Because of uncertainties in manufacturing processes, a mechanical part always shows variations in its geometrical characteristics (ex. form, dimension, orientation and position). Quality then often reflect how well tolerances and hence, functional requirements, are being achieved by the manufacturing processes in the final product. From a design perspective, efficient methods must be made available to compute, from the tolerances on individual parts, the value of the functional requirement on the final assembly. This is known as tolerance analysis. To that end, existing methods, often based on modeling of the open kinematic chains in robotics, are classified as deterministic or statistical. These methods suppose that the assembled parts are not perfect with regard to the nominal geometry and are rigid. The rigidity of the parts implies that the places of contacts are regarded as points. The validation or the determination of a tolerance zone is therefore accomplished by a series of simulation in specific points subjected to assembly constraints. To overcome the limitations and difficulties of point based approaches, the paper proposes the unification of two existing models: the Jacobian’s matrix model, based on the infinitesimal modeling of open kinematic chains in robotics, and the tolerance zone representation model, using small displacement screws and constraints to establish the extreme limits between which points and surfaces can vary. The approach also uses interval algebra as a novel method to take tolerance boundaries into account in tolerance analysis. The approach has been illustrated on a simple two parts assembly, nevertheless demonstrating the capability of the method to handle three-dimensional geometry. The results are then validated geometrically, showing the overall soundness of the approach.

1.
Requicha
,
A. A. G.
,
1983
, “
Towards a Theory of Geometric Tolerancing
,”
Int. J. Robot. Res.
,
2
(
4
), pp.
45
60
.
2.
Ballot, E., Bourdet, P., and Thie´baut, F., 2001, “Determination of Relative Situations of Parts for Tolerance Computation,” 7th CIRP International Seminar on Computer Aided Tolerancing, Cachan, France, April 24–25, pp. 65–74.
3.
Rivest
,
L.
,
Fortin
,
C.
, and
Morel
,
C.
,
1994
, “
Tolerancing a Solid Model With a Kinematic Formulation
,”
Comput.-Aided Des.
,
26
(
6
), pp.
465
485
.
4.
Desrochers, A., Clement, A., and Riviere, A., 1991, “Theory and Practice of 3D Tolerancing for Assembly,” CIRP International Working Seminar on Tolerancing, Penn. State Univ., pp. 25–55.
5.
Wirtz, A., 1989, “Vectorial Tolerancing,” International Conference on CAD/CAM and AMT, CIRP Session on Tolerancing for Function in a CAD/CAM Environment, Vol. 2, Israel, Dec. 11–14.
6.
Ballot, E., 1995, “Lois de comportment ge´ome´trique des me´canismes pour le tole´rancement,” These Ph.D., E´cole Normale Supe´rieure de Cachan.
7.
Gaunet, D., 1993, “Vectorial Tolerancing Model,” 3rd CIRP International Seminar on Computer Aided Tolerancing, April 27–28, pp. 25–50.
8.
Desrochers, A., 1991, “Modele conceptuel du dimensionnement et du tole´rancement des me´canismes: Repre´sentation dans les systemes CFAO,” These Ph.D., E´cole Centrale de Paris.
9.
Adams, J. D., and Whitney, D. E., 1999, “Application of Screw Theory to Motion Analysis of Assemblies of Rigid Parts,” Proceedings of IEEE International Symposium on Assembly and Task Planning, Porto, Portugal, July, pp. 75–80.
10.
Whitney
,
D. E.
,
Gilbert
,
O. L.
, and
Jastrzebski
,
M.
,
1994
, “
Representation of Geometric Variations Using Matrix Transformation for Statistical Tolerance Analysis in Assembly
,”
Res. Eng. Des.
,
6
, pp.
191
210
.
11.
Chase
,
K. W.
,
Gao
,
J.
, and
Magleby
,
S. P.
,
1995
, “
Generalized 2-D Tolerance Analysis of Mechanical Assemblies With Small Kinematic Adjustments
,”
Journal of Design and Manufacturing
,
5
(
4
), pp.
263
274
.
12.
Chase
,
K. W.
,
Gao
,
J.
, and
Magleby
,
S. P.
,
1998
, “
General 3-D Tolerance Analysis of Mechanical Assemblies With Small Kinematic Adjustments
,”
IIE Trans.
,
30
, pp.
367
377
.
13.
Laperriere, L., and Lafond, P., 1998, “Identification of Dispersions Affecting Pre-Defined Functional Requirements of Mechanical Assemblies,” Proceedings of 2nd IDMME Conference, Compiege, France, May 27–29, pp. 721–728.
14.
Laperriere, L., and Lafond, P., 1999, “Tolerances Analysis and Synthesis Using Virtuals Joints,” 6th CIRP International Seminar on Computer Aided Tolerancing, Enschede, Netherlands, March 22–24, pp. 405–414.
15.
Desrochers, A., and Delbart, O., 1997, “Determination of Part Position Uncertainty Within Mechanical Assembly Using Screw Parameters,” 5th CIRP Intrnational Seminar on Computer Aided Tolerancing, Toronto, Canada, April 27–29, pp. 125–136.
16.
Bourdet, P., and Ballot, E., 1995, “Equations formelles et tridimensionnelles des chai^nes de dimensions dans les me´canismes,” Se´minaire Tole´rancement et chai^nes de cotes, fe´v., pp. 135–146.
17.
Desrochers, A., 1999, “Modeling Three-Dimensional Tolerance Zones Using Screw Parameters,” CD-ROM Proceedings of ASME 25th Design Automation Conference, DAC-8587, Las-Vegas.
18.
Turner, J., and Srikanth, S., 1990, “Constraint Representation and Reduction in Assembly Modeling and Analysis,” Rensselear Design Research Center, Tech Report, No. 90027.
19.
Zhang, G., and Porchet, M., 1993, “Some New Developments in Tolerance Design in CAD,” Proceedings of the 19th ASME Annual International Design Automation Conference And Exposition, Albuquerque, 2, pp. 175–185.
20.
Zhang
,
G.
,
1996
, “
Simultaneous Tolerancing for Design and Manufacturing
,”
Int. J. Prod. Res.
,
34
(
12
), pp.
3361
3382
.
21.
Hansen, E., 1992, “Global Optimization Using Interval Analysis,” Marcel Dekker, ISBN 0-8247-8696-3.
22.
Kolev, L., 1993, “Interval Methods for Circuit Analysis,” World Scientific Publishing Company, Incorporated, ISBN 9810214138.
23.
Jaulin, L., Kieffer, M., and Didrit, O., 2001, “Applied Interval Analysis With Examples in Parameter and State Estimation, Robust Control and Robotics,” World Scientific Publishing Company Incorporated, ISBN 1852332190.
24.
Moore, R., 1979, “Methods and Applications of Interval Analysis,” Society for Industrial and Applied Mathematic, ISBN 0898711614.
25.
Agati, P., Lerouge, F., and Rosetto, M., 2001, “Liaisons, me´canismes et assemblages,” Paris, Dunod, ISBN 2-10-004702-7.
26.
Arakelian, V., 1997, “Structure et cine´matique des me´canismes,” Paris, Hermes, ISBN 2-86601-642-47.
27.
Ghie, W., and Laperriere, L., 2000, “Automatic Generation of Tolerance Chains Around Functional Requirements of Mechanical Assemblies,” CD-ROM Proceedings of 3rd IDMME conference, Montreal, Canada.
28.
Khalil, W., and Dombre, E´., 1986, “La robotique pour inge´nieurs,” McGraw-Hill, ISBN 2-7042-1133-7.
29.
Tsai L., 1999, “The Mechanics of Serial and Parallel Manipulators,” New York, N.Y.: J. Wiley and Sons, ISBN 0-471-32593-7.
30.
Ghie, W., Laperriere, L., and Desrochers, A., 2002, “A Unified Jacobian-Torsor Model for Analysis in Computer Aided Tolerancing,” CD-ROM Proceedings of 4th IDMME conference, France, May 14–16.
31.
Laperriere, L., Ghie, W., and Desrochers, A., 2002, “Statistical and Deterministic Tolerance Analysis and Synthesis Using a Unified Jacobian-Torsor Model,” Annals of 52nd CIRP General Assembly, San Sebastian, 51(1) pp. 417–420.
32.
Cle´ment, A., Riviere, A., and Temmerman, M., 1994, “Cotation tridimensionnelle des systemes me´caniques,” the´orie & pratique, PYC, ISBN 2-85330-132-X.