Whether the singularities of a kinematic mapping constitute smoothmanifolds is an important question with significance to mechanism design and robot control. It is thus obvious to ask if this is generically so. In a preceding paper, two kinematically meaningful genericity concepts have been introduced. In this paper, geometric conditions for the manifold property of families of kinematic mappings are addressed, and a sufficient condition is presented. This condition involves the joint screws and screws representing feasible link geometries specific to a class of kinematic mappings. It admits to establish genericity of the manifold property for given classes of kinematic mappings. This is a step forward to prove that singularities of kinematic mappings form generically smooth manifolds. As an example it is shown that the singularities of 3-DOF forward kinematic mappings form generically smooth manifolds. Restricting this condition to a particular geometry allows to check whether singularities of a given kinematic mapping form smooth manifolds.

References

References
1.
Burdick
,
J. W.
, 1995, “
A Classification of 3R Regional Manipulator Singularities and Geometries
,”
Mech. Mach. Theory
,
30
, pp.
71
89
.
2.
Cheng
,
H. H.
, and
Thompson
,
S.
, 1999, “
Singularity Analysis of Spatial Mechanisms Using Dual Polynomials and Complex Dual Numbers
,”
Trans. ASME
,
121
, pp.
200
205
.
3.
Kieffer
,
J.
, 1994, “
Differential Analysis of Bifurcations and Isolated Singularities of Robots and Mechanisms
,”
IEEE Trans. Rob. Autom.
,
10
(
1
), pp.
1
10
.
4.
Pernkopf
,
F.
, and
Husty
,
M.
, 2002, “
Singularity Analysis of Spatial Steward-Gough-Platforms With Planar Base and Platform
,”
Proceedings of the ASME Design Engineering Technical Conference (DETC)
, Montreal, DETC2002/MECH-34267.
5.
Shankar
,
S.
, and
Saraf
,
A.
, 1995, “
Singularities in Mechanisms—The Local Trajectory Tracking Problem
,”
Mech. Mach. Theory
,
30
(
8
), pp.
1139
1148
.
6.
Cheng
,
F. T.
,
Chen
,
J. S.
, and
Kung
,
F. C.
, 1998, “
Study and Resolution of Singularities for a 7-DOF Redundant Manipulator
,”
IEEE Trans. Ind. Electron. Control Instrum.
,
45
(
3
), pp.
469
480
.
7.
Tsumaki
,
Y.
,
Kotera
,
S.
,
Nenchev
,
D. N.
, and
Uchiyama
,
M.
, 1997, “
Singularity-Consistent Inverse Kinematics of a 6-DOF Manipulator With a Non-Spherical Wrist
,”
Proceedings of the IEEE International Conference on Robotics and Automation
, April 20–25,
Vol. 4
, pp.
2980
2985
.
8.
Müller
,
A.
, 2011, “
On the Manifold Property of the Set of Singularities of Kinematic Mappings: Modeling, Classification, and Genericity
,”
ASME J. Mech. Rob.
,
3
(
1
), pp.
011006
-1–011006-
8
.
9.
Lerbet
,
J.
, 2005, “
Stability of Singularities of a Kinematical Chain
,”
Proceedings of ASME International Design Engineering Technical Conference
, September 24–28, 2005, Long Beach, California, DETC2005-84126.
10.
Wenger
,
P.
, and
El Omri
,
K.
, 1997, “
Comments on ‘A classification of 3R Regional Manipulator Geometries and Singularities’
,”
Mech. Mach. Theory
,
32
(
4
), pp.
529
532
.
11.
Wenger
,
P.
, 1998, “
Classification of 3R Positioning Manipulators
,”
ASME J. Mech. Des.
,
120
, pp.
327
332
.
12.
Chablat
,
D.
,
Moroz
,
G.
, and
Wenger
,
P.
, 2011, “
Uniqueness Domains and Non Singular Assembly Mode Changing Trajectories
,”
Proceedings of the IEEE International Conference on Robotics and Automation (ICRA)
, Shanghai, China, May 9–13, pp.
3946
3951
.
13.
Donelan
,
P.
, and
Müller
,
A.
, 2010, “
Singularities of Regional Manipulators Revisited
,”
Advances in Robot Kinematics
J.
Lenarčič
, and
M.
Stanisic
, eds.,
Springer
,
New York
, pp.
509
519
.
14.
Müller
,
A.
, 2009, “
On the Concept of Mobility Used in Robotics
,”
33rd Mechanisms and Robotics Conference, ASME 2007 International Design Engineering Technical Conferences
, Aug. 30–Sept. 2, San Diego, CA.
15.
Gogu
,
G.
, 2005, “
Mobility of Mechanism: A Critical Review
,”
Mech. Mach. Theory
,
40
, pp.
1068
1097
.
16.
Chen
,
C.
, 2011, “
The Order of Local Mobility of Mechanisms
,”
Mech. Mach. Theory
,
46
, pp.
1251
1264
.
17.
Müller
,
A.
, and
Rico
,
J. M.
, 2008, “
Mobility and Higher Order Local Analysis of the Configuration Space of Single-Loop Mechanisms
,”
Advances in Robot Kinematics
J. J.
Lenarcic
, and
P.
Wenger
, eds.,
Springer
,
New York
, pp.
215
224
.
18.
Walter
,
D. R.
, and
Husty
,
M. L.
, 2010, “
On Implicitization of Kinematic Constraint Equations
,”
Mach. Des. Res.
,
26
, pp.
218
226
.
19.
Wampler
,
C. W.
,
Hauenstein
,
J. D.
, and
Sommese
,
A. J.
, 2011, “
Mechanism Mobility and a Local Dimension Test
,”
Mech. Mach. Theory
,
46
, pp.
1193
1206
.
20.
Arimoto
,
S.
, 2011, “
Modeling and Control of Multi-Body Mechanical Systems: Part I A Riemannian Geometry Approach
,”
Advances in the Theory of Control, Signals and Systems With Physical Modeling
,
L.
Jean
, and
M.
Philippe
, eds.,
Springer
,
Berlin/Heidelberg
, pp.
3
16
.
21.
Gan
,
D.
,
Dai
,
J. S.
, and
Caldwell
,
D. G.
, 2011, “
Constraint-Based Limb Synthesis and Mobility-Change-Aimed Mechanism Construction
,”
ASME J. Mech. Des.
,
133
(
5
), pp.
877
887
.
22.
Tchoń
,
K.
, 1991, “
Differential Topology of the Inverse Kinematic Problem for Redundant Robot Manipulators
,”
Int. J. Robot. Res.
,
10
, pp.
492
504
.
23.
Golubitsky
,
M.
, and
Guillemin
,
V.
1973,
Stable mappings and their singularities
,
Springer
,
New York
.
24.
Guillemin
,
V.
, and
Pollack
,
A.
, 1974,
Differential Topology
,
Prentice-Hall
,
New Jersey
.
25.
Donelan
,
P. S.
, 2007, “
Singularity-Theoretic Methods in Robot Kinematics
,”
Robotica
,
25
, pp.
641
659
.
26.
Pai
,
D. K.
, and
Leu
,
M. C.
, “
Genericity and Singularities of Robot Manipulators
,”
IEEE Trans. Rob. Autom.
,
8
(
5
), pp.
545
559
.
27.
Tsai
,
K. Y.
,
Arnold
,
J.
, and
Kohli
,
D.
, 1993, “
Generic Maps of Mechanical Manipulators
,”
Mech. Mach. Theory
,
28
(
1
), pp.
53
64
.
28.
Lerbet
,
J.
, and
Hao
,
K.
, 1999, “
Kinematics of Mechanisms to the Second Order—Application to Closed Mechanisms
,”
Acta Appl. Math.
,
59
, pp.
1
19
.
29.
Müller
,
A.
, 2009, “
A Genericity Condition for General Serial Manipulators
,”
Proceedings of the IEEE International Conference on Robotics and Automation (ICRA)
, Kobe, Japan, May 12–17, pp.
2951
2956
.
30.
Murray
,
R. M.
,
Li
,
Z.
, and
Sastry
,
S. S.
, 1993,
A Mathematical Introduction to Robotic Manipulation
,
CRC Press, Inc. Boca Raton, FL, USA
.
31.
Selig
,
J. M.
, 1996,
Geometrical Methods in Robotics
,
Springer
,
New York
.
32.
Hervé
,
J. M.
, 1982, “
Intrinsic Formulation of Problems of Geometry and Kinematics of Mechanisms
,”
Mech. Mach. Theory
,
17
(
3
), pp.
179
184
.
33.
Wohlhart
,
K.
, 1996, “
Kinematotropic Linkages
,”
Recent Advances in Robot Kinematics
,
J.
Lenarčič
, and
V.
Parent-Castelli
, eds.,
Kluwer, Dortrecht
,
The Netherlands
, pp.
359
368
.
34.
Müller
,
A.
, 2009, “
Generic Mobility of Rigid Body Mechanisms
,”
Mech. Mach. Theory
,
44
(
6
), pp.
1240
1255
.
35.
Karger
,
A.
, 1996, “
Classification of Serial Robot-Manipulators With Non-Removable Singularities
,”
ASME J. Mech. Des.
,
118
, pp.
202
208
.
36.
Wall
,
C. T. C.
, 1976, “
Geometric Properties of Differentiable Manifolds
,”
Geometry and Topology
(Lecture Notes in Mathematics),
Springer
,
Berlin
,
Vol. 597
, pp.
707
774
.
37.
Donelan
,
P.
, 2008, “
Genericity Conditions for Serial Manipulators
,”
Advances in Robot Kinematics
June 22–26, Batz-sur-Mer, France, pp.
185
192
.
38.
Dai
,
J. S.
, and
Jones
,
J. R.
, 2001, “
Interrelationship Between Screw Systems and Corresponding Reciprocal Systems and Applications
,”
Mech. Mach. Theory
,
36
(
5
), pp.
633
651
.
39.
Müller
,
A.
, 2007, “
A Necessary and Sufficient Condition for the Genericity of Serial Manipulators
,”
Proceedings of ASME International Design Engineering Technical Conference
, DETC2007-34620.
40.
Miller
,
W.
, 1964, “
Some Applications of the Representation Theory of the Euclidean Group in Three-Space
,”
Commun. Pure Appl. Math.
,
XVII
, pp.
527
540
.
41.
Hao
,
K.
, 1998, “
Dual Number Method, Rank of a Screw System and Generation of Lie Sub-Algebras
,”
Mech. Mach. Theory
,
33
(
7
), pp.
1063
1084
.
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