A general method for the analytical elastostatic stiffness modeling of overconstrained parallel manipulators (PMs) using geometric algebra and strain energy is proposed. First, an analytical solution of the constraint and actuation wrenches exerted on the moving platform is obtained using the outer product and dual operation of geometric algebra, which avoids solving complex symbolic linear equations. Second, considering the compliances of the limbs, an analytical elastostatic model is established using the strain energy to obtain the stiffness matrices of the limbs. Finally, the deformation compatibility equations are added into equilibrium equations to obtain the overall stiffness matrix of the PM, which has a concise expression and a clear physical meaning. The proposed method is applied to the Tex3 overconstrained PM and the Tex4 overconstrained PM with redundant actuation to prove its validity. Comparable results between the theoretical analysis and the finite-element analysis (FEA) show that the former could be used as an effective alternative to the FEA method in the predesign stage. This new approach is universally applicable to the elastostatic stiffness analysis of overconstrained PMs.

References

References
1.
Jin
,
Y.
,
Kong
,
X.
, and
Higgins
,
C.
,
2012
, “
Kinematic Design of a New Parallel Kinematic Machine for Aircraft Wing Assembly
,”
Proceedings of the 10th IEEE International Conference on Industrial Informatics
,
Beijing, China
,
July 25–27
, pp.
669
674
.
2.
Kiper
,
C.
, and
Bağdadioğlu
,
B.
,
2015
,
“Function Generation Synthesis with a 2-DoF Overconstrained Double-Spherical 7R Mechanism Using the Method of Decomposition and Least Squares Approximation
,”
P.
Flores
, and
F. Viadero
eds.,
New Trends in Mechanism and Machine Science: From Fundamentals to Industrial Applications
,
Springer International Publishing
,
Switzerland
, pp.
175
183
.
3.
Siciliano
,
B.
, and
Khatib
,
O.
,
2008
,
Handbook of Robotics
,
Springer
,
Berlin, Heidelberg
.
4.
Klimchik
,
A.
,
Chablat
,
D.
, and
Pashkevich
,
A.
,
2014
, “
Stiffness Modeling for Perfect and Non-Perfect Parallel Manipulators Under Internal and External Loadings
,”
Mech. Mach. Theory
,
79
, pp.
1
28
.
5.
Khasawneh
,
B. E.
, and
Ferreira
,
P. M.
,
1999
, “
Computation of Stiffness and Stiffness Bounds for Parallel Link Manipulators
,”
Int. J. Mach, Tool. Manuf.
,
39
, pp.
321
342
.
6.
Akin
,
J. E.
,
2005
,
Finite Element Analysis With Error Estimators: An Introduction to the FEM and Adaptive Error Analysis for Engineering Students
,
Butterworth-Heinemann
,
Burlington
.
7.
Klimchik
,
A.
,
Pashkevich
,
A.
, and
Chablat
,
D.
,
2013
, “
CAD-Based Approach for Identification of Elasto-Static Parameters of Robotic Manipulators
,”
Finite. Elem. Anal. Des.
,
75
, pp.
19
30
.
8.
Huang
,
T.
,
Zhao
,
X. Y.
, and
Whitehouse
,
D. J.
,
2002
, “
Stiffness Estimation of a Tripod-Based Parallel Kinematic Machine
,”
IEEE Trans. Robot. Autom.
,
18
(
1
), pp.
50
58
.
9.
Martin
,
H. C.
,
1966
,
Introduction to Matrix Methods of Structural Analysis
,
McGraw-Hill
,
New York
.
10.
Deblaise
,
D.
,
Hernot
,
X.
, and
Maurine
,
P.
,
2006
, “
A Systematic Analytical Method for PKM Stiffness Matrix Calculation
,”
Proceedings of the 2006 IEEE International Conference on Robotics and Automation
,
FL
,
May 15–19
, pp.
4213
4219
.
11.
Pashkevich
,
A.
,
Chablat
,
D.
, and
Wenger
,
P.
,
2009
, “
Stiffness Analysis of Overconstrained Parallel Manipulators
,”
Mech. Mach. Theory
,
44
, pp.
966
982
.
12.
Soares Júnior
,
G. D. L.
,
Carvalho
,
J. C. M.
, and
Goncalves
,
R. S.
,
2016
, “
Stiffness Analysis of Multibody Systems Using Matrix Structural Analysis—MSA
,”
Robotica
,
34
, pp.
2368
2385
.
13.
Yu
,
G.
,
Wang
,
L. P.
,
Wu
,
J.
,
Wang
,
D.
, and
Hu
,
C. J.
,
2018
, “
Stiffness Modeling Approach for a 3-DOF Parallel Manipulator With Consideration of Nonlinear Joint Stiffness
,”
Mech. Mach. Theory
,
123
, pp.
137
152
.
14.
Gosselin
,
C.
,
1990
, “
Stiffness Mapping for Parallel Manipulators
,”
IEEE Trans. Robot. Autom.
,
6
(
3
), pp.
377
382
.
15.
Gosselin
,
C.
, and
Zhang
,
D.
,
2002
, “
Stiffness Analysis of Parallel Mechanisms Using a Lumped Model
,”
Int. J. Robot. Autom.
,
17
(
1
), pp.
17
27
.
16.
Majou
,
F.
,
Cosselin
,
C.
,
Wenger
,
P.
, and
Chablat
,
D.
,
2007
, “
Parametric Stiffness Analysis of the Orthoglide
,”
Mech. Mach. Theory
,
42
, pp.
296
311
.
17.
Hoevenaars
,
A. G.
,
Lambert
,
P.
, and
Herder
,
J. L.
,
2016
, “
Jacobian-Based Stiffness Analysis Method for Parallel Manipulators With Non-Redundant Legs
,”
Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
,
230
(
3
), pp.
341
352
.
18.
Sun
,
T.
,
Lian
,
B. B.
, and
Song
,
Y. M.
,
2016
, “
Stiffness Analysis of a 2-DoF Over-Constrained RPM With an Articulated Traveling Platform
,”
Mech. Mach. Theory
,
96
, pp.
165
178
.
19.
Xu
,
P.
,
Li
,
B.
,
Cheung
,
C.-F.
, and
Zhang
,
J.-F.
,
2017
, “
Stiffness Modeling and Optimization of a 3-DOF Parallel Robot in a Serial-Parallel Polishing Machine
,”
Int. J. Precis. Enf. Man.
,
18
(
4
), pp.
497
507
.
20.
Lian
,
B. B.
,
Sun
,
T.
,
Song
,
Y. M.
,
Jin
,
Y.
, and
Price
,
M.
,
2015
, “
Stiffness Analysis and Experiment of a Novel 5-DoF Parallel Kinematic Machine Considering Gravitational Effects
,”
Int. J. Mach. Tool. Manuf.
,
92
, pp.
82
96
.
21.
Yan
,
S. J.
,
Ong
,
S. K.
, and
Nee
,
A. Y. C.
,
2016
, “
Stiffness Analysis of Parallelogram-Type Parallel Manipulators Using a Strain Energy Method
,”
Robot. Com-Int. Manuf.
,
37
, pp.
13
22
.
22.
Wu
,
J.
,
Wang
,
J. S.
,
Wang
,
L. P.
,
Li
,
T. M.
, and
You
,
Z.
,
2009
, “
Study on the Stiffness of a 5-DOF Hybrid Machine Tool With Actuation Redundancy
,”
Mech. Mach. Theory
,
44
, pp.
289
305
.
23.
Wu
,
J.
,
Li
,
T. M.
,
Wang
,
J. S.
, and
Wang
,
L. P.
,
2013
, “
Stiffness and Natural Frequency of a 3-DOF Parallel Manipulator With Consideration of Additional leg Candidates
,”
Robot. Auton. Syst.
,
61
(
8
), pp.
868
875
.
24.
Xu
,
Y. D.
,
Liu
,
W. L.
,
Yao
,
J. T.
, and
Zhao
,
Y. S.
,
2015
, “
A Method for Force Analysis of the Overconstrained Lower Mobility Parallel Mechanism
,”
Mech. Mach. Theory
,
88
, pp.
31
48
.
25.
Rezaei
,
A.
,
Akbarzadeh
,
A.
, and
Akbarzadeh
,
M. R.
,
2012
, “
An Investigation on Stiffness of a 3-PSP Spatial Parallel Mechanism With Flexible Moving Platform Using Invariant Form
,”
Mech. Mach. Theory
,
51
, pp.
195
216
.
26.
Rezaei
,
A.
, and
Akbarzadeh
,
A.
,
2013
, “
Position and Stiffness Analysis of a New Asymmetric 2PRR-PRR Parallel CNC Machine
,”
Adv. Robot.
,
27
(
2
), pp.
133
145
.
27.
Li
,
X.
,
Zlatonov
,
D.
,
Zoppi
,
M.
, and
Molfino
,
R.
,
2012
, “
Stiffness Estimation and Experiments for the Exechon Parallel Self-Reconfiguring Fixture Mechanism
,”
Proceedings of the ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
,
Chicago, IL
,
Aug. 28–31
, pp.
637
645
.
28.
Bi
,
Z. M.
,
2014
, “
Kinetostatic Modeling of Exechon Parallel Kinematic Machine for Stiffness Analysis
,”
Int. J. Adv. Manuf. Technol.
,
71
(
1–4
), pp.
325
335
.
29.
Zhang
,
J.
,
Zhao
,
Y. Q.
, and
Jin
,
Y.
,
2016
, “
Kinetostatic-Model-Based Stiffness Analysis of Exechon PKM
,”
Robot. Comput. Integr. Manuf.
,
30
, pp.
208
220
.
30.
Cao
,
W. A.
,
Ding
,
H. F.
, and
Yang
,
D. H.
,
2017
, “
A Method for Compliance Modeling of Five Degree-of-Freedom Overconstrained Parallel Robotic Mechanisms With 3T2R Output Motion
,”
ASME J. Mech. Robot.
,
9
, p. 011011.
31.
Briot
,
S.
, and
Khalil
,
W.
,
2015
,
Dynamics of Parallel Robots——From Rigid Bodies to Flexible Elements
,
Springer International Publishing
,
Switzerland
.
32.
Briot
,
S.
, and
Khalil
,
W.
,
2014
, “
Recursive and Symbolic Calculation of the Elastodynamic Model of Flexible Parallel Robots
,”
Int. J. Rob. Res.
,
33
(
3
), pp.
469
483
.
33.
Briot
,
S.
, and
Khalil
,
W.
,
2014
, “
Recursive and Symbolic Calculation of the Stiffness and Mass Matrices of Parallel Robots
,”
Proceedings of the 20th CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators
,
Moscow, Russia
, pp.
123
132
.
34.
Shabana
,
A. A.
,
2013
,
Dynamics of Multibody Systems
,
Cambridge University Press
,
New York
.
35.
Bauchau
,
O. A.
,
2010
,
Flexible Multibody Dynamics
,
Springer Science & Business Media
,
The Netherlands
.
36.
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
82
.
37.
Xu
,
L. M.
,
Li
,
Q. C.
,
Zhang
,
N. B.
, and
Chen
,
Q. H.
,
2017
, “
Mobility, Kinematic Analysis, and Dimensional Optimization of New Three-Degrees-of-Freedom Parallel Manipulator With Actuation Redundancy
,”
ASEM J. Mech. Robot.
,
9
(
4
), p.
041008
.
38.
Xu
,
L. M.
,
Li
,
Q. C.
,
Tong
,
J. H.
, and
Chen
,
Q. H.
,
2018
, “
Tex3: An 2R1T Parallel Manipulator With Minimum DOF of Joints and Fixed Linear Actuators
,”
Int. J. Precis. Eng. Man.
,
19
(
2
), pp.
227
238
.
39.
Li
,
Q. C.
,
Xu
,
L. M.
,
Chen
,
Q. H.
, and
Ye
,
W.
,
2017
, “
New Family of RPR-Equivalent Parallel Mechanisms: Design and Application
,”
Chin. J. Mech. Eng.
,
30
(
2
), pp.
217
221
.
40.
Li
,
Q. C.
,
Chai
,
X. X.
, and
Xiang
,
J. N.
,
2016
, “
Mobility Analysis of Limited-Degrees-of-Freedom Parallel Mechanisms in the Framework of Geometric Algebra
,”
ASME J. Mech. Robot.
,
8
(
4
), p.
041005
.
41.
Chai
,
X. X.
, and
Li
,
Q. C.
,
2017
, “
Analytical Mobility Analysis of Bennett Linkage Using Geometric Algebra
,”
Adv. Appl. Clifford. Al.
,
27
(
3
), pp.
2083
2095
.
42.
Yao
,
H. J.
,
Chen
,
Q. H.
,
Chai
,
X. X.
, and
Li
,
Q. C.
,
2017
, “
Singularity Analysis of 3-RPR Parallel Manipulators Using Geometric Algebra
,”
Adv. Appl. Clifford. Al.
,
27
(
3
), pp.
2097
2113
.
43.
Aristidou
,
A.
, and
Lasenby
,
J.
,
2011
, “
FABRIK: A Fast, Iterative Solver for the Inverse Kinematics Problem
,”
Graph. Models
,
73
(
5
), pp.
243
260
.
44.
Bayro-Corrochano
,
E.
,
2010
, “Geometric Algebra for Modeling in Robot Physics,”
Geometric Computing: For Wavelet Transforms, Robot Vision, Learning, Control and Action
,
Springer
,
London
, pp.
45
59
.
45.
Campos-Macías
,
L.
,
Carbajal-Espinosa
,
O.
,
Loukianov
,
A.
, and
Bayro-Corrochano
,
E.
,
2016
, “
Inverse Kinematics for a 6-DOF Walking Humanoid Robot Leg
,”
Adv. Appl. Clifford. Al.
,
27
(
1
), pp.
581
597
.
46.
Hrdina
,
J.
,
Návrat
,
A.
, and
Vašík
,
P.
,
2015
, “
Control of 3-Link Robotic Snake Based on Conformal Geometric Algebra
,”
Adv. Appl. Clifford. Al.
,
26
(
3
), pp.
1069
1080
.
47.
Hestenes
,
D.
,
1999
,
New Foundations for Classical Mechanics
,
Kluwer Academic Publishers
,
Dordrecht
.
48.
Beer
,
F. P.
,
Johnston
,
J. E. R.
, and
DeWolf
,
J. T.
,
2009
,
Mechanics of Materials
,
McGraw-Hill Press
,
New York
.
49.
Joshi
,
S. A.
, and
Tsai
,
L. W.
,
2002
, “
Jacobian Analysis of Limited-DOF Parallel Manipulators
,”
ASME J. Mech. Des.
,
124
(
2
), pp.
254
258
.
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