This paper deals with a 6R single-loop overconstrained spatial mechanism that has two pairs of revolute joints with intersecting axes and one pair of revolute joints with parallel axes. The 6R mechanism is first constructed from an isosceles triangle and a pair of identical circles. The kinematic analysis of the 6R mechanism is then dealt with using a dual quaternion approach. The analysis shows that the 6R mechanism usually has two solutions to the kinematic analysis for a given input and may have two circuits (closure modes or branches) with one or two pairs of full-turn revolute joints. In two configurations in each circuit of the 6R mechanism, the axes of four revolute joints are coplanar, and the axes of the other two revolute joints are perpendicular to the plane defined by the above four revolute joints. Considering that from one configuration of the 6R mechanism, one can obtain another configuration of the mechanism by simply renumbering the joints, the concept of two-faced mechanism is introduced. The formulas for the analysis of plane symmetric spatial triangle are also presented in this paper. These formulas will be useful for the design and analysis of multiloop overconstrained mechanisms involving plane symmetric spatial RRR triads.

References

1.
Bricard
,
R.
,
1897
, “
Mémoire sur la théorie de l'octaèdre articulé
,”
J. Math. Pures Appl.
,
3
, pp.
113
148
.
2.
Bricard
,
R.
,
1927
, “
Lecons de cinématique
,”
Tome II Cinématique Appliquée
,
Gauthier-Villars
,
Paris, France
, pp.
7
12
.
3.
Goldberg
,
M.
,
1943
, “
New Five-Bar and Six-Bar Linkages in Three Dimensions
,”
Trans. ASME
,
65
, pp.
649
663
.
4.
Waldron
,
K. J.
,
1968
, “
Hybrid Overconstrained Linkages
,”
J. Mech.
, 3(2), pp.
73
78
.
5.
Wohlhart
,
K.
,
1987
, “
A New 6R Space Mechanism
,”
Seventh World Congress on Theory of Machines and Mechanisms
, Seville, Spain, Sept. 17–22, pp.
193
198
.
6.
Wohlhart
,
K.
,
1991
, “
Merging Two General Goldberg 5R Linkages to Obtain a New 6R Space Mechanism
,”
Mech. Mach. Theory
,
26
(
7
), pp.
659
668
.
7.
Mavroidis
,
C.
, and
Roth
,
B.
,
1995
, “
Analysis of Overconstrained Mechanisms
,”
ASME J. Mech. Des.
,
117
(
1
), pp.
69
74
.
8.
Dietmaier
,
P.
,
1995
, “
A New 6R Space Mechanism
,”
Ninth World Congress IFToMM
, Milan, Italy, pp.
52
56.
9.
Six
,
K.
, and
Kecskeméthy
,
A.
,
1999
, “
Steering Properties of a Combined Wheeled and Legged Striding Excavator
,”
Tenth World Congress on the Theory of Machines and Mechanisms
, Oulu, Finland, June 20–24, pp.
135
140
.
10.
Zsombor-Murray
,
P. J.
, and
Gfrerrer
,
A.
,
2002
, “
Robotrac' Mobile 6R Closed Chain
,”
CSME Forum 2002
, Kingston, ON, Canada, May 21–24, Paper No. 02-05.
11.
Baker
,
J. E.
,
2005
, “
A Curious New Family of Overconstrained Six-Bars
,”
ASME J. Mech. Des.
,
127
(
4
), pp.
602
606
.
12.
Baker
,
J. E.
,
2006
, “
On Generating a Class of Foldable Six-Bar Spatial Linkages
,”
ASME J. Mech. Des.
,
128
(
2
), pp.
374
383
.
13.
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
.
14.
Baker
,
J. E.
,
2009
, “
Screw Replacements in Isomeric Variants of Bricard's Line-Symmetric Six-Bar
,”
Proc. Inst. Mech. Eng., Part C
,
223
(
10
), pp.
2391
2398
.
15.
Pfurner
,
M.
,
2009
, “A New Family of Overconstrained 6R-Mechanisms,” Second European Conference on Mechanism Science (
EUCOMES
), Cassino, Italy, Sept. 17–20, pp.
117
124
.
16.
Hegedüs
,
G.
,
Schicho
,
J.
, and
Schröcker
,
H. P.
,
2012
, “
Construction of Overconstrained Linkages by Factorization of Rational Motions
,”
Latest Advances in Robot Kinematics
,
J.
Lenarčič
and
M.
Husty
, eds.,
Springer
,
Dordrecht, The Netherlands
, pp.
213
220
.
17.
Chen
,
Y.
, and
You
,
Z.
,
2012
, “
Spatial Overconstrained Linkages—The Lost Jade
,” Explorations in the History of Machines and Mechanisms (
HMM
2012), Amsterdam, The Netherlands, May 7–11, pp.
535
550
.
18.
Li
,
Z.
, and
Schicho
,
J.
,
2013
, “
Classification of Angle-Symmetric 6R Linkages
,”
Mech. Mach. Theory
,
70
, pp.
372
379
.
19.
Li
,
Z.
, and
Schicho
,
J.
,
2014
, “
Three Types of Parallel 6R Linkages
,”
Computational Kinematics
,
F.
Thomas
and
A.
Perez Gracia
, eds.,
Springer
,
Dordrecht, The Netherlands
, pp.
111
119
.
20.
Li
,
Z.
,
2014
, “
Sharp Linkages
,”
Advances in Robot Kinematics
,
J.
Lenarcic
and
O.
Khatib
, eds.,
Springer
,
Dordrecht, The Netherlands
, pp.
131
138
.
21.
Hegedüs
,
G.
,
Li
,
Z.
,
Schicho
,
J.
, and
Schröcker
,
H. P.
,
2015
, “
The Theory of Bonds II: Closed 6R Linkages With Maximal Genus
,”
J. Symbolic Comput.
,
68
(
2
), pp.
167
180
.
22.
Kong
,
X.
,
2014
, “
Type Synthesis of Single-Loop Overconstrained 6R Mechanisms for Circular Translation
,”
ASME J. Mech. Rob.
,
6
(
4
), p.
041016
.
23.
Lee
,
C. C.
, and
Hervé
,
J. M.
,
2014
, “
Geometric Derivation of 6R Linkages With Circular Translation
,”
Advances in Robot Kinematics
,
J.
Lenarčič
and
O.
Khatib
, eds.,
Springer
, Dordrecht,
The Netherlands
, pp.
59
67
.
24.
Yarullin
,
M. G.
,
Mingazov
,
M. R.
, and
Galiullin
,
I. A.
,
2016
, “
Historical Review of Studies of Spatial nR Linkages
,”
Int. Rev. Mech. Eng.
,
10
(5), pp. 348–356.
25.
Li
,
Z.
, and
Schicho
,
J.
,
2015
, “
A Technique for Deriving Equational Conditions on the Denavit-Hartenberg Parameters of 6R Linkages That are Necessary for Movability
,”
Mech. Mach. Theory
,
94
, pp.
1
8
.
26.
Kong
,
X.
,
2016
, “
Geometric Construction and Kinematic Analysis of a 6R Single-Loop Overconstrained Spatial Mechanism That Has Three Pairs of Revolute Joints With Intersecting Joint Axes
,”
Mech. Mach. Theory
,
102
, pp.
196
202
.
27.
Kong
,
X.
, and
Gosselin
,
C. M.
,
2004
, “
Type Synthesis of Three-Degree-of-Freedom Spherical Parallel Manipulators
,”
Int. J. Rob. Res.
,
23
(
3
), pp.
237
245
.
28.
Kong
,
X.
, and
Gosselin
,
C.
,
2007
,
Type Synthesis of Parallel Mechanisms
,
Springer
,
New York
.
29.
Li
,
B.
,
Huang
,
H.
, and
Deng
,
Z.
,
2015
, “
Mobility Analysis of Symmetric Deployable Mechanisms Involved in a Coplanar 2-Twist Screw System
,”
ASME J. Mech. Rob.
,
8
(
1
), p.
011007
.
30.
Liu
,
C.
,
Yao
,
S.
,
Wang
,
H.
, and
Yao
,
Y.
,
2015
, “
Ground Mobile Schatz Mechanism
,”
ASME J. Mech. Rob.
,
8
(
1
), p.
015002
.
31.
Kong
,
X.
, and
Huang
,
C.
,
2009
, “
Type Synthesis of Single-DOF Single-Loop Mechanisms With Two Operation Modes
,”
ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots
(
ReMAR
), London, June 22–24, pp.
136
141
.
32.
Wohlhart
,
K.
,
2010
, “
Multifunctional 7R Linkages
,”
International Symposium of Mechanisms and Machine Science (AzCIFToMM)
, Izmir, Turkey, Oct. 5–8, pp.
85
91
.
33.
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
.
34.
He
,
X.
,
Kong
,
X.
,
Hao
,
G.
, and
Ritchie
,
J.
,
2016
, “
Design and Analysis of a New 7R Single-Loop Mechanism With 4R, 6R and 7R Operation Modes
,”
Advances in Reconfigurable Mechanisms and Robots II
,
Springer International Publishing
, Cham, Switzerland, pp.
27
37
.
35.
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
.
36.
López-Custodio
,
P. C.
,
Rico
,
J. M.
,
Cervantes-Sánchez
,
J. J.
, and
Pérez-Soto
,
G. I.
,
2016
, “
Reconfigurable Mechanisms From the Intersection of Surfaces
,”
ASME J. Mech. Rob.
,
8
(
2
), p.
021029
.
37.
Kong
,
X.
,
2016
, “Kinematic Analysis of Conventional and Multi-Mode Spatial Mechanisms Using Dual Quaternions,”
ASME
Paper No. DETC2016-59194.
38.
Racila
,
L.
, and
Dahan
,
M.
,
2010
, “
Spatial Properties of Wohlhart Symmetric Mechanism
,”
Meccanica
,
45
(
2
), pp.
153
165
.
39.
Lee
,
C. C.
, and
Yan
,
H. S.
,
1993
, “
Movable Spatial 6R Mechanisms With Three Adjacent Parallel Axes
,”
ASME J. Mech. Des.
,
115
(
3
), pp.
522
529
.
40.
Kong
,
X.
,
2017
, “
Standing on the Shoulders of Giants: A Brief Note From the Perspective of Kinematics
,”
Chin. J. Mech. Eng.
,
30
(
1
), pp.
1
2
.
41.
Mavroidis
,
C.
, and
Roth
,
B.
,
1997
, “
On the Geometry of Spatial Polygons and Screw Polygons
,”
ASME J. Mech. Des.
,
119
(
2
), pp.
246
252
.
42.
Selig
,
J. M.
, and
Husty
,
M.
,
2011
, “
Half-Turns and Line Symmetric Motions
,”
Mech. Mach. Theory
,
46
(
2
), pp.
156
167
.
43.
Kong
,
X.
, and
Yu
,
J.
,
2015
, “
Type Synthesis of Two-Degrees-of-Freedom 3-4R Parallel Mechanisms With Both Spherical Translation Mode and Sphere-on-Sphere Rolling Mode
,”
ASME J. Mech. Rob.
,
7
(
4
), p.
041018
.
44.
Wu
,
Y.
,
Löwe
,
H.
,
Carricato
,
M.
, and
Li
,
Z.
,
2016
, “
Inversion Symmetry of the Euclidean Group: Theory and Application to Robot Kinematics
,”
IEEE Trans. Rob.
,
32
(
2
), pp.
312
326
.
45.
Wang
,
J.
, and
Kong
,
X.
,
2018
, “
Deployable Mechanisms Constructed by Connecting Orthogonal Bricard Linkages, 8R or 10R Single-Loop Linkages Using S Joints
,”
Mech. Mach. Theory
,
120
, pp. 178–191.
You do not currently have access to this content.