The paper presents a computationally efficient method for system reliability analysis of mechanisms. The reliability is defined as the probability that the output error remains within a specified limit in the entire target trajectory of the mechanism. This mechanism reliability problem is formulated as a series system reliability analysis that can be solved using the distribution of maximum output error. The extreme event distribution is derived using the principle maximum entropy (MaxEnt) along with the constraints specified in terms of fractional moments. To optimize the computation of fractional moments of a multivariate response function, a multiplicative form of dimensional reduction method (M-DRM) is developed. The main benefit of the proposed approach is that it provides full probability distribution of the maximal output error from a very few evaluations of the trajectory of mechanism. The proposed method is illustrated by analyzing the system reliability analysis of two planar mechanisms. Examples presented in the paper show that the results of the proposed method are fairly accurate as compared with the benchmark results obtained from the Monte Carlo simulations.

References

References
1.
Mavroidis
,
C.
,
Dubowsky
,
S.
,
Drouet
,
P.
,
Hintersteiner
,
J.
, and
Flanz
,
J.
,
1997
, “
A Systematic Error Analysis of Robotic Manipulators: Application to a High Performance Medical Robot
,”
Proceedings of the 1997 IEEE International Conference of Robotics and Automation
, pp.
980
985
.
2.
Rao
,
S.
, and
Bhatti
,
P.
,
2001
, “
Probabilistic Approach to Manipulator Kinematics and Dynamics
,”
Reliab. Eng. Syst. Safety
,
72
(
1
), pp.
47
58
.10.1016/S0951-8320(00)00106-X
3.
Wu
,
W.
, and
Rao
,
S.
,
2007
, “
Uncertainty Analysis and Allocation of Joint Tolerances in Robot Manipulators Based on Interval Analysis
,”
Reliab. Eng. Syst. Safety
,
92
(
1
), pp.
54
64
.10.1016/j.ress.2005.11.009
4.
Kim
,
J.
,
Song
,
W.
, and
Kang
,
B.
,
2010
, “
Stochastic Approach to Kinematic Reliability of Open-Loop Mechanism With Dimensional Tolerance
,”
Appl. Math. Modell.
,
34
(
5
), pp.
1225
1237
.10.1016/j.apm.2009.08.009
5.
Zhang
,
J.
, and
Du
,
X.
,
2011
, “
Time-Dependent Reliability Analysis for Function Generator Mechanisms
,”
ASME J. Mech. Des.
,
133
(
3
), p.
031005
.10.1115/1.4003539
6.
Zhang
,
J.
,
Wang
,
J.
, and
Du
,
X.
,
2011
, “
Time-Dependent Probabilistic Synthesis for Function Generator Mechanisms
,”
Mech. Mach. Theory
,
46
(
9
), pp.
1236
1250
.10.1016/j.mechmachtheory.2011.04.008
7.
Huang
,
X.
, and
Zhang
,
Y.
,
2010
, “
Reliability Sensitivity Analysis for Rack-and-Pinion Steering Linkages
,”
ASME J. Mech. Des.
,
132
(
7
), p.
071012
.10.1115/1.4001901
8.
Savage
,
G. J.
, and
Son
,
Y. K.
,
2009
, “
Dependability-Based Design Optimization of Degrading Engineering Systems
,”
ASME J. Mech. Des.
,
131
(
1
), p.
011002
.10.1115/1.3013295
9.
Savage
,
G.
, and
Son
,
Y.
,
2011
, “
The Set-Theory Method for Systems Reliability of Structures With Degrading Components
,”
Reliab. Eng. Syst. Safety
,
96
(
1
), pp.
108
116
.10.1016/j.ress.2010.07.009
10.
Bichon
,
B. J.
,
McFarland
,
J. M.
, and
Mahadevan
,
S.
,
2011
, “
Efficient Surrogate Models for Reliability Analysis of Systems With Multiple Failure Modes
,”
Reliab. Eng. Syst. Safety
,
96
(
10
), pp.
1386
1395
.10.1016/j.ress.2011.05.008
11.
Andrieu-Renaud
,
C.
,
Sudret
,
B.
, and
Lemaire
,
M.
,
2004
, “
The PHI2 Method: A Way to Compute Time-Variant Reliability
,”
Reliab. Eng. Syst. Safety
,
84
(
1
), pp.
75
86
.10.1016/j.ress.2003.10.005
12.
Sudret
,
B.
,
2008
, “
Analytical Derivation of the Outcrossing Rate in Time-Variant Reliability Problems
,”
Struct. Infrastruct. Eng.
,
4
(
5
), pp.
353
362
.10.1080/15732470701270058
13.
Rackwitz
,
R.
, and
Flessler
,
B.
,
1978
, “
Structural Reliability Under Combined Random Load Sequences
,”
Comput. Struct.
,
9
(
5
), pp.
489
494
.10.1016/0045-7949(78)90046-9
14.
Pandey
,
M.
, and
Zhang
,
X.
,
2012
, “
System Reliability Analysis of the Robotic Manipulator With Random Joint Clearances
,”
Mech. Mach. Theory
,
58
(
0
), pp.
137
152
.10.1016/j.mechmachtheory.2012.08.009
15.
Zhang
,
X.
, and
Pandey
,
M.
,
2013
, “
An Efficient Method for System Reliability Analysis of Planar Mechanisms
,”
Proc. Inst. Mech. Eng., Part C
,
227
(
2
), pp.
373
386
.10.1177/0954406212448341
16.
Sandor
,
G.
, and
Erdman
,
A.
,
1984
,
Advanced Mechanism Design: Analysis and Synthesis
, Vol.
1
,
Prentice Hall
,
NJ
.
17.
Huang
,
X.
, and
Zhang
,
Y.
,
2010
, “
Robust Tolerance Design for Function Generation Mechanisms With Joint Clearances
,”
Mech. Mach. Theory
,
45
(
9
), pp.
1286
1297
.10.1016/j.mechmachtheory.2010.04.003
18.
Uicker
,
J. J.
,
Pennock
,
G. R.
, and
Shigley
,
J. E.
,
2003
,
Theory of Machines and Mechanisms
,
3rd ed.
,
Oxford University Press
,
New York
.
19.
Meggiolaro
,
M.
,
Dubowsky
,
S.
, and
Mavroidis
,
C.
,
2005
, “
Geometric and Elastic Error Calibration of a High Accuracy Patient Positioning System
,”
Mech. Mach. Theory
,
40
(
4
), pp.
415
427
.10.1016/j.mechmachtheory.2004.07.013
20.
Dubowsky
,
S.
,
Maatuk
,
J.
, and
Perreira
,
N. D.
,
1975
, “
A Parameter Identification Study of Kinematic Errors in Planar Mechanisms
,”
ASME J. Eng. Ind.
,
97
(
2
), pp.
635
642
.10.1115/1.3438628
21.
Chase
,
K.
, and
Greenwood
,
W.
,
1988
, “
Design Issues in Mechanical Tolerance Analysis
,”
ASME Manuf. Rev.
,
1
(
1
), pp.
50
59
. Available at: http://adcats.et.byu.edu/Publication/87-5/WAM2.PDF
22.
Wittwer
,
J. W.
,
Chase
,
K. W.
, and
Howell
,
L. L.
,
2004
, “
The Direct Linearization Method Applied to Position Error in Kinematic Linkages
,”
Mech. Mach. Theory
,
39
(
7
), pp.
681
693
.10.1016/j.mechmachtheory.2004.01.001
23.
Choi
,
J.
,
Lee
,
S.
, and
Choi
,
D.
,
1998
, “
Stochastic Linkage Modeling for Mechanical Error Analysis of Planar Mechanisms
,”
Mech. Struct. Mach.
,
26
(
3
), pp.
257
276
.10.1080/08905459708945494
24.
Briot
,
S.
, and
Bonev
,
I.
,
2008
, “
Accuracy Analysis of 3-dof Planar Parallel Robots
,”
Mech. Mach. Theory
,
43
(
4
), pp.
445
458
.10.1016/j.mechmachtheory.2007.04.002
25.
Huang
,
Z.
,
1987
, “
Error Analysis of Position and Orientation in Robot Manipulators
,”
Mech. Mach. Theory
,
22
(
6
), pp.
577
581
.10.1016/0094-114X(87)90053-X
26.
Li
,
J.
,
Chen
,
J.
, and
Fan
,
W.
,
2007
, “
The Equivalent Extreme-Value Event and Evaluation of the Structural System Reliability
,”
Struct. Safety
,
29
(
2
), pp.
112
131
.10.1016/j.strusafe.2006.03.002
27.
Harbitz
,
A.
,
1983
, “
Efficient and Accurate Probability of Failure Calculation by the Use of the Importance Sampling Technique
,”
Proceedings of Fourth International Conference on Applications of Statistics and Probability in Soil and Structural Engineering
, pp.
825
836
.
28.
Au
,
S.
, and
Beck
,
J.
,
1999
, “
A New Adaptive Importance Sampling Scheme for Reliability Calculations
,”
Struct. Safety
,
21
(
2
), pp.
135
158
.10.1016/S0167-4730(99)00014-4
29.
Gavin
,
H.
, and
Yau
,
S.
,
2008
, “
High-Order Limit State Functions in the Response Surface Method for Structural Reliability Analysis
,”
Struct. Safety
,
30
(
2
), pp.
162
179
.10.1016/j.strusafe.2006.10.003
30.
Rabitz
,
H.
, and
Aliş
,
Ö.
,
1999
, “
General Foundations of High-Dimensional Model Representations
,”
J. Math. Chem.
,
25
(
2
), pp.
197
233
.10.1023/A:1019188517934
31.
Li
,
G.
,
Rosenthal
,
C.
, and
Rabitz
,
H.
,
2001
, “
High Dimensional Model Representations
,”
J. Phys. Chem. A
,
105
(
33
), pp.
7765
7777
.10.1021/jp010450t
32.
Rahman
,
S.
, and
Xu
,
H.
,
2004
, “
A Univariate Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics
,”
Prob. Eng. Mech.
,
19
(
4
), pp.
393
408
.10.1016/j.probengmech.2004.04.003
33.
Xu
,
H.
, and
Rahman
,
S.
,
2004
, “
A Generalized Dimension-Reduction Method for Multidimensional Integration in Stochastic Mechanics
,”
Int. J. Numer. Methods Eng.
,
61
(
12
), pp.
1992
2019
.10.1002/nme.1135
34.
Montgomery
,
D.
, and
Myers
,
R.
,
2002
,
Response Surface Methodology: Process and Product Optimization Using Designed Experiments
,
2nd ed.
,
John Wiley & Sons Ltd.
,
New York
.
35.
Zhang
,
X.
, and
Pandey
,
M. D.
,
2013
, “
Structural Reliability Analysis Based on the Concepts of Entropy, Fractional Moment and Dimensional Reduction Method
,”
Struct. Safety
,
43
(
4
), pp.
28
40
.10.1016/j.strusafe.2013.03.001
36.
Zhang
,
X.
, and
Pandey
,
M. D.
,
2014
, “
An Effective Approximation for Variance-Based Global Sensitivity Analysis
,”
Reliab. Eng. Syst. Safety
,
121
, pp.
164
174
.10.1016/j.ress.2013.07.010
37.
Taufer
,
E.
,
Bose
,
S.
, and
Tagliani
,
A.
,
2009
, “
Optimal Predictive Densities and Fractional Moments
,”
Appl. Stochastic Models Bus. Ind.
,
25
(
1
), pp.
57
71
.10.1002/asmb.721
38.
Tagliani
,
A.
,
1999
, “
Hausdorff Moment Problem and Maximum Entropy: A Unified Approach
,”
Appl. Math. Comput.
,
105
(
2–3
), pp.
291
305
.10.1016/S0096-3003(98)10084-X
39.
Inverardi
,
P.
, and
Tagliani
,
A.
,
2003
, “
Maximum Entropy Density Estimation From Fractional Moments
,”
Commun. Statistics: Theory Methods
,
32
(
2
), pp.
327
345
.
40.
Milev
,
M.
,
Inverardi
,
P. N.
, and
Tagliani
,
A.
,
2012
, “
Moment Information and Entropy Evaluation for Probability Densities
,”
Appl. Math. Comput.
,
218
(
9
), pp.
5782
5795
.10.1016/j.amc.2011.11.093
41.
Gzyl
,
H.
, and
Tagliani
,
A.
,
2010
, “
Hausdorff Moment Problem and Fractional Moments
,”
Appl. Math. Comput.
,
216
(
11
), pp.
3319
3328
.10.1016/j.amc.2010.04.059
42.
Graham
,
R. L.
,
Knuth
,
D. E.
, and
Patashnik
,
O.
,
1988
,
Concrete Mathematics: A Foundation for Computer Science
,
Addison-Wesley Publishing Company
,
Menlo Park
, Chap. 5, p.
154
.
43.
Jaynes
,
E.
,
1957
, “
Information Theory and Statistical Mechanics
,”
Phys. Rev.
,
108
(
2
), pp.
171
190
.10.1103/PhysRev.108.171
44.
Kapur
,
J.
, and
Kesavan
,
H.
,
1992
,
Entropy Optimization Principles With Applications
,
Academic Press, Inc.
,
New York
.
45.
Pandey
,
M.
,
2002
, “
An Adaptive Exponential Model for Extreme Wind Speed Estimation
,”
J. Wind Eng. Ind. Aerodyn.
,
90
(
7
), pp.
839
866
.10.1016/S0167-6105(02)00161-7
46.
Pandey
,
M.
,
Van Gelder
,
P.
, and
Vrijling
,
J.
,
2001
, “
Assessment of an l-Kurtosis-Based Criterion for Quantile Estimation
,”
J. Hydrol. Eng.
,
6
(
4
), pp.
284
292
.10.1061/(ASCE)1084-0699(2001)6:4(284)
47.
Novi Inverardi
,
P.
,
Petri
,
A.
,
Pontuale
,
G.
, and
Tagliani
,
A.
,
2005
, “
Stieltjes Moment Problem Via Fractional Moments
,”
Appl. Math. Comput.
,
166
(
3
), pp.
664
677
.10.1016/j.amc.2004.06.060
48.
Lagarias
,
J.
,
Reeds
,
J.
,
Wright
,
M.
, and
Wright
,
P.
,
1998
, “
Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions
,”
SIAM J. Optim.
,
9
(
1
), pp.
112
147
.10.1137/S1052623496303470
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