This paper introduces the three-dimensional (3D) dual Kennedy theorem in statics, and demonstrates its application to characterize the singular configuration of the 6/6 Stewart Platform (6/6 SP). The proposed characterization is articulated as a simple geometric relation that is easy to apply and check. We find two lines that cross four of the six legs of the platform. Each one of these two lines has a parallel line that crosses the remaining two legs. Each pair of parallel lines defines a plane. The 6/6 SP is in a singular position if the intersection of these two planes is perpendicular to the common normal of the remaining two legs. The method developed for the singular characterization is also used for the analysis of the mobility and forces of the SP. Finally, the proposed method is compared to some known singular configurations, such as Hunt's and Fichter's singular configurations and the 3/6 Stewart Platform (3/6 SP) singularity. The relation between the reported characterizations of the 6/6 SP and other reported works is highlighted. Moreover, it is shown that the known 3/6 SP characterization is a special case of the results reported in the paper. Finally, a characterization of a platform that does not appear in the literature, 5/6 SP, is developed based on the new approach to demonstrate its utility.

References

1.
Gough
,
V. E.
, and
Whitehall
,
S. G.
,
1962
, “
Universal Tyre Test Machine
,”
FISITA 9th International Technical Congress
, London, Apr. 30–May 5, pp.
117
137
.
2.
Stewart
,
D.
,
1965
, “
A Platform With Six Degrees of Freedom
,”
Proc. Inst. Mech. Eng.
,
180
(
1
), pp.
371
386
.
3.
Fichter
,
E. F.
,
1986
, “
A Stewart Platform-Based Manipulator: General Theory and Practical Construction
,”
Int. J. Rob. Res.
,
5
(
2
), pp.
157
182
.
4.
Shai
,
O.
,
Tehori
,
I.
,
Bronfeld
,
A.
,
Slavutin
,
M.
, and
Ben-Hanan
,
U.
,
2009
, “
Adjustable Tensegrity Robot Based on Assur Graph Principle
,”
ASME
Paper No. IMECE2009-11301.
5.
Fichter
,
E. F.
, and
McDowell
,
E. D.
,
1980
, “
A Novel Design for a Robot Arm
,”
Computer Technology Conference
, New York, Aug. 12–15, pp.
250
256
.
6.
Laryushkin
,
P. A.
, and
Glazunov
,
V. A.
,
2015
, “
On the Estimation of Closeness to Singularity for Parallel Mechanisms Using Generalized Velocities and Reactions
,”
The 14th IFToMM World Congress
, Taipei, Taiwan, Oct. 25–30, pp. 286–291.
7.
Liu
,
X.-J.
,
Wu
,
C.
, and
Wang
,
J.
,
2012
, “
A New Approach for Singularity Analysis and Closeness Measurement to Singularities of Parallel Manipulators
,”
ASME J. Mech. Rob.
,
4
(
4
), p.
041001
.
8.
Bandyopadhyay
,
S.
, and
Ghosal
,
A.
,
2006
, “
Geometric Characterization and Parametric Representation of the Singularity Manifold of a 6–6 Stewart Platform Manipulator
,”
Mech. Mach. Theory
,
41
(
11
), pp.
1377
1400
.
9.
Aleshin
,
A. K.
,
Glazunov
,
V. A.
,
Rashoyan
,
G. V.
, and
Shai
,
O.
,
2016
, “
Analysis of Kinematic Screws That Determine the Topology of Singular Zones of Parallel-Structure Robots
,”
J. Mach. Manuf. Reliab.
,
45
(
4
), pp.
291
296
.
10.
Orki
,
O.
,
Shai
,
O.
,
Ayali
,
A.
, and
Ben-Hanan
,
U.
,
2011
, “
A Model of Caterpillar Locomotion Based on Assur Tensegrity Structures
,”
ASME
Paper No. DETC2011-47708.
11.
Gosselin
,
C. M.
, and
Angeles
,
J.
,
1990
, “
Singularity Analysis of Closed-Loop Kinematic Chains
,”
IEEE Trans. Rob. Autom.
,
6
(
3
), pp.
281
290
.
12.
Ma
,
O.
, and
Angeles
,
J.
,
1992
, “
Architecture Singularities of Parallel Manipulators
,”
Int. J. Rob. Autom.
,
7
(
1
), pp.
23
29
.
13.
Hao
,
F.
, and
McCarthy
,
J. M.
,
1998
, “
Conditions for Line-Based Singularities in Spatial Platform Manipulators
,”
J. Rob. Syst.
,
15
(
1
), pp.
43
55
.
14.
Wolf
,
A.
, and
Shoham
,
M.
,
2003
, “
Investigation of Parallel Manipulators Using Linear Complex Approximation
,”
ASME J. Mech. Des.
,
125
(
3
), pp.
564
572
.
15.
Merlet
,
J.-P.
,
1989
, “
Singular Configurations of Parallel Manipulators and Grassmann Geometry
,”
Int. J. Rob. Res.
,
8
(
5
), pp.
45
56
.
16.
Ben-Horin
,
P.
, and
Shoham
,
M.
,
2006
, “
Singularity Condition of Six-Degree-of-Freedom Three-Legged Parallel Robots Based on Grassmann-Cayley Algebra
,”
IEEE Trans. Rob.
,
22
(
4
), pp.
577
590
.
17.
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
.
18.
Gan
,
D.
,
Dai
,
J. S.
,
Dias
,
J.
, and
Seneviratne
,
L.
,
2016
, “
Variable Motion/Force Transmissibility of a Metamorphic Parallel Mechanism With Reconfigurable 3T and 3R Motion
,”
ASME J. Mech. Rob.
,
8
(
5
), p.
051001
.
19.
Downing
,
D. M.
,
Samuel
,
A. E.
, and
Hunt
,
K. H.
,
2002
, “
Identification of the Special Configurations of the Octahedral Manipulator Using the Pure Condition
,”
Int. J. Rob. Res.
,
21
(
2
), pp.
147
159
.
20.
Ben-Horin
,
P.
, and
Shoham
,
M.
,
2009
, “
Application of Grassmann-Cayley Algebra to Geometrical Interpretation of Parallel Robot Singularities
,”
Int. J. Rob. Res.
,
28
(
1
), pp.
127
141
.
21.
Hunt
,
K. H.
,
1978
,
Kinematic Geometry of Mechanisms
,
University Press
,
Oxford, UK
.
22.
Ben-Horin
,
P.
, and
Shoham
,
M.
,
2007
, “
Singularity of Gough-Stewart Platforms With Collinear Joints
,”
12th IFToMM World Congress
, Besançon, France, June 17–21, pp.
743
748
.
23.
Huang
,
Z.
,
Li
,
Q.
, and
Ding
,
H.
,
2014
, “
Basics of Screw Theory
,”
Theory of Parallel Mechanisms
,
Springer
,
New York
, pp.
1
16
.
24.
Zhao
,
J.
,
Feng
,
Z.
,
Chu
,
F.
, and
Ma
,
N.
,
2014
, “
A Brief Introduction to Screw Theory
,”
Advanced Theory of Constraint and Motion Analysis for Robot Mechanisms
,
Elsevier
,
Oxford, UK
, pp.
28
79
.
25.
Beggs
,
J. S.
,
1966
,
Advanced Mechanisms
,
The Macmillan Company
,
New York
.
26.
Angeles
,
J.
,
1982
,
Spatial Kinematic Chains
,
Springer-Verlag, Berlin
.
27.
Ramahi
,
A.
, and
Tokad
,
Y.
,
1998
, “
On the Kinematics of Three-Link Spatial Cam Mechanism
,”
Meccanica
,
33
(
4
), pp.
349
361
.
28.
Li
,
B.
,
Cao
,
Y.
,
Zhang
,
Q.
, and
Huang
,
Z.
,
2013
, “
Position-Singularity Analysis of a Special Class of the Stewart Parallel Mechanisms With Two Dissimilar Semi-Symmetrical Hexagons
,”
Robotica
,
31
(
1
), pp.
123
136
.
29.
Mayer St-Onge
,
B.
, and
Gosselin
,
C. M.
,
2000
, “
Singularity Analysis and Representation of the General Gough-Stewart Platform
,”
Int. J. Rob. Res.
,
19
(
3
), pp.
271
288
.
30.
Kong
,
X.
, and
Gosselin
,
C. M.
,
2002
, “
Generation of Architecturally Singular 6-SPS Parallel Manipulators With Linearly Related Planar Platforms
,”
Electron. J. Comput. Kinematics
,
1
(
1
), p.
9
.http://www-sop.inria.fr/coprin/EJCK/Vol1-1/07_KongGos.pdf
31.
Servatius
,
B.
,
Shai
,
O.
, and
Whiteley
,
W.
,
2010
, “
Geometric Properties of Assur Graphs
,”
Eur. J. Combinatorics
,
31
(
4
), pp.
1105
1120
.
32.
Hahn
,
E.
, and
Shai
,
O.
,
2016
, “
The Unique Engineering Properties of Assur Groups/Graphs, Assur Kinematic Chains, Baranov Trusses and Parallel Robots
,”
ASME
Paper No. DETC2016-59135.
33.
Hilbert
,
D.
, and
Cohn-Vossen
,
S.
,
1952
,
Geometry and the Imagination
,
Chelsea
,
New York
.
34.
Dörrie
,
H.
,
1965
,
100 Great Problems of Elementary Mathematics: Their History and Solutions
,
Dover Publication
,
Mineola, NY
.
You do not currently have access to this content.