This paper deals with the singularity analysis of parallel manipulators with identical limb structures performing Schönflies motions, namely, three independent translations and one rotation about an axis of fixed direction (3T1R). Eleven architectures obtained from a recent type synthesis of such manipulators are analyzed. The constraint analysis shows that these architectures are all overconstrained and share some common properties between the actuation and the constraint wrenches. The singularities of such manipulators are examined through the singularity analysis of the 4-RUU parallel manipulator. A wrench graph representing the constraint wrenches and the actuation forces of the manipulator is introduced to formulate its superbracket. Grassmann–Cayley Algebra is used to obtain geometric singularity conditions. Based on the concept of wrench graph, Grassmann geometry is used to show the rank deficiency of the Jacobian matrix for the singularity conditions. Finally, this paper shows the general aspect of the obtained singularity conditions and their validity for 3T1R parallel manipulators with identical limb structures.

References

References
1.
Zlatanov
,
D.
,
Bonev
,
I.
, and
Gosselin
,
C. M
., 2002, “
Constraint Singularities of Parallel Mechanisms
,” IEEE International Conference on Robotics and Automation, pp.
496
502
.
2.
Zlatanov
,
D.
,
Fenton
,
R. G.
, and
Benhabib
,
B
., 1994, “
Singularity Analysis of Mechanisms and Robots via a Velocity-Equation Model of the Instantaneous Kinematics
,” IEEE International Conference on Robotics and Automation, pp.
986
991
.
3.
Ball
,
R. S.
, 1900,
A Treatise on the Theory of Screws
,
Cambridge University Press
,
Cambridge, CA
.
4.
Waldron
,
K. J
., 1969, “
The Mobility of Linkages
,” Ph.D. thesis,
Stanford University
,
Cambrid
ge, CA
.
5.
Hunt
,
K. H.
, 1978,
Kinematic Geometry of Mechanisms
,
Clarendon
,
Oxford
.
6.
Kong
,
X.
, and
Gosselin
,
C.
, 2007,
Type Synthesis of Parallel Mechanisms
,
Springer
,
Heidelberg
, Vol.
33
.
7.
Amine
,
S.
,
Kanaan
,
D.
,
Caro
,
S.
, and
Wenger
,
P
., 2010, “
Constraint and Singularity Analysis of Lower-Mobility Parallel Manipulators With Parallelogram Joints
,”
ASME 2010 International Design Engineering Technical Conferences
, Paper No. 28483 in DETC2010.
8.
Joshi
,
S. A.
, and
Tsai
,
L. W
., 2002, “
Jacobian Analysis of Limited-DOF Parallel Manipulators
,”
ASME J. Mech. Des.
,
124
(
2
), pp.
254
258
.
9.
Gosselin
,
C.
, and
Angeles
,
J
., 1990, “
Singularity Analysis of Closed-Loop Kinematic Chains
,”
IEEE Trans. Rob. Autom.
,
6
(
3
), pp.
281
290
.
10.
Fang
,
Y.
, and
Tsai
,
L. W
., 2002, “
Structure Synthesis of a Class of 4-DoF and 5-DoF Parallel Manipulators With Identical Limb Structures
,”
Int. J. Robot. Res.
,
21
(
9
), pp.
799
810
.
11.
Conconi
,
M.
, and
Carricato
,
M
., 2008, “
A New Assessment of Singularities of Parallel Kinematic Chains
,”
Advances in Robot Kinematics: Analysis and Design
, eds.
J.
Lenarcˇicˇ
and
P.
Wenger
,
Springer
,
Dordrecht
, pp.
3
12
.
12.
Downing
,
D.
,
Samuel
,
A.
, and
Hunt
,
K
., 2002, “
Identification of the Special Configurations of the Octahedral Manipulator Using the Pure Condition
,”
Int. J. Robot. Res.
,
21
(
2
), pp.
147
159
.
13.
Ben-Horin
,
P.
, and
Shoham
,
M
., 2006, “
Singularity Analysis of a Class of Parallel Robots Based on Grassmann–Cayley Algebra
,”
Mech. Mach. Theory
,
41
(
8
), pp.
958
970
.
14.
Ben-Horin
,
P.
, and
Shoham
,
M
., 2009, “
Application of Grassmann–Cayley Algebra to Geometrical Interpretation of Parallel Robot Singularities
,”
Int. J. Robot. Res.
,
28
(
1
), pp.
127
141
.
15.
Kanaan
,
D.
,
Wenger
,
P.
,
Caro
,
S.
, and
Chablat
,
D
., 2009, “
Singularity Analysis of Lower-Mobility Parallel Manipulators Using Grassmann–Cayley Algebra
,”
IEEE Trans. Robot.
,
25
, pp.
995
1004
.
16.
Merlet
,
J. P
., 1988, “
Parallel Manipulators. Part 2: Theory. Singular Configurations and Grassmann Geometry
,”
Sophia Antipolis
,
France
, Technical Report No. 791, INRIA.
17.
Merlet
,
J. P
., 1989, “
Singular Configurations of Parallel Manipulators and Grassmann Geometry
,”
Int. J. Robot. Res.
,
8
(
5
), pp.
45
56
.
18.
Merlet
,
J. P.
, 2006,
Parallel Robots
(Solid Mechanics and Its Applications),
Springer
,
New York
, Vol.
128
.
19.
Mbarek
,
T.
,
Lonij
,
G.
, and
Corves
,
B
., 2007, “
Singularity Analysis of a Fully Parallel Manipulator With Five-Degrees-of-Freedom Based on Grassmann Line Geometry
,”
12th IFToMM World Congress
.
20.
Tale Masouleh
,
M.
, and
Gosselin
,
C
., 2009, “
Singularity Analysis of 5-RPUR Parallel Mechanisms (3T2R)
,” The International Journal of Advanced Manufacturing Technology,
57
(
9–12
), pp.
1107
1121
.
21.
St-Onge
,
B. M.
, and
Gosselin
,
C. M
., 2000, “
Singularity Analysis and Representation of the General Gough-Stewart Platform
,”
Int. J. Robot. Res.
,
19
(
3
), pp.
271
288
.
22.
Amine
,
S.
,
Kanaan
,
D.
,
Caro
,
S.
, and
Wenger
,
P
., 2010, “
Singularity Analysis of Lower-Mobility Parallel Robots With an Articulated Nacelle
,”
Advances in Robot Kinematics: Motion in Man and Machine
,
Springer
,
New York
, pp.
273
282
.
23.
Amine
,
S.
,
Tale Masouleh
,
M.
,
Caro
,
S.
,
Wenger
,
P.
, and
Gosselin
,
C
., 2011, “
Singularity Analysis of 5-DOF Parallel Mechanisms 3T2R Using Grassmann-Cayley Algebra
,”
13th IFToMM World Congress in Mechanism and Machine Science
.
24.
Caro
,
S.
,
Khan
,
W. A.
,
Pasini
,
D.
, and
Angeles
,
J
., 2010, “
The Rule-Based Conceptual Design of the Architecture of Serial Schönflies-Motion Generators
,”
Mech. Mach. Theory
,
45
(
2
), pp.
251
260
.
25.
Kong
,
X.
, and
Gosselin
,
C
., 2004, “
Type Synthesis of 3T1R 4-DOF Parallel Manipulators Based on Screw Theory
,”
IEEE Trans. Robo. Autom.
,
20
(
2
), pp.
181
190
.
26.
Gogu
,
G
., 2007, “
Structural Synthesis of Fully-Isotropic Parallel Robots With Schoenflies Motions via Theory of Linear Transformations and Evolutionary Morphology
,”
Euro. J. Mech. A/Solids
,
26
(
2
), pp.
242
269
.
27.
White
,
N. L
., 2005, “
Grassmann-Cayley Algebra and Robotics Applications
,”
Handbook of Geometric Computing
,
Springer-Verlag Berlin and Heidelberg
, Vol.
VIII
.
28.
McMillan
,
T
., 1990, “
Invariants of Antisymmetric Tensors
,” Ph.D. thesis,
University of Florida
.
29.
Zlatanov
,
D.
,
Bonev
,
I.
, and
Gosselin
,
C. M
., 2002, “
Constraint Singularities as C-Space Singularities
,”
8th International Symposium on Advances in Robot Kinematics
,
Caldes de Malavella
,
Spain
, June 24–28, pp.
183
192
.
This content is only available via PDF.
You do not currently have access to this content.