This paper considers single degree-of-freedom (DOF), closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has input singularities, that is, turning points with respect to the input angle, which break the motion curve into branches. Motion of the linkage along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. Allowing the design parameter to vary, the singularities form a curve called the critical curve, whose projection is the singularity trace. Many critical points are the singularities of the critical curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. This paper presents a general method to compute the singularity trace and its critical points. As an example, the method is used on a Stephenson III linkage, and a range of the design parameter is found where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with critical points that appear as cusps on the singularity trace.

References

References
1.
Erdman
,
A.
Sandor
,
G.
, and
Kota
,
S.
,
2001
,
Mechanism Design: Analysis and Synthesis
, Vol.
1
, No.
4/e
,
Prentice–Hall
,
Englewood Cliffs, NJ
.
2.
Kovacs
,
P.
, and
Mommel
,
G.
,
1993
, “
On the Tangent-Half-Angle Substitution
,”
Computational Kinematics
,
J.
Angeles
, ed.,
Kluwer Academic Publishers
,
Norwell, MA
, pp.
27
40
.
3.
Porta
J.
,
Ros
,
L.
,
Creemers
,
T.
, and
Thomas
,
F.
,
2007
, “
Box Approximations of Planar Linkage Configuration Spaces
,”
ASME J. Mech. Des.
,
129
(
4
), pp.
393
405
.10.1115/1.2437808
4.
Wampler
,
C. W.
,
1999
, “
Solving the Kinematics of Planar Mechanisms
,”
ASME J. Mech. Des.
,
121
(
3
), pp.
387
391
.10.1115/1.2829473
5.
Wampler
,
C. W.
,
2001
, “
Solving the Kinematics of Planar Mechanisms by Dixon Determinant and a Complex-Plane Formulation
,”
ASME J. Mech. Des.
,
123
(
3
), pp.
382
387
.10.1115/1.1372192
6.
Wampler
,
C. W.
,
1996
, “
Isotropic Coordinates, Circularity, and Bezout Numbers: Planar Kinematics From a New Perspective
,”
Proceedings of the ASME Design Technical Conference
, Paper No. 96-DETC/MECH-1210.
7.
Nielsen
,
J.
, and
Roth
,
B.
,
1999
, “
Solving the Input/Output Problem for Planar Mechanisms
,”
ASME J. Mech. Des.
,
121
(
2
), pp.
206
211
.10.1115/1.2829445
8.
Wampler
,
C. W.
, and
Sommese
,
A. J.
,
2011
, “
Numerical Algebraic Geometry and Algebraic Kinematics
,”
Acta Numer
., pp.
469
567
.10.1017/S0962492911000067
9.
Chase
,
T.
, and
Mirth
,
J.
,
1993
, “
Circuits and Branches of Single-Degree-of-Freedom Planar Linkages
,”
ASME J. Mech. Des.
,
115
(
2
), pp.
223
230
.10.1115/1.2919181
10.
Foster
,
D.
, and
Cipra
,
R.
,
1998
, “
Assembly Configurations and Branches of Planar Single-Input Dyadic Mechanisms
,”
ASME J. Mech. Des.
,
120
(
3
), pp.
381
386
.10.1115/1.2829162
11.
Foster
,
D.
, and
Cipra
,
R.
,
2002
, “
An Automatic Method for Finding the Assembly Configurations of Planar Non-Single-Input Dyadic Mechanisms
,”
ASME J. Mech. Des.
,
124
(
3
), pp.
58
67
.10.1115/1.1425813
12.
Mirth
,
J.
, and
Chase
,
T.
,
1993
, “
Circuit Analysis of Watt Chain Six-Bar Mechanisms
,”
ASME J. Mech. Des.
,
115
(
2
), pp.
214
222
.10.1115/1.2919180
13.
Davis
,
H.
, and
Chase
,
T.
,
1994
, “
Circuit Analysis of Stephenson Chain Six-Bar Mechanisms
,”
Proceedings of the ASME Design Technical Conferences
, DE-Vol. 70, pp.
349
358
.
14.
Wantanabe
,
K.
, and
Katoh
,
H.
,
2004
, “
Identification of Motion Domains of Planar Six-Link Mechanisms of the Stephenson-Type
,”
Mech. Mach. Theory
,
39
, pp.
1081
1099
.10.1016/j.mechmachtheory.2003.12.003
15.
Ting
,
K. L.
, and
Dou
,
X.
,
1996
, “
Classification and Branch Identification of Stephenson Six-Bar Chains
,”
Mech. Mach. Theory
,
31
(
3
), pp.
283
295
.10.1016/0094-114X(95)00075-A
16.
Litvin
,
F.
, and
Tan
,
J.
,
1989
, “
Singularities in Motion and Displacement Functions of Constrained Mechanical Systems
,”
Int. J. Robot. Res.
,
8
(
2
), pp.
30
43
.10.1177/027836498900800203
17.
Gosselin
,
C.
, and
Angeles
,
J.
,
1990
, “
Singularity Analysis of Closed-Loop Kinematic Chains
,”
IEEE Trans. Rob. Autom.
,
6
(
3
), pp.
281
290
.10.1109/70.56660
18.
Murray
,
A.
,
Turner
,
M.
, and
Martin
,
D.
,
2008
, “
Synthesizing Single DOF Linkages via Transition Linkage Identification
,”
ASME J. Mech. Des.
,
130
(
2
), p.
022301
.10.1115/1.2812418
19.
Innocenti
,
C.
, and
Parenti-Castelli
,
V.
,
1998
, “
Singularity-Free Evolution From One Configuration to Another in Serial and Fully Parallel Manipulators
,”
ASME J. Mech. Des.
,
120
(
1
), pp.
73
99
.10.1115/1.2826679
20.
McAree
,
P. R.
, and
Daniel
,
R. W.
,
1999
, “
An Explanation of Never-Special Assembly Changing Motions for 3-3 Parallel Manipulators
,”
Int. J. Robot. Res.
,
18
(
6
), pp.
556
574
.10.1177/02783649922066394
21.
Wenger
,
P.
, and
Chablat
,
D.
,
1998
, “
Workspace and Assembly-Modes in Fully Parallel Manipulators: A Descriptive Study
,”
Advances in Robot Kinematics
,
Kluwer Academic Publishers
,
Norwell, MA
, pp.
117
126
.
22.
Wenger
,
P.
,
Chablat
,
D.
, and
Zein
,
M.
,
2007
, “
Degeneracy Study of the Forward Kinematics of Planar 3-RPR Parallel Manipulators
,”
ASME J. Mech. Des.
,
129
(
12
), pp.
1265
1268
.10.1115/1.2779893
23.
Zein
,
M.
,
Wenger
,
P.
, and
Chablat
,
D.
,
2006
, “
Singular Curves and Cusp Points in the Joint Space of 3-RPR Parallel Manipulators
,”
Proceedings of IEEE International Conference on Robotics and Automation
.
24.
Zein
,
M.
,
Wenger
,
P.
, and
Chablat
,
D.
,
2008
, “
Non-Singular Assembly Mode Changing Motions for 3-RPR Parallel Manipulators
,”
Adv. Rob. Kinematics
,
43
(
4
), pp.
480
490
.
25.
Macho
,
E.
,
Altuzarra
,
O.
,
Pinto
,
C.
, and
Hernandez
,
A.
,
2008
, “
Transitions Between Multiple Solutions of the Direct Kinematic Problem
,”
Advances in Robot Kinematics
,
Springer
,
The Netherlands
, Part 5, pp.
301
310
.
26.
Merlet
J.-P.
,
2007
, “
A Formal-Numerical Approach for Robust In-Workspace Singularity Detection
,”
IEEE Trans. Rob.
,
23
(
3
), pp.
393
402
.10.1109/TRO.2007.898981
27.
Bohigas
O.
,
Manubens
,
M.
, and
Ros
,
L.
,
2013
, “
Singularities of Non-Redundant Manipulators: A Short Account and a Method for Their Computation in the Planar Case
,”
Mech. Mach. Theory
,
68
(
1
), pp.
1
17
.10.1016/j.mechmachtheory.2013.03.001
28.
Lu
,
Y.
,
Bates
,
D. J.
,
Sommese
,
A. J.
, and
Wampler
,
C. W.
,
2007
, “
Finding all Real Points of a Complex Curve
,”
Proceedings of Midwest Algebra, Geometry and Its Interactions Conference, Contemporary Mathematics, AMS
, Vol.
448
, pp.
183
205
.
29.
Besana
,
G. M.
,
Di Rocco
,
S.
,
Hauenstein
,
J. D.
,
Sommese
,
A. J.
, and
Wampler
,
C. W.
,
2013
, “
Cell Decomposition of Almost Smooth Real Algebraic Surfaces
,
Numerical Algorithms
” (in press).
30.
Bates
,
D. J.
,
Hauenstein
,
J. D.
,
Sommese
,
A. J.
, and
Wampler
,
C. W.
, Bertini, “
Software for Numerical Algebraic Geometry
,” Available at http://www.nd.edu/∼sommese/bertini
31.
Zhou
,
H.
, and
Cheung
E. H. M.
,
2004
, “
Adjustable Four-Bar Linkages for Multi-Phase Motion Generation
,”
Mech. Mach. Theory
,
39
(
3
), pp.
261
279
.10.1016/j.mechmachtheory.2003.07.001
32.
Naik
,
D. P.
, and
Amarnath
,
C.
,
1989
, “
Synthesis of Adjustable Four Bar Function Generators through Five Bar Loop Closure Equations
,”
Mech. Mach. Theory
,
24
(
6
), pp.
523
526
.10.1016/0094-114X(89)90009-8
33.
Lin
,
L.
,
Myszka
D.
,
Murray
,
A.
, and
Wampler
,
C.
,
2013
, “
Using the Singularity Trace to Understand Linkage Motion Characteristics
,”
Proceedings of the ASME International Design Engineering Technical Conferences
, Paper No. DETC2013-13244.
34.
Sommese
,
A. J.
, and
Wampler
,
C. W.
,
2005
,
Numerical Solution of Systems of Polynomials Arising In Engineering and Science
,
World Scientific Press
,
Singapore
.
35.
Allgower
,
E. L.
, and
Georg
,
K.
,
1997
, “
Numerical Path Following
,”
Handbook of Numerical Analysis
, Vol.
V
,
North-Holland, Amsterdam
.
36.
Pennock
G.
, and
Kassner
,
D.
,
1992
, “
Kinematic Analysis of a Planar Eight-Bar Linkage: Application to a Platform-Type Robot
,”
ASME J. Mech. Des.
,
114
(
1
), pp.
87
95
.10.1115/1.2916930
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