Abstract

In this paper, we provide a general framework to determine inner and outer approximations to the singularity-free workspace of fully actuated robotic manipulators, subject to Type-I and Type-II singularities. This framework utilizes the sum-of-squares optimization technique, which is numerically implemented by semidefinite programming. In order to apply the sum-of-squares optimization technique, we convert the trigonometric functions in the kinematics of the manipulator to polynomial functions with an additional constraint. We define two quadratic forms, describing two ellipsoids, whose volumes are optimized to yield inner and outer approximations of the singularity-free workspace.

References

References
1.
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
2.
Merlet
,
J.-P.
,
2006
,
Parallel Robots
, Vol.
128
,
Springer Science & Business Media
,
Netherlands
.
3.
Ryu
,
S.-J.
,
Kim
,
J. W.
,
Hwang
,
J. C.
,
Park
,
C.
,
Cho
,
H. S.
,
Lee
,
K.
,
Lee
,
Y.
,
Cornel
,
U.
,
Park
,
F.
, and
Kim
,
J.
,
1999
, “Eclipse: An Overactuated Parallel Mechanism for Rapid Machining,”
Parallel Kinematic Machines
,
C. R.
Boler
,
L.
Molinari-Tosatti
and
K. S.
Smith
, eds.,
Springer
,
London
, pp.
441
455
.
4.
Kim
,
J.
,
Hwang
,
J. C.
,
Kim
,
J. S.
,
Iurascu
,
C. C.
,
Park
,
F. C.
, and
Cho
,
Y. M.
,
2002
, “
Eclipse II: A New Parallel Mechanism Enabling Continuous 360-Degree Spinning Plus Three-Axis Translational Motions
,”
IEEE Trans. Rob. Autom.
,
18
(
3
), pp.
367
373
. 10.1109/TRA.2002.1019472
5.
Kock
,
S.
, and
Schumacher
,
W.
,
2000
, “
A Mixed Elastic and Rigid-Body Dynamic Model of An Actuation Redundant Parallel Robot With High-Reduction Gears
,”
Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings
,
San Francisco, CA
,
Apr. 24–28
, Vol.
2
,
IEEE
, pp.
1918
1923
.
6.
Kock
,
S.
, and
Schumacher
,
W.
,
1998
, “
A Parallel Xy Manipulator with Actuation Redundancy for High-Speed and Active-Stiffness Applications
,”
Proceedings of the 1998 IEEE International Conference on Robotics and Automation
,
Leuven, Belgium
,
May
, Vol.
3
,
IEEE
, pp.
2295
2300
.
7.
Ebrahimi
,
I.
,
Carretero
,
J. A.
, and
Boudreau
,
R.
,
2008
, “
A Family of Kinematically Redundant Planar Parallel Manipulators
,”
ASME J. Mech. Des.
,
130
(
6
), p.
062306
. 10.1115/1.2900723
8.
Dash
,
A. K.
,
Chen
,
I. -M.
,
Yeo
,
S. H.
, and
Yang
,
G.
,
2003
, “
Singularity-Free Path Planning of Parallel Manipulators Using Clustering Algorithm and Line Geometry
,”
2003 IEEE International Conference on Robotics and Automation
,
Taipei, Taiwan
,
Sept. 14–19
, Vol.
1
,
IEEE
, pp.
761
766
.
9.
Sen
,
S.
,
Dasgupta
,
B.
, and
Mallik
,
A. K.
,
2003
, “
Variational Approach for Singularity-Free Path-Planning of Parallel Manipulators
,”
Mech. Mach. Theory
,
38
(
11
), pp.
1165
1183
. 10.1016/S0094-114X(03)00065-X
10.
Merlet
,
J.-P.
,
1994
, “
Trajectory Verification in the Workspace for Parallel Manipulators
,”
Int. J. Rob. Res.
,
13
(
4
), pp.
326
333
. 10.1177/027836499401300404
11.
Bhattacharya
,
S.
,
Hatwal
,
H.
, and
Ghosh
,
A.
,
1998
, “
Comparison of An Exact and An Approximate Method of Singularity Avoidance in Platform Type Parallel Manipulators
,”
Mech. Mach. Theory
,
33
(
7
), pp.
965
974
. 10.1016/S0094-114X(97)00066-9
12.
Dasgupta
,
B.
, and
Mruthyunjaya
,
T.
,
1998
, “
Singularity-Free Path Planning for the Stewart Platform Manipulator
,”
Mech. Mach. Theory
,
33
(
6
), pp.
711
725
. 10.1016/S0094-114X(97)00095-5
13.
Kaloorazi
,
M.-H. F.
,
Masouleh
,
M. T.
, and
Caro
,
S.
,
2015
, “
Determination of the Maximal Singularity-Free Workspace of 3-dof Parallel Mechanisms With a Constructive Geometric Approach
,”
Mech. Mach. Theory
,
84
, pp.
25
36
. 10.1016/j.mechmachtheory.2014.10.003
14.
Kaloorazi
,
M. H. F.
,
Masouleh
,
M. T.
, and
Caro
,
S.
,
2016
, “
Determining the Maximal Singularity-Free Circle Or Sphere of Parallel Mechanisms Using Interval Analysis
,”
Robotica
,
34
(
1
), pp.
135
149
. 10.1017/S0263574714001271
15.
Jiang
,
Q.
, and
Gosselin
,
C. M.
,
2006
, “
The Maximal Singularity-Free Workspace of Planar 3-RPR Parallel Mechanisms
,”
2006 International Conference on Mechatronics and Automation
,
Luoyang, Henan, China
,
June 25–28
,
IEEE
, pp.
142
146
.
16.
Li
,
H.
,
Gosselin
,
C. M.
, and
Richard
,
M. J.
,
2006
, “
Determination of Maximal Singularity-Free Zones in the Workspace of Planar Three-Degree-of-Freedom Parallel Mechanisms
,”
Mech. Mach. Theory
,
41
(
10
), pp.
1157
1167
. 10.1016/j.mechmachtheory.2005.12.003
17.
Li
,
H.
,
Gosselin
,
C. M.
, and
Richard
,
M. J.
,
2007
, “
Determination of the Maximal Singularity-Free Zones in the Six-Dimensional Workspace of the General Gough–Stewart Platform
,”
Mech. Mach. Theory
,
42
(
4
), pp.
497
511
. 10.1016/j.mechmachtheory.2006.04.006
18.
Abbasnejad
,
G.
,
Daniali
,
H. M.
, and
Kazemi
,
S.
,
2012
, “
A New Approach to Determine the Maximal Singularity-Free Zone of 3-RPR Planar Parallel Manipulator
,”
Robotica
,
30
(
6
), pp.
1005
1012
. 10.1017/S0263574711001238
19.
Saadatzi
,
M.
,
Masouleh
,
M. T.
,
Taghirad
,
H.
,
Gosselin
,
C.
, and
Teshnehlab
,
M.
,
2011
, “
Multi-Objective Scale Independent Optimization of 3-RPR Parallel Mechanisms
,”
Proc. IFToMM
,
Guanajuato, Mexico
. 2019/07/2011
20.
Saadatzi
,
M. H.
,
Masouleh
,
M. T.
,
Taghirad
,
H. D.
,
Gosselin
,
C.
, and
Cardou
,
P.
,
2011
, “
On the Optimum Design of 3-RPR Parallel Mechanisms
,”
2011 19th Iranian Conference on Electrical Engineering (ICEE)
,
Tehran, Iran
,
May 17–19
,
IEEE
, pp.
1
6
.
21.
Arsenault
,
M.
, and
Boudreau
,
R.
,
2004
, “
The Synthesis of Three-Degree-of-Freedom Planar Parallel Mechanisms with Revolute Joints (3-r Rr) for An Optimal Singularity-Free Workspace
,”
J. Rob. Syst.
,
21
(
5
), pp.
259
274
. 10.1002/rob.20013
22.
Gallant
,
M.
, and
Boudreau
,
R.
,
2002
, “
The Synthesis of Planar Parallel Manipulators with Prismatic Joints for An Optimal, Singularity-Free Workspace
,”
J. Rob. Syst.
,
19
(
1
), pp.
13
24
. 10.1002/rob.8118
23.
Caro
,
S.
,
Chablat
,
D.
,
Goldsztejn
,
A.
,
Ishii
,
D.
, and
Jermann
,
C.
,
2014
, “
A Branch and Prune Algorithm for the Computation of Generalized Aspects of Parallel Robots
,”
Artif. Intell.
,
211
, pp.
34
50
. 10.1016/j.artint.2014.02.001
24.
Stamper
,
R. E.
,
Tsai
,
L. -W.
, and
Walsh
,
G. C.
,
1997
, “
Optimization of a Three Dof Translational Platform for Well-Conditioned Workspace
,”
Proceedings of International Conference on Robotics and Automation
,
Albuquerque, NM
,
Apr. 25–27
, Vol.
4
,
IEEE
, pp.
3250
3255
.
25.
Mousavi
,
M. A.
,
Masouleh
,
M. T.
, and
Karimi
,
A.
,
2014
, “
On the Maximal Singularity-Free Ellipse of Planar 3-RPR Parallel Mechanisms Via Convex Optimization
,”
Rob. Comput.-Integrated Manuf.
,
30
(
2
), pp.
218
227
. 10.1016/j.rcim.2013.09.012
26.
Karimi
,
A.
,
Masouleh
,
M. T.
, and
Cardou
,
P.
,
2014
, “
Singularity-Free Workspace Analysis of General 6-ups Parallel Mechanisms Via Convex Optimization
,”
Mech. Mach. Theory
,
80
, pp.
17
34
. 10.1016/j.mechmachtheory.2014.04.005
27.
Blekherman
,
G.
,
Parrilo
,
P.
, and
Thomas
,
R.
,
2013
,
Semidefinite Optimization and Convex Algebraic Geometry
(
MOS-SIAM Series on Optimization
),
Society for Industrial and Applied Mathematics
.
28.
Boyd
,
S.
,
El Ghaoui
,
L.
,
Feron
,
E.
, and
Balakrishnan
,
V.
,
1994
,
Linear Matrix Inequalities in System and Control Theory
, Vol.
15
,
SIAM
,
Philadelphia, PA
.
29.
Zlatanov
,
D.
,
Bonev
,
I. A.
, and
Gosselin
,
C. M.
,
2002
, “
Constraint Singularities of Parallel Mechanisms
,”
Proceedings 2002 IEEE International Conference on Robotics and Automation
,
Washington, DC
,
May 11–15
, Vol.
1
,
IEEE
, pp.
496
502
.
30.
Murray
,
R.
,
2017
,
A Mathematical Introduction to Robotic Manipulation
,
CRC Press
,
Boca Raton, FL
.
31.
Apostol
,
T.
,
1974
,
Mathematical Analysis
(
Addison-Wesley Series in Mathematics
),
Addison-Wesley
.
32.
Folland
,
G.
,
2013
,
Real Analysis: Modern Techniques and Their Applications
(
Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
),
Wiley
.
33.
Boyd
,
S.
, and
Vandenberghe
,
L.
,
2004
,
Convex Optimization
,
Cambridge University Press
,
Cambridge, UK
.
34.
Weisstein
,
E. W.
Ellipse
.
From MathWorld—A Wolfram Web Resource
.
35.
Parrilo
,
P. A.
,
2000
, “
Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization
,”
Ph.D. thesis
,
California Institute of Technology
,
Pasadena, CA
.
36.
Bohigas
,
O.
,
Manubens
,
M.
, and
Ros
,
L.
,
2016
,
Singularities of Robot Mechanisms: Numerical Computation and Avoidance Path Planning
, Vol.
41
,
Springer
,
Switzerland
.
37.
Bezanson
,
J.
,
Edelman
,
A.
,
Karpinski
,
S.
, and
Shah
,
V. B.
,
2017
, “
Julia: A Fresh Approach to Numerical Computing
,”
SIAM Rev.
,
59
(
1
), pp.
65
98
. 10.1137/141000671
38.
MOSEK
,
A.
,
2019
,
MOSEK Optimizer API for Julia 9.0.88
.
39.
Dunning
,
I.
,
Huchette
,
J.
, and
Lubin
,
M.
,
2017
, “
Jump: A Modeling Language for Mathematical Optimization
,”
SIAM Rev.
,
59
(
2
), pp.
295
320
. 10.1137/15M1020575
40.
Porta
,
J. M.
,
Ros
,
L.
,
Bohigas
,
O.
,
Manubens
,
M.
,
Rosales
,
C.
, and
Jaillet
,
L.
,
2014
, “
The Cuik Suite: Analyzing the Motion Closed-Chain Multibody Systems
,”
IEEE Rob. Autom. Mag.
,
21
(
3
), pp.
105
114
. 10.1109/MRA.2013.2287462
41.
W. R.
Mathematica
,
Version 12.0. Champaign, IL, 2019
.
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