Abstract
This work focuses on the synthesis of non-redundant planar kinematic chains applied to parallel manipulators, where the mobility of the kinematic chains is constrained by the planar workspace, allowing for kinematic chains with up to three degrees-of-freedom. The synthesis process consists of two stages. First, a graph generator enumerates non-isomorphic biconnected graphs representing planar kinematic chains with up to three degrees-of-freedom and seven loops. Subsequently, graphs associated to chains with rigid subchains are removed using a degeneracy identification method based on matroid theory. Concise results are presented for chains with up to five loops. For chains with six and seven loops, results are categorized based on link partitions for a direct comparison with previous results. These findings align with prior research, with minor variations in the reported chain numbers. These variations can be attributed to several factors, including graph generation, isomorphism testing, fractionation, and subchain identification. These factors are comprehensively discussed and examined.
Graphical Abstract Figure
References
1.
Merlet
, J.-P.
, Gosselin
, C.
, and Huang
, T.
, 2016
, Parallel Mechanisms
, Springer International Publishing
, Cham
, pp. 443
–462
, Chapter 18.2.
Riaza
, H. F. Q.
, Calderón
, L. A. M.
, and Rodríguez
, M. D.
, 2019
, “Synthesis of Planar Parallel Manipulators Including Dexterity, Force Transmission and Stiffness Index
,” Mech. Based Des. Struct. Mach.
, 47
(6
), pp. 680
–702
.3.
Nakamura
, Y.
, Hanafusa
, H.
, and Yoshikawa
, T.
, 1987
, “Task-Priority Based Redundancy Control of Robot Manipulators
,” Int. J. Robot. Res.
, 6
(2
), pp. 3
–15
. 4.
Conkur
, E. S.
, and Buckingham
, R.
, 1997
, “Clarifying the Definition of Redundancy as Used in Robotics
,” Robotica
, 15
(5
), pp. 583
–586
. 5.
Mohamed
, M.
, and Gosselin
, C.
, 2005
, “Design and Analysis of Kinematically Redundant Parallel Manipulators With Configurable Platforms
,” IEEE Trans. Robot.
, 21
(3
), pp. 277
–287
. 6.
Wang
, J.
, and Gosselin
, C. M.
, 2004
, “Kinematic Analysis and Design of Kinematically Redundant Parallel Mechanisms
,” ASME J. Mech. Des.
, 126
(1
), pp. 109
–118
. 7.
Ebrahimi
, I.
, Carretero
, J. A.
, and Boudreau
, R.
, 2008
, “Kinematic Analysis and Path Planning of a New Kinematically Redundant Planar Parallel Manipulator
,” Robotica
, 26
(3
), pp. 405
–413
. 8.
Morlin
, F. V.
, Carboni
, A. P.
, and Martins
, D.
, 2021
, “Reconciling Enumeration Contradictions: Complete List of Baranov Chains With Up to 15 Links With Mathematical Proof
,” ASME J. Mech. Des.
, 143
(8
), p. 083304
. 9.
Piga Carboni
, A.
, Simas
, H.
, and Martins
, D.
, 2020
, “Actuation Scheme Enumeration and Optimal Selection for Parallel Mechanisms Based on Matroid Theory
,” Mech. Mach. Theory
, 151
(9
), p. 103891
. 10.
Gosselin
, C.
, Laliberté
, T.
, and Veillette
, A.
, 2015
, “Singularity-Free Kinematically Redundant Planar Parallel Mechanisms With Unlimited Rotational Capability
,” IEEE Trans. Robot.
, 31
(2
), pp. 457
–467
. 11.
Gordon
, G.
, and McNulty
, J.
, 2012
, Matroids: A Geometric Introduction
, Cambridge University Press
, Cambridge, UK
.12.
13.
Welsh
, D. J.
, 2010
, Matroid Theory
, Courier Corporation
, New York
.14.
The Sage Developers
, 2022
, SageMath, the Sage Mathematics Software System (Version 9.7)
.15.
Oxley
, J. G.
, 1981
, “On Connectivity in Matroids and Graphs
,” Trans. Am. Math. Soc.
, 265
(1
), pp. 47
–58
. 16.
Oxley
, J. G.
, 2001
, “On the Interplay Between Graphs and Matroids,” Surveys in Combinatorics, 2001
, Cambridge University Press
, Cambridge
, pp. 199
–239
.17.
McKay
, B. D.
, and Piperno
, A.
, 2014
, “Practical Graph Isomorphism, II
,” J. Symb. Comput.
, 60
(1
), pp. 94
–112
. 18.
Stolee
, D.
, 2011
, “Isomorph-Free Generation of 2-Connected Graphs With Applications
,” arXiV
. 19.
Hartke
, S. G.
, and Radcliffe
, A.
, 2009
, “Mckay’s Canonical Graph Labeling Algorithm,” Communicating Mathematics
, Vol. 479
, American Mathematical Society
, Providence, RI
, pp. 99
–111
.20.
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
(2
), pp. 75
–93
. 21.
Butcher
, E. A.
, and Hartman
, C.
, 2005
, “Efficient Enumeration and Hierarchical Classification of Planar Simple-Jointed Kinematic Chains: Application to 12- and 14-Bar Single Degree-of-Freedom Chains
,” Mech. Mach. Theory
, 40
(9
), pp. 1030
–1050
. 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
. 23.
Peisakh
, E. E.
, 2008
, “Structural Analysis of Planar Jointed Mechanisms: Current State and Problems
,” J. Mach. Manuf. Reliab.
, 37
(3
), pp. 207
–212
. 24.
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
(1
), pp. 1
–15
. 25.
Tuttle
, E.
, 1996
, “Generation of Planar Kinematic Chains
,” Mech. Mach. Theory
, 31
(6
), pp. 729
–748
. 26.
Sun
, L.
, Ye
, Z.
, Cui
, R.
, Huang
, X.
, and Wu
, C.
, 2022
, “Eliminating Isomorphism Identification Method for Synthesizing Nonfractionated Kinematic Chains Based on Graph Similarity
,” Mech. Mach. Theory
, 167
(1
), p. 104500
. 27.
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
. 28.
Ding
, H.
, Cao
, W.
, Kecskeméthy
, 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
. 29.
Sun
, L.
, Huang
, X.
, Zheng
, G.
, Ye
, Z.
, Ye
, B.
, and Wu
, C.
, 2023
, “Automatic Identification Method of Rigid Sub-chains Based on Branch Chain Disassembly and Replacement
,” Mech. Mach. Theory
, 179
(1
), p. 105097
. 30.
Yang
, W.
, Ding
, H.
, and Kecskeméthy
, A.
, 2022
, “Structural Synthesis Towards Intelligent Design of Plane Mechanisms: Current Status and Future Research Trend
,” Mech. Mach. Theory
, 171
(5
), p. 104715
. 31.
Ding
, H.
, Huang
, P.
, Liu
, J.
, and Kecskeméthy
, A.
, 2013
, “Automatic Structural Synthesis of the Whole Family of Planar 3-Degrees of Freedom Closed Loop Mechanisms
,” ASME J. Mech. Rob.
, 5
(4
), p. 041006
. 32.
Lee
, H.-J.
, and Yoon
, Y.-S.
, 1994
, “Automatic Method for Enumeration of Complete Set of Kinematic Chains
,” JSME Int. J. Ser. C Dyn. Control Robot. Des. Manuf.
, 37
(4
), pp. 812
–818
. 33.
Jonnalagadda
, S.
, and Krishnamurty
, S.
, 1996
, “Modified Standard Codes in Enumeration and Automatic Sketching of Mechanisms
,” International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Vol. 97577
, Irvine, CA
, Aug. 18–22
, American Society of Mechanical Engineers
, p. V02AT02A030
.34.
Butcher
, E.
, and Hartman
, C.
, 2002
, “Enumeration and Classification of 12-Bar Planar Simple-Jointed Kinematic Chains Using a Hierarchical Approach
,” 27th Biennial Mechanisms and Robotics Conferenc
e, Montreal, Quebec, Canada
, Sept. 29–Oct. 2
, pp. 471
–480
.35.
Yan
, H.-S.
, and Hwang
, Y.-W.
, 1990
, “Number Synthesis of Kinematic Chains Based on Permutation Groups
,” Math. Comput. Model.
, 13
(8
), pp. 29
–42
. 36.
Hwang
, W.-M.
, and Hwang
, Y.-W.
, 1991
, “An Algorithm for the Detection of Degenerate Kinematic Chains
,” Math. Comput. Model.
, 15
(11
), pp. 9
–15
. 37.
Hwang
, W.-M.
, and Hwang
, Y.-W.
, 1992
, “Computer-Aided Structural Synthesis of Planar Kinematic Chains With Simple Joints
,” Mech. Mach. Theory
, 27
(2
), pp. 189
–199
. 38.
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
.39.
Rao
, A.
, and Deshmukh
, P. B.
, 2001
, “Computer Aided Structural Synthesis of Planar Kinematic Chains Obviating the Test for Isomorphism
,” Mech. Mach. Theory
, 36
(4
), pp. 489
–506
. 40.
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
(12
), pp. 179
–191
. 
