This paper presents algorithms for workpiece positioning analysis under locating errors. Workpiece constraint equations are first constructed using the method of homogenous coordinate transformation. These constraint equations are solved numerically for exact workpiece positional deviations by means of deterministic analysis (using the Newton–Raphson method) and variation analysis (i.e., random analysis using a Monte Carlo simulation). To enhance numerical efficiency in variation analysis, we further propose a quadratic approximation solution using the method of moments instead of the Monte Carlo method. Several case studies are presented to exemplify the proposed algorithms, with comparisons to prior literature results on linear and quadratic analyses. The criterion for using the proposed quadratic variation analysis versus the linear method and Monte Carlo simulation is also presented. By using the proposed algorithms, the exact workpiece positioning errors or quadratic variation approximations can be calculated, with consideration of workpiece surface nonlinearity, interactions between locating errors, and the impact of noninfinitesimal locating errors. This paper represents algorithmic advancement in the field where exact solutions and approximations can all be obtained at users’ choice.

1.
Weill
,
R.
,
Darel
,
I.
, and
Laloum
,
M.
, 1991, “
The Influence of Fixture Positioning Errors on the Geometric Accuracy of Mechanical Parts
,”
Proceedings of the CIRP Conference on PE & MS
, Tianjin, China, Sept., pp.
215
225
.
2.
Rong
,
Y.
, and
Zhu
,
Y.
, 1999,
Computer-Aided Fixture Design
,
Dekker
,
New York
.
3.
Xiong
,
C. H.
,
Li
,
Y. F.
,
Ding
,
H.
, and
Xiong
,
Y. L.
, 1999, “
On the Dynamic Stability of Grasping
,”
Int. J. Robot. Res.
,
18
(
9
), pp.
951
958
. 0278-3649
4.
DeMeter
,
E. C.
, 1994, “
Restraint Analysis of Fixtures Which Rely on Surface Contact
,”
ASME J. Eng. Ind.
0022-0817,
116
(
2
), pp.
207
215
.
5.
Sayeed
,
Q. A.
, and
DeMeter
,
E. C.
, 1994, “
Machining Fixture Design and Analysis Software
,”
Int. J. Prod. Res.
0020-7543,
32
(
7
), pp.
1655
1674
.
6.
Cai
,
W.
,
Hu
,
S. J.
, and
Yuan
,
J. X.
, 1996, “
Deformable Sheet Metal Fixturing: Principles, Algorithms and Simulations
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
118
(
3
), pp.
318
324
.
7.
Camelio
,
J.
,
Hu
,
S. J.
, and
Ceglarek
,
D.
, 2004, “
Impact of Fixture Design on Sheet Metal Assembly Variation
,”
J. Manuf. Syst.
0278-6125,
23
(
3
), pp.
182
193
.
8.
Kong
,
Z.
,
Huang
,
W.
, and
Ceglarek
,
D.
, 2005, “
Visibility Analysis for Assembly Fixture Calibration Using Screen Space Transformation
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
127
(
3
), pp.
622
634
.
9.
Liao
,
Y. J. G.
, and
Hu
,
S. J.
, 2000, “
Flexible Multibody Dynamics Based Fixture-Workpiece Analysis Model for Fixturing Stability
,”
Int. J. Mach. Tools Manuf.
0890-6955,
40
, pp.
343
362
.
10.
Cai
,
W.
,
Hu
,
S. J.
, and
Yuan
,
J. X.
, 1997, “
A Variational Method of Robust Fixture Configuration Design for 3D Workpieces
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
119
(
4
), pp.
593
602
.
11.
Cai
,
W.
, 2006, “
Robust Pin Layout Design for Sheet Metal Locating
,”
,
28
(
5-6
), pp.
622
634
. 0268-3768
12.
Carlson
,
J. S.
, 2001, “
Quadratic Sensitivity Analysis of Fixtures and Locating Schemes for Rigid Parts
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
123
(
3
), pp.
462
472
.
13.
Liu
,
T.
, and
Wang
,
M. Y.
, 2003, “
An Approximate Quadratic Analysis of Fixture Locating Schemes
,”
Proceedings of the Automation
, National Chung Cheng University, Chia-Yi, Taiwan, Sept. 12–14.
14.
Wang
,
M. Y.
,
Liu
,
T.
, and
Pelinescu
,
D. M.
, 2003, “
Fixture Kinematic Analysis Based on the Full Contact Model of Rigid Bodies
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
125
(
2
), pp.
316
324
.
15.
Loose
,
J.
,
Zhou
,
S.
, and
Ceglarek
,
D.
, 2007, “
Kinematic Analysis of Dimensional Variation Propagation for Multistage Machining Processes With General Fixture Layouts
,”
IEEE Trans. Autom. Sci. Eng.
,
4
(
2
), pp.
141
152
. 1545-5955
16.
Ceglarek
,
D.
, and
Shi
,
J.
, 1996, “
Fixture Failure Diagnosis for the Autobody Assembly Using Pattern Recognition
,”
ASME J. Eng. Ind.
0022-0817,
118
(
1
), pp.
55
66
.
17.
Ceglarek
,
D.
, and
Shi
,
J.
, 1999, “
Fixture Failure Diagnosis for Sheet Metal Assembly With Consideration of Measurement Noise
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
121
, pp.
771
777
.
18.
Camelio
,
J. A.
, and
Hu
,
S. J.
, 2004, “
Multiple Fault Diagnosis for Sheet Metal Fixtures Using Designated Component Analysis
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
126
, pp.
91
97
.
19.
Camelio
,
J. A.
,
Hu
,
S. J.
, and
Ceglarek
,
D.
, 2004, “
Impact of Fixture Design on Sheet Metal Assembly Variation
,”
J. Manuf. Syst.
0278-6125,
23
(
3
), pp.
182
193
.
20.
Liu
,
Y. G.
, and
Hu
,
S. J.
, 2005, “
Assembly Fixture Fault Diagnosis Using Designated Component Analysis
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
127
(
2
), pp.
358
368
.
21.
Shapiro
,
S. S.
, and
Gross
,
A. J.
, 1981,
Statistical Modeling Techniques
,
Dekker
,
New York
.
22.
Cox
,
N. D.
, 1984,
How to Perform Statistical Tolerance Analysis
, Vol.
11
, American Society for Quality Control, Statistical Division, Minneapolis, MN.
23.
Greenwood
,
W. H.
, and
Chase
,
K. W.
, 1987, “
A New Tolerance Analysis Method for Designers and Manufacturers
,”
ASME J. Eng. Ind.
0022-0817,
109
, pp.
112
116
.
24.
Chase
,
K. W.
, and
Parkinson
,
A. R.
, 1991, “
A Survey of Research in the Application of Tolerance Analysis to the Design of Mechanical Assemblies
,”
Res. Eng. Des.
0934-9839,
3
, pp.
23
37
.
25.
Glancy
,
C. G.
, 1994, “
A Second-Order Method for Assembly Tolerance Analysis
,” M.S. thesis, Brigham Young University.
26.
Haug
,
E. J.
, 1989,
Intermediate Dynamics
,
Prentice-Hall
,
Englewood Cliffs, NJ
.