In a classical mobility-one single loop linkage, the motion begins from an original position determined by the assembled condition and runs in cycles. In normal circumstances, the linkage experiences a full cycle when the input-joint completes a full revolution. However, there are some linkages that accomplish a whole cycle with the input-joint having to go through multiple revolutions. Their motion cycle covers multiple revolutions of the input-joint. This paper investigates this typical phenomenon that the output angle is in a different motion cycle of the input angle that we coin this as the multiple input-joint revolution cycle. The paper then presents the configuration torus for presenting the motion cycle and reveals both bifurcation and double points of the linkage, using these mathematics-termed curve characteristics for the first time in mechanism analysis. The paper examines the motion cycle of the Bennett plano-spherical hybrid linkage that covers an 8π range of an input-joint revolution, reveals its four double points in the kinematic curve, and presents two motion branches in the configuration torus where double points give bifurcations of the linkage. The paper further examines the Myard plane-symmetric 5R linkage with its motion cycle covering a 4π range of the input-joint revolution. The paper, hence, presents a way of mechanism cycle and reconfiguration analysis based on the configuration torus.

References

References
1.
Bennett
,
G. T.
,
1903
, “
A New Mechanism
,”
Engineering
,
76
, pp.
777
778
.
2.
Bennett
,
G. T.
,
1905
, “
The Parallel Motion of Sarrut and Some Allied Mechanisms
,”
Philos. Mag.
,
6
(
54
), pp.
803
810
.
3.
Dai
,
J. S.
,
Huang
,
Z.
, and
Lipkin
,
H.
,
2006
, “
Mobility of Overconstrained Parallel Mechanisms
,”
ASME J. Mech. Des.
,
128
(
1
), pp.
220
229
.
4.
Dai
,
J. S.
,
2014
,
Geometrical Foundations and Screw Algebra for Mechanisms and Robotics
,
Higher Education Press
,
Beijing, China
(translated from Dai, J. S. 2019, Screw Algebra and Kinematic Approaches for Mechanisms and Robotics, Springer, London).
5.
Kutzbach
,
K.
,
1929
, “
Mechanische Leitungsverzweigung, Ihre Gesetze Und Anwendungen
,”
Maschinenbau
,
8
(
21
), pp.
710
716
.
6.
Myard
,
F. E.
,
1931
, “
Contribution à La Géométrie Des Systèmes Articulés
,”
Soc. Math. France
,
59
, pp.
183
210
.
7.
Goldberg
,
M.
,
1943
, “
New Five-Bar and Six-Bar Linkages in Three Dimensions
,”
ASME Trans.
,
65
, pp.
649
661
.
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.
Bricard
,
R.
,
1897
, “
Mémoire Sur La Théorie De L'octaedre Articulé
,”
J. Pure Appl. Math.
,
3
, pp.
113
148
.
10.
Bricard
,
R.
,
1927
,
Leçons De Cinématique
,
Gauthier-Villars
,
Paris, France
.
11.
Schatz
,
P.
,
1942
, “
Mechanism Producing Wavering and Rotating Movements of Receptacles
,” U.S. Patent No.
US2302804
.
12.
Lee, C. C.
, and
Dai, J. S.
, 2003, “
Configuration Analysis of the Schatz Linkage
,”
J. Mech. Eng. Sci.
,
217
(7), pp. 779–786.
13.
Grodzinski, P.
, and
M'Ewen, E.
,
1954
, “
Link Mechanisms in Modern Kinematics
,”
Proc. Inst. Mech. Eng.
,
168
(
1
), pp.
877
896
.
14.
Waldron
,
K. J.
,
1968
, “
Hybrid Overconstrained Linkages
,”
J. Mech
,
3
(
2
), pp.
73
78
.
15.
Wohlhart
,
K.
,
1991
, “
Merging Two General Goldberg 5R Linkages to Obtain a New 6R Space Mechanism
,”
Mech. Mach. Theory
,
26
(
7
), pp.
659
668
.
16.
Chen
,
Y.
, and
You
,
Z.
,
2007
, “
Spatial 6R Linkages Based on the Combination of Two Goldberg 5R Linkages
,”
Mech. Mach. Theory
,
42
(
11
), pp.
1484
1498
.
17.
Song
,
C. Y.
,
Chen
,
Y.
, and
Chen
,
I. M.
,
2013
, “
A 6R Linkage Reconfigurable Between the Line-Symmetric Bricard Linkage and the Bennett Linkage
,”
Mech. Mach. Theory
,
70
, pp.
278
292
.
18.
Zhang
,
K.
, and
Dai
,
J. S.
,
2014
, “
Trifurcation of the Evolved Sarrus-Motion Linkage Based on Parametric Constraints
,”
Advances in Robot Kinematics
,
Springer International Publishing
, Cham,
Switzerland
, pp.
345
353
.
19.
Baker
,
J. E.
,
1979
, “
The Bennett, Goldberg and Myard Linkages—in Perspective
,”
Mech. Mach. Theory
,
14
(
4
), pp.
239
253
.
20.
Baker
,
J. E.
,
1980
, “
An Analysis of the Bricard Linkages
,”
Mech. Mach. Theory
,
15
(
4
), pp.
267
286
.
21.
Baker
,
J. E.
,
2002
, “
Displacement–Closure Equations of the Unspecialised Double-Hooke's-Joint Linkage
,”
Mech. Mach. Theory
,
37
(
10
), pp.
1127
1144
.
22.
Cui
,
L.
, and
Dai
,
J. S.
,
2011
, “
Axis Constraint Analysis and Its Resultant 6R Double-Centered Over-Constrainted Mechanisms
,”
ASME J. Mech. Rob.
,
3
(
3
), p.
031004
.
23.
Kong
,
X.
,
2014
, “
Type Synthesis of Single-Loop Over-Constrainted 6R Spatial Mechanisms for Circular Translation
,”
ASME J. Mech. Rob.
,
6
(
4
), p.
041016
.
24.
Zhang
,
K.
,
Müller
,
A.
, and
Dai
,
J. S.
,
2016
, “
A Novel Reconfigurable 7R Linkage With Multifurcation
,”
Advances in Reconfigurable Mechanisms and Robots II
,
Springer International Publishing
,
Cham, Switzerland
, pp.
15
25
.
25.
Aimedee
,
F.
,
Gogu
,
G.
,
Dai
,
J. S.
,
Bouzgarrou
,
C.
, and
Bouton
,
N.
,
2016
, “
Systematization of Morphing in Reconfigurable Mechanisms
,”
Mech. Mach. Theory
,
96
, pp.
215
224
.
26.
Zhang
,
K.
, and
Dai
,
J. S.
,
2014
, “
A Kirigami-Inspired 8R Linkage and Its Evolved Overconstrained 6R Linkages With the Rotational Symmetry of Order Two
,”
ASME J. Mech. Rob.
,
6
(
2
), p.
021007
.
27.
Barton
,
L. O.
,
1993
,
Mechanism Analysis: Simplified Graphical and Analytical Techniques
, Marcel Dekker,
New York
.
28.
Waldron
,
K. J.
, and
Kinzel
,
G. L.
,
1998
,
Kinematics, Dynamics, and Design of Machinery
,
Wiley
,
Hoboken, NJ
.
29.
Qin
,
Y.
,
Dai
,
J. S.
, and
Gogu
,
G.
,
2014
, “
Multi-Furcation in a Derivative Queer-Square Mechanism
,”
Mech. Mach. Theory
,
81
, pp.
36
53
.
30.
Kong
,
X.
,
2015
, “
Kinematic Analysis of a 6R Single-Loop Overconstrained Spatial Mechanism for Circular Translation
,”
Mech. Mach. Theory
,
93
, pp.
163
174
.
31.
Zlatanov
,
D.
,
1999
, “
Generalized Singularity Analysis of Mechanisms
,” Ph.D. dissertation, University of Toronto, Toronto, Canada.
32.
Zlatanov
,
D.
,
Bonev
,
I. A.
, and
Gosselin
,
C. M.
,
2002
, “
Constraint Singularities as C-Space Singularities
,”
Advances in Robot Kinematics
,
Springer
,
Dordrecht, The Netherlands
, pp.
183
192
.
33.
Harold Hilton
,
M. A.
,
1920
,
Plane Algebraic Curves
,
The Clarendon Press
,
Oxford, UK
.
34.
Dhami
,
H. S.
,
2009
,
Differential Calculus
,
New Age International Pvt Ltd Publishers
,
New Delhi, India
.
35.
Zhang
,
K.
, and
Dai
,
J. S.
,
2015
, “
Screw-System-Variation Enabled Reconfiguration of the Bennett Plano-Spherical Hybrid Linkage and Its Evolved Parallel Mechanism
,”
ASME J. Mech. Des.
,
137
(
6
), p.
062303
.
36.
Dai
,
J. S.
,
2012
, “
Finite Displacement Screw Operators With Embedded Chasles' Motion
,”
ASME J. Mech. Rob.
,
4
(
4
), p.
041002
.
37.
Dai
,
J. S.
,
2014
,
Screw Algebra and Lie Groups and Lie Algebra
,
Higher Education Press
,
Beijing, China
.
38.
McCarthy
,
J. M.
,
1990
,
An Introduction to Theoretical Kinematics
,
MIT Press
,
Cambridge, MA
.
39.
Baker
,
J. E.
,
2003
, “
Overconstrained Six-Bars With Parallel Adjacent Joint-Axes
,”
Mech. Mach. Theory
,
38
(
2
), pp.
103
117
.
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