
,
Boros
,
E.
,
Elbassioni
,
K.
,
Gurvich
,
V.
, and
Makino
,
K.
,
2005
, “
On the Complexity of Some Enumeration Problems for Matroids
,”
SIAM J. Discrete Math.
,
19
(
4
), pp.
966
984
. 10.1137/S0895480103428338
43.
Mayhew
,
D.
,
2008
, “
Matroid Complexity and Non-Succinct Descriptions
,”
SIAM J. Discrete Math.
,
22
(
2
), pp.
455
466
. 10.1137/050640576
44.
Khachiyan
,
L.
,
Boros
,
E.
,
Elbassioni
,
K.
,
Gurvich
,
V.
, and
Makino
,
K.
,
2005
, “
On the Complexity of Some Enumeration Problems for Matroids
,”
SIAM J. Discrete Math.
,
19
(
4
), pp.
966
984
. 10.1137/S0895480103428338
45.
Assur
,
L.
,
1913
, “
Investigation of Plane Hinged Mechanisms With Lower Pairs From the Point of View of Their Structure and Classification (in Russian): Part I, Ii
,”
Bull. Petrograd Polytech. Inst
,
20
, pp.
329
386
.
46.
Manolescu
,
N.
, and
Manafu
,
V.
,
1963
, “
Sur La Détermination Du Degré De Mobilité Des Mécanismes
,”
Buletin Institut Politehnic Bucuresti
,
XXV
(
5
), pp.
45
66
.
47.
McKay
,
B. D.
, and
Piperno
,
A.
,
2014
, “
Practical Graph Isomorphism, Ii
,”
J. Symb. Comput.
,
60
, pp.
94
112
. 10.1016/j.jsc.2013.09.003
48.
The Sage Developers
,
2019
,
SageMath, the Sage Mathematics Software System (Version 8.9)
.
49.
Nie
,
S.
,
Liao
,
A.
,
Qiu
,
A.
, and
Gong
,
S.
,
2012
, “
Addition Method With 2 Links and 3 Pairs of Type Synthesis to Planar Closed Kinematic Chains
,”
Mech. Mach. Theory.
,
58
, pp.
179
191
. 10.1016/j.mechmachtheory.2012.08.006
You do not currently have access to this content.