This paper presents a novel metamorphic 8R linkage extracted from a kirigami-fold with pregrooved creases and two overconstrained 6R linkages evolved from the 8R linkage by taking the concept of metamorphosis in the sense of structural evolution. The geometric characteristics and the parametric constraints of the evolved 6R linkages are identified following the structural evolution of the 8R linkage. The paper reveals that the evolved 6R linkages are special line-symmetric Bricard 6R loops characterized by the rotational symmetry of order two. The joint space of the overconstrained 6R linkages is analyzed and the relationship between motion parameters of the evolved 6R linkages and reconfiguration parameters of the metamorphic 8R linkage are derived. The motion characteristics of the overconstrained 6R linkages are further verified in terms of screw theory. The bifurcation and trifurcation associated with various transitory positions of the evolved 6R linkages having distinct parametric constraints are consequently identified based on constraint analysis.

References

References
1.
Bunker
,
P. R.
,
1979
,
Molecular Symmetry and Spectroscopy
,
Academic Press
,
New York
.
2.
Miller
,
W.
, Jr.
,
1972
,
Symmetry Groups and Their Applications
,
Academic Press
,
New York
.
3.
Burns
,
G.
, and
Glazer
,
A. M.
,
1990
,
Space Groups for Scientists and Engineers
,
2nd ed.
,
Academic Press, Inc.
,
Boston
, MA.
4.
Raghavan
,
M.
, and
Roth
,
B.
,
1993
, “
Inverse Kinematics of the General 6R Manipulator and Related Linkages
,”
ASME J. Mech. Des.
,
115
(
3
), pp.
502
509
.10.1115/1.2919218
5.
Baker
,
J. E.
,
2002
, “
Displacement-Closure Equations of the Unspecialised Double-Hooke's-Joint Linkage
,”
Mech. Mach. Theory
,
37
(
10
), pp.
1127
1144
.10.1016/S0094-114X(02)00042-3
6.
Cui
,
L.
, and
Dai
,
J. S.
,
2011
, “
Axis Constraint Analysis and Its Resultant 6R Double-Centered Overconstrained Mechanisms
,”
ASME J. Mech. Rob.
,
3
(
3
), p.
031004
.10.1115/1.4004225
7.
Dai
,
J. S.
,
Huang
,
Z.
, and
Lipkin
,
H.
,
2006
, “
Mobility of Overconstrained Parallel Mechanisms
,”
ASME J. Mech. Des.
,
128
(
1
), pp.
220
229
.10.1115/1.1901708
8.
Sarrus
,
P. T.
,
1853
, “
Note sur la transformation des mouvements rectilignes alternatifs, en mouvements circulaires; et reciproquement
,”
Acad. Sci.
,
36
, pp.
1036
1038
.
9.
Phillips
,
J.
,
1990
,
Freedom in Machinery: Vol. 2, Screw Theory Exemplified
,
Cambridge University Press
,
Cambridge, NY
.
10.
Bennett
,
G. T.
,
1903
, “
A New Mechanism
,”
Engineering
,
76
, pp.
777
778
.
11.
Bennett
,
G. T.
,
1914
, “
The Skew Isogram Mechanism
,”
Proc. London Math. Soc.
,
s2–13
(
2
), pp.
151
173
.10.1112/plms/s2-13.1.151
12.
Myard
,
F. E.
,
1931
, “
Contribution La Géométrie
,”
Societe mathématiques de France
,
59
, pp.
183
210
.
13.
Goldberg
,
M.
,
1943
, “
New Five-Bar and Six-Bar Linkages in Three Dimensions
,”
Trans. ASME
,
65
,
649
663
.
14.
Waldron
,
K. J.
,
1969
, “
The Mobility of Linkages
,” Ph.D. dissertation, Stanford University, Stanford, CA.
15.
Bricard
,
R.
,
1897
, “
Mémoire sur la théorie de l'octaedre articulé
,”
Journal de mathématiques pures et appliquées, Liouville
3
, pp.
113
148
.
16.
Bricard
,
R.
,
1927
,
Leçons De Cinématique
,
Tome II Cinématique Appliquée
, Gauthier-Villars, Paris, France.
17.
Schatz
,
P.
,
1975
,
Rhythmusforschung Und Technik
,
Verlag Freies Geistesleben
, Stuttgart, Germany.
18.
Lee
,
C.-C.
, and
Dai
,
J. S.
,
2003
, “
Configuration Analysis of the Schatz Linkage
,”
Proc. Inst. Mech. Eng., C: J. Mech. Eng. Sci.
,
217
(
7
), pp.
779
786
.10.1243/095440603767764426
19.
Altmann
,
P. G.
,
1954
, “
Communications to Grodzinski, P., and M'ewen, E, Link Mechanisms in Modern Kinematics
,”
Proc. Inst. Mech. Eng.
,
168
,
889
896
.
20.
Waldron
,
K. J.
,
1968
, “
Hybrid Overconstrained Linkages
,”
J. Mech.
,
3
, pp.
73
78
.10.1016/0022-2569(68)90016-5
21.
Baker
,
J. E.
,
1980
, “
An Analysis of Bricard Linkages
,”
Mech. Mach. Theory
,
15
, pp.
267
286
.10.1016/0094-114X(80)90021-X
22.
Chen
,
Y.
, and
You
,
Z.
,
2006
, “
Square Deployable Frames for Space Applications. Part 1: Theory
,”
Proc. Inst. Mech. Eng., Part G: J. Aerosp. Eng.
,
220
, pp.
347
354
.10.1243/09544100JAERO68
23.
Chen
,
Y.
, and
You
,
Z.
,
2009
, “
Two-Fold Symmetrical 6R Foldable Frames and Their Bifurcations
,”
Int. J. Solids Struct.
,
46
, pp.
4504
4514
.10.1016/j.ijsolstr.2009.09.012
24.
Rodriguez-Leal
,
E.
, and
Dai
,
J. S.
,
2007
, “
From Origami to a New Class of Centralized 3-DOF Parallel Mechanisms
,”
31st ASME Mechanisms and Robotics Conference, Proceedings of the ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Las Vegas, Nevada, September 4–7, 2007,
ASME
, Paper No. DETC2007-35516, pp.
1183
1193
.10.1115/DETC2007-35516
25.
Dai
,
J. S.
, and
Rees Jones
,
J.
,
1999
, “
Configuration Transformations in Metamorphic Mechanisms of Foldable/Erectable Kinds
,”
Proceedings of 10th World Congress on the Theory of Machine and Mechanisms
, Oulu, Finland, pp.
20
24
.
26.
Dai
,
J. S.
, and
Rees
,
J. J.
,
2002
, “
Kinematics and Mobility Analysis of Carton Folds in Packing Manipulation
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
216
(
C10
), pp.
959
970
.10.1243/095440602760400931
27.
Zhang
,
K.
,
Fang
,
Y.
,
Dai
,
J. S.
, and
Fang
,
H.
,
2010
, “
Geometry and Constraint Analysis of the 3-Spherical Kinematic Chain Based Parallel Mechanism
,”
ASME J. Mech. Rob.
,
2
(
3
), p.
031014
.10.1115/1.4001783
28.
Wei
,
G.
, and
Dai
,
J. S.
,
2014
, “
Origami-Inspired Integrated Planar-Spherical Overconstrained Mechanisms
,”
ASME J. Mech. Des
,
136
(
3
), p.
031004
.
29.
Zhang
,
K.
,
Dai
,
J. S.
, and
Fang
,
Y.
,
2012
, “
Constraint Analysis and Bifurcated Motion of the 3PUP Parallel Mechanism
,”
Mech. Mach. Theory
,
49
, pp.
256
269
.10.1016/j.mechmachtheory.2011.10.004
30.
Winder
,
B. G.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2009
, “
Kinematic Representations of Pop-Up Paper Mechanisms
,”
ASME J. Mech. Rob.
,
1
(
2
), p.
021009
.10.1115/1.3046128
31.
Balkcom
,
D. J.
, and
Mason
,
M. T.
,
2008
, “
Robotic Origami Folding
,”
Int. J. Rob. Res.
,
27
(
5
), pp.
613
627
.10.1177/0278364908090235
32.
Dai
,
J. S.
,
2010
, “
Surgical Robotics and Its Development and Progress, Special Issue on Surgical Robotics, System Development, Application Study and Performance Analysis
,”
Robotica
,
28
(
3
), p.
1
.10.1017/S0263574709990877
33.
Dai
,
J. S.
, and
Rees Jones
,
J.
,
1998
, “
Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds
,”
ASME J. Mech. Des.
,
121
(
3
), pp.
375
382
10.1115/1.2829470.
34.
Zhang
,
L. P.
, and
Dai
,
J. S.
,
2008
. “
Reconfiguration of Spatial Metamorphic Mechanisms
,”
ASME J. Mech. Rob.
,
1
(
1
), p.
011012
.10.1115/1.2963025
35.
Liu
,
C. H.
, and
Yang
,
T. L.
,
2004
. “
Essence and Characteristics of Metamorphic Mechanisms and Their Metamorphic Ways
,”
Proceedings of the 11th World Congress in Mechanism and Machine Science
, Tianjin, China, pp.
1285
1288
.
36.
Zhang
,
K.
,
Dai
,
J. S.
, and
Fang
,
Y.
,
2009
, “
A New Metamorphic Mechanism With Ability for Platform Orientation Switch and Mobility Change
,”
ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots
, London, UK, pp.
626
632
.
37.
Zhang
,
K.
,
Dai
,
J. S.
, and
Fang
,
Y.
,
2010
, “
Topology and Constraint Analysis of Phase Change in the Metamorphic Chain and Its Evolved Mechanism
,”
ASME J. Mech. Des.
,
132
(
12
), p.
121001
.10.1115/1.4002691
38.
Temko
,
F.
, and
Takahama
,
T.
,
1978
,
The Magic of Kirigami: Happenings With Paper and Scissors
,
Japan Publications
, Tokyo.
39.
Hunt
,
K. H.
,
1990
,
Kinematic Geometry of Mechanisms
,
Oxford University Press
,
New York
.
40.
Dai
,
J. S.
, and
Rees
,
J. J.
,
2001
, “
Interrelationship Between Screw Systems and Corresponding Reciprocal Systems and Applications
,”
Mech. Mach. Theory
,
36
(
5
), pp.
633
651
.10.1016/S0094-114X(01)00004-0
41.
McCarthy
,
J. M.
,
2000
,
Geometric Design of Linkages
,
Springer-Verlag
,
New York
.
42.
Dai
,
J. S.
,
2014
,
Screw Algebra and Kinematic Approaches for Mechanisms and Robotics
, Springer, London.
43.
Cundy
,
H. M.
, and
Rollett
,
A. P.
,
1981
,
Mathematical Models
,
Tarquin
,
Suffolk, UK
.
44.
Zhang
,
K.
,
Dai
,
J. S.
,
Fang
,
Y.
, and
Zeng
,
Q.
,
2011
, “
String Matrix Based Geometrical and Topological Representation of Mechanisms
,”
13th World Congress in Mechanism and Machine Science
, Guanajuato, Mexico.
45.
Denavit
,
J.
, and
Hartenberg
,
R. S.
,
1955
, “
A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices
,”
ASME J. Appl. Mech.
,
77
, pp.
215
221
.
46.
Cui
,
L.
,
Wang
,
D.
, and
Dai
,
J. S.
2009
, “
Kinematic Geometry of Circular Surfaces With a Fixed Radius Based on Euclidean Invariants
,”
ASME J. Mech. Des.
,
131
(
10
), p.
101009
.10.1115/1.3212679
47.
Zhang
,
K.
, and
Dai
,
J. S.
2012
, “
Kinematics of an Overconstrained 6R Linkage With 2-Fold Rotational Symmetry
,”
Latest Advances in Robot Kinematics
,
J.
Lenarčič
and
M.
Husty
,
Springer
The Netherlands
, pp.
229
236
.
48.
Tsai
,
L. W.
,
1999
.
Robot Analysis: The Mechanics of Serial and Parallel Manipulators
,
John Wiley
,
New York
.
49.
Gogu
,
G.
,
2009
, “
Branching Singularities in Kinematotropic Parallel Mechanisms
,”
Proceedings of the 5th International Workshop on Computational Kinematics
,
Duisburg
,
Germany
, pp.
341
348
.
50.
Huang
,
Z.
, and
Li
,
Q. C.
,
2002
, “
General Methodology for Type Synthesis of Symmetrical Lower-Mobility Parallel Manipulators and Several Novel Manipulators
,”
Int. J. Rob. Res.
,
21
(
9
), pp.
131
145
.10.1177/027836402760475342
51.
Gan
,
D.
,
Dai
,
J. S.
, and
Liao
,
Q.
,
2009
, “
Mobility Change in Two Types of Metamorphic Parallel Mechanisms
,”
ASME J. Mech. Rob.
,
1
(
4
), p.
041007
.10.1115/1.3211023
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