In this paper, an algorithm to help designers to integrate the elastodynamics analysis along with the inverse positioning and orienting problems of a parallel kinematic machine (PKM) into a single package is conceived. The proposed algorithm applies concepts from the matrix structural analysis (MSA) and finite element analysis (FEA) to determine the generalized stiffness matrix and the linearized elastodynamics equations of a PKM with only lower kinematic pairs. A PKM is modeled as a system of flexible links and rigid bodies connected by means of joints. Three cases are analyzed to consider the combinations between flexible and rigid bodies in order to find the local stiffness matrices. The latter are combined to obtain the limb matrices and, then, the global stiffness matrix of the whole robotic system. The same nodes coming from the links discretization are considered to include joint masses/inertias into the model. Finally, a case study is proposed to show some feasible applications and to compare results to commercial software for validation.

References

References
1.
Zienkiewicz
,
O. C.
, and
Taylor
,
R. L.
,
2000
,
Solid Mechanics—Volume 2
,
Butterworth Heinemann
,
London
.
2.
Corradini
,
C.
,
Fauroux
,
J. C.
,
Krut
,
S.
, and
Company
,
O.
,
2004
, “
Evaluation of a 4 Degree of Freedom Parallel Manipulator Stiffness
,”
Proceedings of the 11th World Congress in Mechanism and Machine Science, IFTOMM2004
,
Tianjin, China
,
April 1–4
.
3.
Bouzgarrou
,
B. C.
,
Fauroux
,
J. C.
,
Gogu
,
G.
, and
Heerah
,
Y.
,
2004
, “
Rigidity Analysis of T3R1 Parallel Robot With Uncoupled Kinematics
,”
Proceedings of the 35th International Symposium on Robotics
,
Paris, France
,
March
.
4.
Zhao
,
Y.
,
2011
, “
Kineto-Elastodynamic Characteristics of the Six-Degree-of-Freedom Parallel Structure Seismic Simulator
,”
J. Rob.
,
2011
, pp.
1
17
.10.1155/2011/489695
5.
Martin
,
H. C.
,
1966
,
Introduction to Matrix Methods of Structural Analysis
,
McGraw-Hill Book Company
, New York, 29(9), p. 4.
6.
Wang
,
C. K.
,
1966
,
Matrix Methods of Structural Analysis
,
International Textbook Company
, Scranton, PA.
7.
Przemieniecki
,
J. S.
,
1985
,
Theory of Matrix Structural Analysis
,
Dover Publications, Inc
,
New York
.
8.
Huang
,
T.
,
Zhao
,
X.
, and
Withehouse
,
D. J.
,
2001
, “
Stiffness Estimation of a Tripod-Based Parallel Kinematic Machine
,” Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164),
4
, p. 3280.
9.
Li
,
Y. W.
,
Wang
,
J. S.
, and
Wang
,
L. P.
,
2002
, “
Stiffness Analysis of a Stewart Platform-Based Parallel Kinematic Machine
,”
Proceedings of IEEE ICRA: International Conference on Robotics and Automation
,
Washington, USA
,
May 11–15
.
10.
Clinton
,
C. M.
,
Zhang
,
G.
, and
Wavering
,
A. J.
,
1997
, “
Stiffness Modeling of a Stewart-Platform-Based Milling Machine
,”
Transactions of the North America Manufacturing Research Institution of SME
,
Lincoln, NB, USA
,
May 20–23
, Vol.
XXV
, pp.
335
340
.
11.
Al Bassit
,
L.
,
Angeles
,
J.
,
Al-Wydyan
,
K.
, and
Morozov
,
A.
,
2002
, “
The Elastodynamics of a Schönflies -Motion Generator
,”
Centre for Intelligent Machines, McGill University
,
Montreal, Canada
, Technical Report No. TR-CIM-02-06.
12.
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
,
Orlando, FL
,
May
.
13.
Gonalves
,
R. S.
, and
Carvalho
,
J. C. M.
,
2008
, “
Stiffness Analysis of Parallel Manipulator Using Matrix Structural Analysis
,”
Second European Conference on Mechanism Science, EUCOMES 2008
,
Cassino, Italy
.
14.
Wittbrodt
,
E.
,
Adamiec-Wójcik
,
I.
, and
Wojciech
,
S.
,
2006
,
Dynamics of Flexible Multibody Systems
,
Springer
,
New York
.
15.
Gosselin
,
C. M.
,
1990
, “
Stiffness Mapping for Parallel Manipulator
,”
IEEE Trans. Rob. Autom.
,
6
, pp.
377
382
.10.1109/70.56657
16.
Briot
,
S.
,
Pashkevich
,
A.
, and
Chablat
,
D.
,
2009
, “
On the Optimal Design of Parallel Robots Taking Into Account Their Deformations and Natural Frequencies
,”
Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE
,
San Diego, CA, USA
,
Aug. 30–Sept. 2
.
17.
Yoon
,
W. K.
,
Suehiro
,
T.
,
Tsumaki
,
Y.
, and
Uchiyama
,
M.
,
2004
, “
Stiffness Analysis and Design of a Compact Modified Delta Parallel Mechanism
,”
Robotica
,
22
(
5
), pp.
463
475
.10.1017/S0263574704000050
18.
Gosselin
,
C. M.
, and
Zhang
,
D.
,
2002
, “
Stiffness Analysis of Parallel Mechanisms Using a Lumped Model
,”
Int. J. Rob. Autom.
,
17
, pp.
17
27
.
19.
Pashkevich
,
A.
,
Chablat
,
D.
, and
Wenger
,
P.
,
2009
, “
Stiffness Analysis of Overconstrained Parallel Manipulators
,”
Mech. Mach. Theory
,
44
, pp.
966
982
.10.1016/j.mechmachtheory.2008.05.017
20.
Pashkevich
,
A.
,
Klimchik
,
A.
, and
Chablat
,
D.
,
2011
, “
Enhanced Stiffness Modeling of Manipulators With Passive Joints
,”
Mech. Mach. Theory
,
46
(
5
), pp.
662
679
.10.1016/j.mechmachtheory.2010.12.008
21.
Angeles
,
J.
,
2007
,
Fundamentals of Robotic Mechanical Systems
,
Springer
,
New York
.
22.
Pashkevich
,
A.
,
Chablat
,
D.
, and
Wenger
,
P.
,
2009
, “
Stiffness Analysis of Multi-Chain Parallel Robotic Systems
,”
J. Autom. Mobile Rob. Intell. Syst.
,
3
(
3
), pp.
75
82
.
23.
Shabana
,
A. A.
,
2005
,
Dynamics of Multibody Systems
,
3rd ed.
,
Cambridge University Press
, Cambridge, MA.
24.
Cammarata
,
A.
,
2012
, “
On the Stiffness Analysis and Elastodynamics of Parallel Kinematic Machines
,”
Serial and Parallel Robot Manipulators: Kinematic Dynamics and Control
, Serdar Küçük, ed., In-Tech, Rijeka, Croatia.
25.
Zhou
,
Z.
,
Mechefske
,
C. K.
, and
Xi
,
F.
,
2007
, “
Nonstationary Vibration of a Fully Flexible Parallel Kinematic Machine
,”
ASME J. Vibr. Acoust.
,
129
, pp.
623
630
.10.1115/1.2748477
26.
Gosselin
,
C. M.
,
Tale-Masouleh
,
M.
,
Duchaine
,
V.
,
Richard
,
P.-L.
,
Foucault
,
S.
, and
Kong
,
A.
,
2007
, “
Tripteron, Quadrupteron, and Pentapteron: Kinematic Analysis and Benchmarking of a Family of Parallel Mechanisms
,”
Proceeding of the IEEE International Conference on Robotics and Automation (ICRA07)
,
April 10–14
,
Rome, Italy
.
You do not currently have access to this content.