For a platform connected to its base through two chains forming a single loop, the instantaneous mobility may be expressed by a set of motion screws that is in the intersection of the sets of motion screws for each of the two chains. A recent work shows that the platform remains mobile after differential displacement along all mobile paths if the Lie closures of the screw sets of the two chains are each within the span of the union of screw sets of those chains. If this union span is one dimension short of containing the Lie closures of the two chains, a quadratic form determines whether the reference pose is at a constraint singularity and resolves the mobile paths at that singularity. Those results are now extended to a platform manipulator with more than two chains, using a recursive procedure for updating velocity, acceleration, and higher-order descriptions of platform mobility after adding successive chains. The new analytical technique characterizes the bifurcation of the mobility at constraint singularity of 3RSR, 3RER, and 3UPU platform mechanisms proposed for use in constant-velocity couplings, robotic wrists, and translational manipulators.

1.
Gogu
,
G.
, 2005, “
Mobility of Mechanisms: A Critical Review
,”
Mech. Mach. Theory
0094-114X,
40
(
9
), pp.
1068
1097
.
2.
Adams
,
J. D.
, and
Whitney
,
D. E.
, 2001, “
Application of Screw Theory to Constraint Analysis of Mechanical Assemblies Joined by Features
,”
ASME J. Mech. Des.
0161-8458,
123
, pp.
26
32
.
3.
Dai
,
J. S.
,
Huang
,
Z.
, and
Lipkin
,
H.
, 2006, “
Mobility of Overconstrained Parallel Mechanisms
,”
ASME J. Mech. Des.
0161-8458,
128
, pp.
220
229
.
4.
Zhao
,
J. S.
,
Feng
,
Z. J.
, and
Wang
,
L. P.
, 2006, “
The Free Mobility of a Parallel Manipulator
,”
Robotica
0263-5747,
24
(
5
), pp.
635
641
.
5.
Wohlhart
,
K.
, 2000, “
Architectural Shakiness or Architectural Mobility of Platforms
,”
Advances in Robot Kinematics
,
J.
Lenarcic
and
M. M.
Stanisic
, eds.,
Kluwer
,
Dordrecht
, pp.
365
374
.
6.
Roschel
,
O.
, and
Mick
,
S.
, 1998, “
Characterization of Architecturally Shaky Platforms
,”
Advances in Robot Kinematics: Analysis and Control
,
J.
Lenarcic
and
M. L.
Husty
, eds.,
Kluwer
,
Dordrecht
, pp.
465
474
.
7.
Muller
,
A.
, 2002, “
Higher Order Local Analysis of Singularities in Parallel Mechanisms
,”
ASME
Paper No. DETC2002/MECH-34258.
8.
Zlatanov
,
D.
,
Bonev
,
I. A.
, and
Gosselin
,
C. M.
, 2002, “
Constraint Singularities of Parallel Mechanisms
,”
Proceedings of the ICRA ’02, IEEE International Conference on Robotics and Automation
, Vol.
1
, pp.
980
985
.
9.
Kong
,
X.
, and
Gosselin
,
C. M.
, 2005, “
Mobility Analysis of Parallel Mechanisms Based on Screw Theory and the concept of Equivalent Serial Kinematic Chains
,”
ASME
Paper No. DETC2005.
10.
Chan
,
V. K.
, and
Ebert-Uphoff
,
I.
, 2001, “
Investigation of the Deficiencies of Parallel Manipulators in Singular Configurations Through the Jacobian Nullspace
,”
IEEE International Conference on Robotics and Automation
, Vol.
2
, pp.
1313
1320
.
11.
Wolf
,
A.
, and
Shoham
,
M.
, 2003, “
Investigation of Parallel Manipulators Using Linear Complex Approximation
,”
ASME J. Mech. Des.
0161-8458,
125
, pp.
564
572
.
12.
Wang
,
Y. X.
, and
Wang
,
Y. M.
, 2005, “
Configuration Bifurcations Analysis of Six Degree-of-Freedom Symmetrical Stewart Parallel Mechanisms
,”
ASME J. Mech. Des.
0161-8458,
127
, pp.
70
77
.
13.
Rico
,
J. M.
, and
Gallardo
,
J.
, 1996, “
Acceleration Analysis, via Screw Theory, and Characterization of Singularities of Closed Chains
,”
Recent Advances in Robot Kinematics
,
J.
Lenarcic
and
V.
Parenti-Castelli
, eds.,
Kluwer
,
Dordrecht
, pp.
139
148
.
14.
Kieffer
,
J.
, 1992, “
Manipulator Inverse Kinematics for Untimed End-Effector Trajectories With Ordinary Singularities
,”
Int. J. Robot. Res.
0278-3649,
11
(
3
), pp.
225
237
.
15.
Kieffer
,
J.
, 1994, “
Differential Analysis of Bifurcations and Isolated Singularities for Robots and Mechanisms
,”
IEEE Trans. Rob. Autom.
1042-296X,
10
(
1
), pp.
1
10
.
16.
Lerbet
,
J.
, and
Fayet
,
M.
, 2003, “
Singularities of Mechanisms and the Degree of Mobility
,”
Proc. Inst. Mech. Eng., Part K: J. Multi-Body Dynamics
,
217
(
2
), pp.
111
119
.
17.
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
.
18.
Abbaspour
,
H.
, and
Moskowitz
,
M.
, 2007,
Basic Lie Theory
,
World Scientific
,
Singapore
, p.
427
.
19.
Donelan
,
P. S.
, 2007, “
Singularity-Theoretic Methods in Robot Kinematics
,”
Robotica
0263-5747,
25
(
06
), pp.
641
659
.
20.
Donelan
,
P. S.
, 2008, “
Genericity Conditions for Serial Manipulators
,”
Advances in Robot Kinematics: Analysis and Design
,
J.
Lenarcic
and
P.
Wenger
, eds.,
Springer
,
New York
, pp.
185
192
.
21.
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
.
22.
Park
,
F. C.
, 1994, “
Computational Aspects of the Product-of-Exponentials Formula for Robot Kinematics
,”
IEEE Trans. Autom. Control
0018-9286,
39
(
3
), pp.
643
647
.
23.
Milenkovic
,
P.
, 2010, “
Mobility of Single-Loop Kinematic Mechanisms Under Differential Displacement
,”
ASME J. Mech. Des.
0161-8458,
132
(
4
), p.
041001
.
24.
Howe
,
R.
, 1983, “
Very Basic Lie Theory
,”
Am. Math. Monthly
0002-9890,
90
(
9
), pp.
600
623
.
25.
Rico
,
J. M.
, and
Ravani
,
B.
, 2007, “
On Calculating the Degrees of Freedom or Mobility of Overconstrained Linkages: Single-Loop Exceptional Linkages
,”
ASME J. Mech. Des.
0161-8458,
129
, pp.
301
311
.
26.
Yefimov
,
N. V.
, and
Shenitzer
,
A.
, 1964,
Quadratic Forms and Matrices: An Introductory Approach
,
Academic
,
New York
.
27.
Haykin
,
S.
, 1991,
Adaptive Filter Theory
,
2nd ed.
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
28.
Hunt
,
K. H.
, 1973, “
Constant-Velocity Shaft Couplings: A General Theory
,”
ASME J. Eng. Ind.
0022-0817,
95
, pp.
455
464
.
29.
Clemens
,
M.
, 1869, “
Improvement in Apparatus for Transmitting Rotary Motion
,” U.S. Patent No. 96,395.
30.
Canfield
,
S. L.
,
Reinholtz
,
C. F.
,
Salerno
,
R. J.
, 1997, “
Spatial, Parallel-Architecture Robotic Carpal Wrist
,” U.S. Patent No. 5,699,695.
31.
Salerno
,
R. J.
,
Canfield
,
S. L.
, and
Ganino
,
A. J.
, 1995, “
Parallel, Four Degree-of-Freedom Robotic Wrist
,”
Proceedings of the 1995 ASME Design Engineering Technical Conferences
, Boston, MA, Vol.
82
, pp.
765
771
.
32.
Milenkovic
,
P.
, 2009, “
Triangle Pseudocongruence in Constraint Singularity of Constant-Velocity Couplings
,”
ASME J. Mech. Rob.
1942-4302,
1
, p.
021006
.
33.
Seymour
,
K.
, 2007, “
JLAPACK Source and Class Files Version 0.8
,” http://www.Netlib.org/java/f2j/http://www.Netlib.org/java/f2j/
34.
Di Gregorio
,
R.
, 2004, “
Kinematics of the 3-RSR Wrist
,”
IEEE Trans. Robot.
,
20
(
4
), pp.
750
753
.
35.
Myard
,
F. E.
, 1933, “
Theorie Generale Des Joints De Transmission De Rotation—a Couples d’Emboitement
,”
Le Genie Civil
,
102
, pp.
345
348
.
36.
Milenkovic
,
V.
, 1977, “
A New Constant Velocity Coupling
,”
ASME J. Eng. Ind.
0022-0817,
99
, pp.
367
374
.
37.
Aravind
,
P. K.
, 1989, “
Geometrical Interpretation of the Simultaneous Diagonalization of Two Quadratic Forms
,”
Am. J. Phys.
0002-9505,
57
(
4
), pp.
309
311
.
38.
Guan
,
L. W.
,
Wang
,
J. S.
, and
Wang
,
L. P.
, 2004, “
Mobility Analysis of the 3-UPU Parallel Mechanism Based on Screw Theory
,”
Proceedings of the 2004 International Conference on Intelligent Mechatronics and Automation
, pp.
309
314
.
39.
Di Gregorio
,
R.
, and
Parenti-Castelli
,
V.
, 2002, “
Mobility Analysis of the 3-UPU Parallel Mechanism Assembled for a Pure Translational Motion
,”
ASME J. Mech. Des.
0161-8458,
124
, pp.
259
264
.
40.
Gogu
,
G.
, 2008, “
Constraint Singularities and the Structural Parameters of Parallel Robots
,”
Advances in Robot Kinematics: Analysis and Design
,
J.
Lenarcic
and
P.
Wenger
, eds.,
Springer
,
The Netherlands
, pp.
21
28
.
41.
Zlatanov
,
D.
,
Bonev
,
I. A.
, and
Gosselin
,
C. M.
, 2002, “
Constraint Singularities as C-space Singularities
,”
Advances in Robot Kinematics: Theory and Applications
,
J.
Lenarcic
and
F.
Thomas
, eds.,
Kluwer
,
The Netherlands
, pp.
183
192
.
42.
Bonev
,
I.
, and
Zlatanov
,
D.
, 2001 “
The Mystery of the Singular SNU Translational Robot
,” http://www.parallemic.org/Reviews/Review004.htmlhttp://www.parallemic.org/Reviews/Review004.html
43.
Walter
,
D. R.
,
Husty
,
M. L.
, and
Pfurner
,
M.
, 2009, “
A Complete Kinematic Analysis of the SNU 3-UPU Parallel Robot
,”
Interactions of Classical and Numerical Algebraic Geometry: A Conference in Honor of A.J. Sommese
,
D. J.
Bates
,
G. M.
Besana
,
S.
Di Rocco
, and
C. W.
Wampler
, eds.,
American Mathematical Society
,
Providence, RI
, pp.
331
346
.
44.
Zlatanov
,
D.
,
Bonev
,
I. A.
, and
Gosselin
,
C. M.
, 2001, “
Constraint Singularities as Configuration Space Singularities
,” http://www.parallemic.org/Reviews/Review008.htmlhttp://www.parallemic.org/Reviews/Review008.html
You do not currently have access to this content.