A general accuracy analysis method of redundantly actuated and overconstrained parallel mechanisms is proposed. Coupled effects of actuation errors and internal elastic forces arose from the elastic deformation are both considered. The accuracy analysis approach is based on the Lie-group theory and screw theory, and it includes three steps. First, stiffness matrices of serial legs are obtained. Second, the movement of each leg is modeled based on group theory and the elastic forces arose from the deformation are represented using the stiffness matrices, following which the multiclosed-loop structure constraint and self-balanced force constraint are modeled. Finally, the error pose is estimated. The proposed method is illustrated by the accuracy study of a redundantly actuated and overconstrained Stewart platform. The error modeling is easy as the use of stiffness matrix can model the passive joint motions and deformation together. Moreover, the proposed method is computationally cheap as all computations are linear.

References

References
1.
Sugimoto
,
K.
,
1990
, “
Existence Criteria for Over-Constrained Mechanisms: An Extension of Motor Algebra
,”
ASME J. Mech. Des.
,
112
(
3
), pp.
295
298
.
2.
Li
,
Y.
,
Sun
,
Y.
,
Peng
,
B.
, and
Hu
,
F.
,
2016
, “
Dynamic Modeling of a High-Speed Over-Constrained Press Machine
,”
J. Mech. Sci. Technol.
,
30
(
7
), pp.
3051
3059
.
3.
Maaroof
,
O. W.
, and
Dede
,
M. İ. C.
,
2014
, “
Kinematic Synthesis of Over-Constrained Double-Spherical Six-Bar Mechanism
,”
Mech. Mach. Theory
,
73
, pp.
154
168
.
4.
Fu
,
J.
,
Gao
,
F.
,
Chen
,
W.
,
Pan
,
Y.
, and
Lin
,
R.
,
2016
, “
Kinematic Accuracy Research of a Novel Six-Degree-of-Freedom Parallel Robot With Three Legs
,”
Mech. Mach. Theory
,
102
, pp.
86
102
.
5.
Ropponen
,
T.
, and
Arai
,
T.
,
1995
, “
Accuracy Analysis of a Modified Stewart Platform Manipulator
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Nagoya, Japan, May 21–27, pp.
521
525
.
6.
Hao
,
F.
, and
Merlet
,
J.-P.
,
2005
, “
Multi-Criteria Optimal Design of Parallel Manipulators Based on Interval Analysis
,”
Mech. Mach. Theory
,
40
(
2
), pp.
157
171
.
7.
Wang
,
S.-M.
, and
Ehmann
,
K. F.
,
2002
, “
Error Model and Accuracy Analysis of a Six-Dof Stewart Platform
,”
ASME J. Manuf. Sci. Eng.
,
124
(
2
), pp.
286
295
.
8.
Ma
,
X.
,
Zhang
,
L.
,
Zhu
,
L.
,
Yang
,
W.
, and
Hu
,
P.
,
2016
, “
The Slider Motion Error Analysis by Positive Solution Method in Parallel Mechanism
,”
Proc. SPIE
,
99603
, p.
99030
.
9.
Patel
,
A. J.
, and
Ehmann
,
K.
,
1997
, “
Volumetric Error Analysis of a Stewart Platform-Based Machine Tool
,”
CIRP Ann.-Manuf. Technol.
,
46
(
1
), pp.
287
290
.
10.
Li
,
H.
,
Zhang
,
X.
,
Zeng
,
L.
, and
Wu
,
H.
,
2017
, “
Vision-Aided Online Kinematic Calibration of a Planar 3 R RR Manipulator
,”
Mechanism and Machine Science
(Lecture Notes in Electrical Engineering, Vol.
408
),
Springer
,
Singapore
, pp.
963
972
.
11.
Liu
,
H.
,
Huang
,
T.
, and
Chetwynd
,
D. G.
,
2011
, “
A General Approach for Geometric Error Modeling of Lower Mobility Parallel Manipulators
,”
ASME J. Mech. Rob.
,
3
(
2
), p.
021013
.
12.
Briot
,
S.
, and
Bonev
,
I. A.
,
2008
, “
Accuracy Analysis of 3-DOF Planar Parallel Robots
,”
Mech. Mach. Theory
,
43
(
4
), pp.
445
458
.
13.
Briot
,
S.
, and
Bonev
,
I. A.
,
2010
, “
Accuracy Analysis of 3T1R Fully-Parallel Robots
,”
Mech. Mach. Theory
,
45
(
5
), pp.
695
706
.
14.
Wang
,
Y.
, and
Chirikjian
,
G. S.
,
2006
, “
Error Propagation on the Euclidean Group With Applications to Manipulator Kinematics
,”
IEEE Trans. Rob.
,
22
(
4
), pp.
591
602
.
15.
Li
,
X.
,
Ding
,
X.
, and
Chirikjian
,
G. S.
,
2015
, “
Analysis of Angular-Error Uncertainty in Planar Multiple-Loop Structures With Joint Clearances
,”
Mech. Mach. Theory
,
91
, pp.
69
85
.
16.
Chen
,
G.
,
Wang
,
H.
, and
Lin
,
Z.
,
2013
, “
A Unified Approach to the Accuracy Analysis of Planar Parallel Manipulators Both With Input Uncertainties and Joint Clearance
,”
Mech. Mach. Theory
,
64
, pp.
1
17
.
17.
Pucheta
,
M. A.
, and
Gallardo
,
A. G.
,
2017
, “
Synthesis of Precision Flexible Mechanisms Using Screw Theory With a Finite Elements Validation
,”
International Symposium on Multibody Systems and Mechatronics
, pp.
3
14
.
18.
Jiang
,
Y.
,
Li
,
T.
,
Wang
,
L.
, and
Chen
,
F.
,
2018
, “
Improving Tracking Accuracy of a Novel 3-Dof Redundant Planar Parallel Kinematic Machine
,”
Mech. Mach. Theory
,
119
, pp.
198
218
.
19.
Shang
,
W.
, and
Cong
,
S.
,
2014
, “
Motion Control of Parallel Manipulators Using Acceleration Feedback
,”
IEEE Trans. Control Syst. Technol.
,
22
(
1
), pp.
314
321
.
20.
Wang
,
K.
,
Luo
,
M.
,
Mei
,
T.
,
Zhao
,
J.
, and
Cao
,
Y.
,
2013
, “
Dynamics Analysis of a Three-DOF Planar Serial-Parallel Mechanism for Active Dynamic Balancing With Respect to a Given Trajectory
,”
Int. J. Adv. Rob. Syst.
,
10
(
1
), p.
23
.
21.
Ciblak
,
N.
, and
Lipkin
,
H.
,
1994
, “
Centers of Stiffness, Compliance, and Elasticity in the Modelling of Robotic Systems
,”
Robotics: Kinematics, Dynamics and Control
, Minneapolis, MN, pp.
185
194
.
22.
Ciblak
,
N.
,
1994
, “
Asymmetric Cartesian Stiffness for the Modeling of Compliant Robotic Systems
,”
23rd Biennial ASME Mechanisms Conference
, Minneapolis, MN, pp.
197
204
.
23.
Selig
,
J.
, and
Ding
,
X.
,
2001
, “
A Screw Theory of Static Beams
,”
IEEE/RSJ
International Conference on Intelligent Robots and Systems,
Maui, HI, Oct. 29–Nov. 3, pp.
312
317
.
24.
Ciblak
,
N.
, and
Lipkin
,
H.
,
1999
, “
Synthesis of Cartesian Stiffness for Robotic Applications
,”
IEEE
International Conference on Robotics and Automation,
Detroit, MI, May 10–15, pp.
2147
2152
.
25.
Liu
,
H.
,
Huang
,
T.
,
Chetwynd
,
D. G.
, and
Kecskeméthy
,
A.
,
2017
, “
Stiffness Modeling of Parallel Mechanisms at Limb and Joint/Link Levels
,”
IEEE Trans. Rob.
,
33
(
3
), pp.
734
741
.
26.
Ding
,
X.
, and
Selig
,
J.
,
2002
, “
Analysis of Spatial Compliance Behavior of Coiled Springs Via Screw Theory
,”
Chin. J. Mech. Eng.
,
15
(
4
), pp.
293
297
.
27.
Ding
,
X.
, and
Mark
,
S. J.
,
2005
, “
Lie Groups and Lie Algebras on Dynamic Analysis of Beam With Spatial Compliance
,”
Chin. J. Mech. Eng.
,
41
(
1
), pp.
16
23
.
28.
Li
,
Y.
, and
Xu
,
Q.
,
2008
, “
Stiffness Analysis for a 3-PUU Parallel Kinematic Machine
,”
Mech. Mach. Theory
,
43
(
2
), pp.
186
200
.
29.
Kim
,
H. S.
, and
Lipkin
,
H.
,
2014
, “
Stiffness of Parallel Manipulators With Serially Connected Legs
,”
ASME J. Mech. Rob.
,
6
(
3
), p.
031001
.
30.
Sun
,
T.
,
Lian
,
B.
, and
Song
,
Y.
,
2016
, “
Stiffness Analysis of a 2-DOF Over-Constrained Rpm With an Articulated Traveling Platform
,”
Mech. Mach. Theory
,
96
, pp.
165
178
.
31.
Liu
,
H.
,
Huang
,
T.
,
Kecskeméthy
,
A.
,
Chetwynd
,
D. G.
, and
Li
,
Q.
,
2017
, “
Force/Motion Transmissibility Analyses of Redundantly Actuated and Overconstrained Parallel Manipulators
,”
Mech. Mach. Theory
,
109
, pp.
126
138
.
You do not currently have access to this content.