For many single-loop closed-chain mechanisms, mobility may be characterized by the closure of sets in the theory of Lie groups. The four-revolute (4R) Bennett mechanism remains a persistent exception, requiring the formulation and expression of solutions to the loop closure relations, either directly or indirectly through spatial geometric figures. The simpler loop closure relations of the revolute-revolute-revolute-spherical (RRRS) loop, however, place conditions on the mobility of the 4R mechanism. That loop closure in turn may be interpreted as the congruence of a pair of ellipses. This new result is applied to proving the uniqueness of the Bennett mechanism along with deriving conditions where it is free from singularities. Design parameters are also identified for overconstrained RRRS mechanisms with 1DOF that are neither plane nor line symmetric. Such mechanisms, however, place the S-joint along the revolute axis of an underlying Bennett mechanism.

1.
Hervé
,
J. M.
, 1978, “
Analyse Structurelle Des Mécanismes Par Groupe Des Déplacements
,”
Mech. Mach. Theory
0094-114X,
13
(
4
), pp.
437
450
.
2.
Hunt
,
K. H.
, 1978,
Kinematic Geometry of Mechanisms
,
Clarendon
,
Oxford
.
3.
Angeles
,
J.
, 1982,
Spatial Kinematic Chains: Analysis, Synthesis, Optimization
,
Springer
,
New York
.
4.
Gogu
,
G.
, 2005, “
Mobility of Mechanisms: A Critical Review
,”
Mech. Mach. Theory
0094-114X,
40
(
9
), pp.
1068
1097
.
5.
Angeles
,
J.
, 2004, “
The Qualitative Synthesis of Parallel Manipulators
,”
ASME J. Mech. Des.
0161-8458,
126
(
4
), pp.
617
624
.
6.
Hervé
,
J.
, 1999, “
The Lie Group of Rigid Body Displacements, a Fundamental Tool for Mechanism Design
,”
Mech. Mach. Theory
0094-114X,
34
(
5
), pp.
719
730
.
7.
Selig
,
J. M.
, 1996,
Geometrical Methods in Robotics
,
Springer
,
New York
.
8.
Bennett
,
G. T.
, 1903, “
A New Mechanism
,”
Engineering (London)
0013-7782,
76
, pp.
777
778
.
9.
Parkin
,
I. A.
, and
Preston
,
J.
, 2000, “
Analysis of the Bennett Mechanism by Means of Finite Displacement Screws
,”
Proceedings of the Symposium Commemorating the Legacy, Works, and Life of Sir Robert Stawell Ball Upon the 100th Anniversary of a Treatise on the Theory of Screws
, Trinity College, University of Cambridge, UK, pp.
9
22
.
10.
Huang
,
Z.
,
Liu
,
J. F.
, and
Li
,
Q. C.
, 2008, “
Unified Methodology for Mobility Analysis Based on Screw Theory
,”
Smart Devices and Machines for Advanced Manufacturing
,
L.
Wang
and
J.
Xi
, eds.,
Springer
,
London
, pp.
49
78
.
11.
Rico
,
J. M.
, and
Ravani
,
B.
, 2002, “
Group Theory Can Explain the Mobility of Paradoxical Linkages
,”
Advances in Robot Kinematics
,
J.
Lenarcic
and
F.
Thomas
, eds.,
Kluwer Academic
,
The Netherlands
, pp.
245
254
.
12.
Rico
,
J. M.
, and
Ravani
,
B.
, 2005, “
Mobility Determination of Paradoxical Linkages
,”
Proceedings of the ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, New York, ASME Paper No. DETC2005-84936, pp.
1057
1073
.
13.
Rico
,
J. M.
,
Gallardo
,
J.
, and
Ravani
,
B.
, 2003, “
Lie Algebra and the Mobility of Kinematic Chains
,”
J. Rob. Syst.
0741-2223,
20
(
8
), pp.
477
499
.
14.
Milenkovic
,
P.
, 2010, “
Mobility of Single-Loop Kinematic Mechanisms Under Differential Displacement
,”
ASME J. Mech. Des.
0161-8458,
132
(
4
), p.
041001
.
15.
Lerbet
,
J.
, and
Fayet
,
M.
, 2003, “
Singularities of Mechanisms and the Degree of Mobility
,”
Proc. Inst. Mech. Eng., Part K: Journal of Multi-Body Dynamics
,
217
(
2
), pp.
111
119
.
16.
Waldron
,
K. J.
, 1969, “
Symmetric Overconstrained Linkages
,”
ASME J. Eng. Ind.
0022-0817,
91
, pp.
158
162
.
17.
Yu
,
H. -C.
, 1981, “
The Bennett Linkage, Its Associated Tetrahedron and the Hyperboloid of Its Axes
,”
Mech. Mach. Theory
0094-114X,
16
(
2
), pp.
105
114
.
18.
Bennett
,
G. T.
, 1914, “
The Skew Isogram Mechanism
,”
Proc. London Math. Soc.
0024-6115,
s2-13
, pp.
151
173
.
19.
Guest
,
S. D.
, and
Fowler
,
P. W.
, 2005, “
A Symmetry-Extended Mobility Rule
,”
Mech. Mach. Theory
0094-114X,
40
(
9
), pp.
1002
1014
.
20.
Bottema
,
O.
, and
Roth
,
B.
, 1990,
Theoretical Kinematics
,
North-Holland
,
Amsterdam
.
21.
Huang
,
C.
, 1997, “
The Cylindroid Associated With Finite Motions of the Bennett Mechanism
,”
ASME J. Mech. Des.
0161-8458,
119
(
4
), pp.
521
524
.
22.
Baker
,
J. E.
, 2006, “
Investigation of a Cylindroid Engendered by the Bennett Linkage
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
0954-4062,
220
(
7
), pp.
945
951
.
23.
Perez
,
A.
, and
McCarthy
,
J. M.
, 2002, “
Bennett’s Linkage and the Cylindroid
,”
Mech. Mach. Theory
0094-114X,
37
(
11
), pp.
1245
1260
.
24.
Hunt
,
K. H.
, 1967, “
Screw Axes and Mobility in Spatial Mechanisms via the Linear Complex
,”
J. Mech.
0022-2569,
2
(
3
), pp.
307
327
.
25.
Baker
,
J. E.
, 2005, “
On Certain Surfaces and Curves Associated With the Bennett Linkage
,”
Proc. Inst. Mech. Eng., Part K: Journal of Multi-Body Dynamics
,
219
(
3
), pp.
217
224
.
26.
Baker
,
J. E.
, 2008, “
A Kinematic Representation of Bennett’s Tetrahedron of Reference
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
0954-4062,
222
(
9
), pp.
1821
1827
.
27.
Delassus
,
E.
, 1922, “
Les Chaînes Articulées Fermées Et Déformables à Quatre Membres
,”
Bull. Sci. Math.
0007-4497,
46
, pp.
283
304
.
28.
Husty
,
M. L.
,
Pfurner
,
M.
, and
Schröcker
,
H. P.
, 2007, “
Algebraic Methods in Mechanism Analysis and Synthesis
,”
Robotica
0263-5747,
25
(
06
), pp.
661
675
.
29.
Pfurner
,
M.
, 2009, “
A New Family of Overconstrained 6-R Mechanisms
,”
Proceedings of EUCOMES 08
,
M.
Ceccarelli
, ed.,
Springer
,
New York
, pp.
117
124
.
30.
Smith
,
D. R.
, and
Lipkin
,
H.
, 1990, “
Analysis of Fourth Order Manipulator Kinematics Using Conic Sections
,”
Proceedings of the 1990 IEEE International Conference on Robotics and Automation
, pp.
274
278
.
31.
Waldron
,
K. J.
, 1973, “
A Study of Overconstrained Linkage Geometry by Solution of Closure Equations—Part I. Method of Study
,”
Mech. Mach. Theory
0094-114X,
8
, pp.
95
104
.
32.
Waldron
,
K. J.
, 1973, “
A Study of Overconstrained Linkage Geometry by Solution of Closure Equations—Part II. Four-Bar Linkages With Lower Pair Joints Other Than Screw Joints
,”
Mech. Mach. Theory
0094-114X,
8
(
2
), pp.
233
247
.
33.
Jin
,
Q.
, and
Yang
,
T.
, 2002, “
Overconstraint Analysis on Spatial 6-Link Loops
,”
Mech. Mach. Theory
0094-114X,
37
(
3
), pp.
267
278
.
34.
Dimentberg
,
F. M.
, 1959, A General Method for the Investigation of Finite Displacements of Spatial Mechanisms and Certain Cases of Passive Joints, Purdue Translation 436, Purdue University, Lafayette, IN.
35.
Bil
,
T.
, 2010, “
Mechanizm Bennetta w Geometrii Torusów
,”
Acta Mechanica Et Automatica
,
4
(
1
), pp.
5
8
, see http://www.actawm.pb.edu.plhttp://www.actawm.pb.edu.pl/.
36.
Bil
,
T.
, 2010, “
Geometry of a Mechanism With a Higher Pair in the Form of Two Elliptical Tori
,”
Mech. Mach. Theory
0094-114X,
45
(
2
), pp.
185
192
.
37.
Fichter
,
E.
, and
Hunt
,
K.
, 1975, “
The Fecund Torus, Its Bitangent-Circles and Derived Linkages
,”
Mech. Mach. Theory
0094-114X,
10
(
2–3
), pp.
167
176
.
38.
Alizade
,
R.
,
Selvi
,
O.
, and
Gezgin
,
E.
, 2010, “
Structural Design of Parallel Manipulators With General Constraint One
,”
Mech. Mach. Theory
0094-114X,
45
(
1
), pp.
1
14
.
39.
Hunt
,
K. H.
, 1973, “
Constant-Velocity Shaft Couplings: A General Theory
,”
ASME J. Eng. Ind.
0022-0817,
95
, pp.
455
464
.
40.
Milenkovic
,
P.
, 2009, “
Triangle Pseudocongruence in Constraint Singularity of Constant-Velocity Couplings
,”
ASME J. Mech. Rob.
1942-4302,
1
(
2
), p.
021006
.
41.
Myard
,
F. E.
, 1933, “
Theorie Generale Des Joints De Transmission De Rotation-a Couples d’Emboitement
,”
Le Génie Civil
,
102
, pp.
345
348
.
42.
Milenkovic
,
V.
, 1977, “
A New Constant Velocity Coupling
,”
ASME J. Eng. Ind.
0022-0817,
99
, pp.
367
374
.
43.
Bix
,
R.
, 2006,
Conics and Cubics: A Concrete Introduction to Algebraic Curves
,
Springer
,
New York
.
44.
Graustein
,
W. C.
, 1930,
Introduction to Higher Geometry
,
Macmillan
,
New York
.
45.
Farin
,
G. E.
, and
Hansford
,
D.
, 1998,
The Geometry Toolbox for Graphics and Modeling
,
AK Peters
,
Natick, MA
.
46.
Beggs
,
J. S.
, 1966,
Advanced Mechanism
,
Macmillan
,
New York
.
47.
Woo
,
L.
, and
Freudenstein
,
F.
, 1970, “
Application of Line Geometry to Theoretical Kinematics and the Kinematic Analysis of Mechanical Systems
,”
J. Mech.
0022-2569,
5
, pp.
417
460
.
48.
Muller
,
A.
, 2002, “
Higher Order Local Analysis of Singularities in Parallel Mechanisms
,”
ASME 2002 Design Engineering Technical Conferences and Computer and Information in Engineering Conference
, New York, ASME Paper No. DETC2002/MECH-34258, pp.
515
522
.
49.
Eisenhart
,
L. P.
, 1918, “
Surfaces Which Can be Generated in More Than One Way by the Motion of an Invariable Curve
,”
Ann. Math.
0003-486X,
19
(
3
), pp.
217
230
.
50.
Baker
,
J. E.
, 2001, “
The Axodes of the Bennett Linkage
,”
Mech. Mach. Theory
0094-114X,
36
(
1
), pp.
105
116
.
You do not currently have access to this content.