The theory of screws plays a fundamental role in the field of mechanisms and robotics. Based on the rank-one decomposition of positive semidefinite (PSD) matrices, this paper presents a new algorithm to identify the canonical basis of high-order screw systems. Using the proposed approach, a screw system can be decomposed into the direct sum of two subsystems, which are referred to as the general and special subsystems, respectively. By a particular choice of the general subsystem, the canonical basis of the original system can be obtained by the direct combination of the subsystems' principal elements. In the proposed decomposition, not only the canonical form of the screw system but also the corresponding distribution of all those possible base elements can be determined in a straightforward manner.

References

References
1.
Joshi
,
S. A.
, and
Tsai
,
L. W.
,
2002
, “
Jacobian Analysis of Limited-DOF Parallel Manipulators
,”
ASME J. Mech. Des.
,
124
(
2
), pp.
254
258
.
2.
Li
,
Q. C.
, and
Huang
,
Z.
,
2003
, “
Mobility Analysis of Lower-Mobility Parallel Manipulators Based on Screw Theory
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Taipei, Taiwan, Sept. 14–19, pp.
1179
1184
.
3.
Dai
,
J. S.
,
Huang
,
Z.
, and
Lipkin
,
H.
,
2006
, “
Mobility of Overconstrained Parallel Mechanisms
,”
ASME J. Mech. Des.
,
128
(
1
), pp.
220
229
.
4.
Kong
,
X. W.
, and
Gosselin
,
C. M.
,
2002
, “
Kinematics and Singularity Analysis of a Novel Type of 3-CRR 3-DOF Translational Parallel Manipulator
,”
Int. J. Rob. Res.
,
21
(
9
), pp.
791
798
.
5.
Liu
,
X. J.
,
Wu
,
C.
, and
Wang
,
J. S.
,
2012
, “
A New Approach for Singularity Analysis and Closeness Measurement to Singularities of Parallel Manipulators
,”
ASME J. Mech. Rob.
,
4
(
4
), p.
041001
.
6.
Kong
,
X. W.
, and
Gosselin
,
C. M.
,
2004
, “
Type Synthesis of 3T1R 4-DOF Parallel Manipulators Based on Screw Theory
,”
IEEE Trans. Rob. Autom.
,
20
(
2
), pp.
181
190
.
7.
Carricato
,
M.
, and
Parenti-Castelli
,
V.
,
2002
, “
Singularity-Free Fully-Isotropic Translational Parallel Mechanisms
,”
Int. J. Rob. Res.
,
21
(
2
), pp.
161
174
.
8.
Huang
,
Z.
, and
Li
,
Q. C.
,
2003
, “
Type Synthesis of Symmetrical Lower-Mobility Parallel Mechanisms Using the Constraint-Synthesis Method
,”
Int. J. Rob. Res.
,
22
(
1
), pp.
59
79
.
9.
Su
,
H. J.
,
Dorozhkin
,
D. V.
, and
Vance
,
J. M.
,
2009
, “
A Screw Theory Approach for the Conceptual Design of Flexible Joints for Compliant Mechanisms
,”
ASME J. Mech. Rob.
,
1
(
4
), p.
041009
.
10.
Yu
,
J. J.
,
Li
,
S. Z.
,
Su
,
H. J.
, and
Culpepper
,
M. L.
,
2011
, “
Screw Theory Based Methodology for the Deterministic Type Synthesis of Flexure Mechanisms
,”
ASME J. Mech. Rob.
,
3
(
3
), p.
031008
.
11.
Ball
,
R. S.
,
1900
,
A Treatise on the Theory of Screws
,
Cambridge University Press
,
Cambridge, UK
.
12.
Hunt
,
K. H.
,
1978
,
Kinematic Geometry of Mechanisms
,
Oxford University Press
,
London
.
13.
Gibson
,
C. G.
, and
Hunt
,
K. H.
,
1990
, “
Geometry of Screw Systems—1: Screws: Genesis and Geometry
,”
Mech. Mach. Theory
,
25
(
1
), pp.
1
10
.
14.
Gibson
,
C. G.
, and
Hunt
,
K. H.
,
1990
, “
Geometry of Screw Systems—2: Classification of Screw Systems
,”
Mech. Mach. Theory
,
25
(
1
), pp.
11
27
.
15.
Donelan
,
P. S.
, and
Gibson
,
C. G.
,
1993
, “
On the Hierarchy of Screw Systems
,”
Acta Appl. Math.
,
32
(
3
), pp.
267
296
.
16.
Rico
,
J. M.
, and
Duffy
,
J.
,
1992
, “
Classification of Screw Systems—I: One- and Two-Systems
,”
Mech. Mach. Theory
,
27
(
4
), pp.
459
470
.
17.
Rico
,
J. M.
, and
Duffy
,
J.
,
1992
, “
Classification of Screw Systems—II: Three-Systems
,”
Mech. Mach. Theory
,
27
(
4
), pp.
471
490
.
18.
Tsai
,
M. J.
, and
Lee
,
H. W.
,
1993
, “
On the Special Bases of Two- and Three-Screw Systems
,”
ASME J. Mech. Des.
,
115
(
3
), pp.
540
546
.
19.
Dai
,
J. S.
,
1993
, “
Screw Image Space and Its Application to Robotic Grasping
,”
Ph.D. thesis
, University of Salford, Manchester, UK.
20.
Bandyopadhyay
,
S.
, and
Ghosal
,
A.
,
2009
, “
An Eigenproblem Approach to Classical Screw Theory
,”
Mech. Mach. Theory
,
44
(
6
), pp.
1256
1269
.
21.
Fang
,
Y. F.
, and
Huang
,
Z.
,
1998
, “
Analytical Identification of the Principal Screws of the Third Order Screw System
,”
Mech. Mach. Theory
,
33
(
7
), pp.
987
992
.
22.
Zhao
,
J. S.
,
Zhou
,
H. X.
,
Feng
,
Z. J.
, and
Dai
,
J. S.
,
2009
, “
An Algebraic Methodology to Identify the Principal Screws and Pitches of Screw Systems
,”
J. Mech. Eng. Sci.
,
223
(
8
), pp.
1931
1941
.
23.
Altuzarra
,
O.
,
Salgado
,
O.
,
Pinto
,
C.
, and
Hernández
,
A.
,
2013
, “
Analytical Determination of the Principal Screws for General Screw Systems
,”
Mech. Mach. Theory
,
60
, pp.
28
46
.
24.
Dai
,
J. S.
, and
Jones
,
J. R.
,
2003
, “
A Linear Algebraic Procedure in Obtaining Reciprocal Screw Systems
,”
J. Rob. Syst.
,
20
(
7
), pp.
401
412
.
25.
Kim
,
D.
, and
Chung
,
W. K.
,
2003
, “
Analytic Formulation of Reciprocal Screws and Its Application to Nonredundant Robot Manipulators
,”
ASME J. Mech. Des.
,
125
(
1
), pp.
158
164
.
26.
Dai
,
J.
,
2014
,
Screw Algebra and Lie Groups and Lie Algebra
,
Higher Education Press
,
Beijing, China
.
27.
Murray
,
R. M.
,
Sastry
,
S. S.
, and
Li
,
Z. X.
,
1994
,
A Mathematical Introduction to Robotic Manipulation
,
CRC Press
, Boca Raton, FL.
28.
Selig
,
J. M.
,
2005
,
Geometric Fundamentals of Robotics
,
Springer
,
New York
.
29.
Chen
,
G. L.
,
Wang
,
H.
,
Lin
,
Z. Q.
, and
Lai
,
X. M.
,
2015
, “
The Principal Axes Decomposition of Spatial Stiffness Matrices
,”
IEEE Trans. Rob.
,
31
(
1
), pp.
191
207
.
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