Mobility is a basic property of a mechanism. The aim of mobility analysis is to determine the number of degrees-of-freedom (DOF) and the motion pattern of a mechanism. The existing methods for mobility analysis have some drawbacks when being applied to limited-DOF parallel mechanisms (PMs). Particularly, it is difficult to obtain a symbolic or closed-form expression of mobility and its geometric interpretations are not always straightforward. This paper presents a general method for mobility analysis of limited-DOF PMs in the framework of geometric algebra. The motion space and constraint space of each limb are expressed using geometric algebra. Then the mobility of the PM can be calculated based on the orthogonal complement relationship between the motion space and the constraint space. The detailed mobility analyses of a 3-RPS PM and a 3-RPC PM are presented. It is shown that this method can obtain a symbolic expression of mobility with straightforward geometric interpretations and is applicable to limited-DOF PMs with or without redundant constraints. Without solving complicated symbolic linear equations, this method also has computational advantages.

References

1.
Merlet
,
J. P.
,
2006
,
Parallel Robots
,
Kluwer Academic Publishers
,
Dordrecht, The Netherlands
.
2.
Gogu
,
G.
,
2005
, “
Mobility of Mechanisms: A Critical Review
,”
Mech. Mach. Theory
,
40
(
9
), pp.
1068
1097
.
3.
Clavel
,
R.
,
1988
, “
Delta: A Fast Robot With Parallel Geometry
,”
18th International Symposium on Industrial Robots
, pp.
91
100
.
4.
Wahl
,
J.
,
2002
, “
Articulated Tool Head
,”
U.S. Patent No. 6,431,802
.
5.
Neumann
,
K. E.
,
1988
, “
Robot
,”
U.S. Patent No. 4,732,525
.
6.
Gogu
,
G.
,
2008
, “
Constraint Singularities and the Structural Parameters of Parallel Robots
,”
Advances in Robot Kinematics: Analysis and Design
,
Springer
,
Dordrecht, The Netherlands
, pp.
21
28
.
7.
Huang
,
Z.
,
Kong
,
L.
, and
Fang
,
Y.
,
1997
,
Mechanism Theory of Parallel Robotic Manipulator and Control
,
China Mechanical
,
Beijing, China
.
8.
Huang
,
Z.
, and
Li
,
Q.
,
2003
, “
Type Synthesis of Symmetrical Lower-Mobility Parallel Mechanisms Using the Constraint-Synthesis Method
,”
Int. J. Rob. Res.
,
22
(
1
), pp.
59
82
.
9.
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
.
10.
Dai
,
J. S.
,
Huang
,
Z.
, and
Lipkin
,
H.
,
2006
, “
Mobility of Overconstrained Parallel Mechanisms
,”
ASME J. Mech. Des.
,
128
(
1
), pp.
220
229
.
11.
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
.
12.
Kong
,
X.
, and
Gosselin
,
C.
,
2007
,
Type Synthesis of Parallel Mechanisms
,
Springer
,
Heidelberg, Germany
.
13.
Clifford
,
W. K.
,
1882
, “
On the Classification of Geometric Algebras
,”
Mathematical Papers
,
T.
Robert
, ed.,
American Mathematical Society
,
Providence, RI
, pp.
397
401
.
14.
Clifford
,
W. K.
,
1878
,
Elements of Dynamic: An Introduction to the Study of Motion and Rest in Solid and Fluid Bodies
,
MacMillan and Co.
,
London, UK
.
15.
Clifford
,
W. K.
,
1976
, “
On the Space-Theory of Matter
,”
The Concepts of Space and Time
,
Springer
,
Dordrecht, The Netherlands
, pp.
295
296
.
16.
Hestenes
,
D.
,
1999
,
New Foundations for Classical Mechanics
,
Kluwer Academic Publishers
,
Dordrecht, The Netherlands
.
17.
Hitzer
,
E.
,
Nitta
,
T.
, and
Kuroe
,
Y.
,
2013
, “
Applications of Clifford's Geometric Algebra
,”
Adv. Appl. Clifford Algebras
,
23
(
2
), pp.
377
404
.
18.
Zamora
,
J.
, and
Bayro-Corrochano
,
E.
,
2004
, “
Inverse Kinematics, Fixation and Grasping Using Conformal Geometric Algebra
,” 2004
IEEE/RSJ
International Conference on Intelligent Robots and Systems
, Sendai, Japan, Sept. 28–Oct. 2, pp.
3841
3846
.
19.
Aristidou
,
A.
, and
Lasenby
,
J.
,
2011
, “
Inverse Kinematics Solutions Using Conformal Geometric Algebra
,”
Guide to Geometric Algebra in Practice
,
L.
Dorst
, and
J.
Lasenby
, eds.,
Springer-Verlag
,
London, UK
, pp.
47
62
.
20.
Aristidou
,
A.
, and
Lasenby
,
J.
,
2011
, “
Fabrik: A Fast, Iterative Solver for the Inverse Kinematics Problem
,”
Graphical Models
,
73
(
5
), pp.
243
260
.
21.
Lasenby
,
J.
,
Fitzgerald
,
W. J.
,
Lasenby
,
A. N.
, and
Doran
,
C.
,
1998
, “
New Geometric Methods for Computer Vision: An Application to Structure and Motion Estimation
,”
Int. J. Comput. Vision
,
26
(
3
), pp.
191
213
.
22.
Tanev
,
T. K.
,
2006
, “
Singularity Analysis of a 4-DOF Parallel Manipulator Using Geometric Algebra
,”
Advances in Robot Kinematics
,
Springer
,
Dordrecht, The Netherlands
, pp.
275
284
.
23.
Tanev
,
T. K.
,
2008
, “
Geometric Algebra Approach to Singularity of Parallel Manipulators With Limited Mobility
,”
Advances in Robot Kinematics: Analysis and Design
,
Springer
,
Dordrecht, The Netherlands
, pp.
39
48
.
24.
Selig
,
J.
,
1996
,
Geometrical Methods in Robotics
,
Springer Science + Business Media
,
New York
.
25.
McCarthy
,
J. M.
, and
Soh
,
G. S.
,
2010
,
Geometric Design of Linkages
,
Springer Science + Business Media
,
New York
.
26.
Dorst
,
L.
,
Fontijne
,
D.
, and
Mann
,
S.
,
2009
,
Geometric Algebra for Computer Science
(Revised Edition): An Object-Oriented Approach to Geometry,
Morgan Kaufmann
,
Burlington, VT
.
27.
Lipkin
,
H.
, and
Duffy
,
J.
,
1985
, “
The Elliptic Polarity of Screws
,”
ASME J. Mech. Des.
,
107
(
3
), pp.
377
386
.
28.
Bonev
,
I. A.
,
2003
, “
Geometric Analysis of Parallel Mechanisms
,” Dissertation, Laval University, Quebec, QC, Canada.
29.
Hervé
,
J.
, and
Sparacino
,
F.
,
1991
, “
Structural Synthesis of Parallel Robots Generating Spatial Translation
,” 5th
IEEE
International Conference on Advanced Robotics
, Pisa, Italy, June 19–22, pp.
808
813
.
You do not currently have access to this content.