Abstract
This paper presents a unified model for computer-aided tolerancing. It combines the benefits of the Jacobian matrix model and the torsor (or screw) model. The former is based on small displacements modeling of points using 6x6 transformation matrices of open kinematic chains in robotics. The latter models the boundaries of 3-D tolerance zones resulting from a feature’s small displacements using a torsor representation with constraints. The proposed unified model expands the functionalities of the Jacobian model under two important aspects. First, the punctual small displacement variables of the former Jacobian formulation are now considered as intervals formulated and solved using interval-based arithmetic. The equations describing the bounds within which the feature is permitted to move, which are the constraint equations of the torsor formulation, are applied on the unified model. Second, some of the small displacement variables used in the model are eliminated due to the invariant nature of the movements they generate with respect to the toleranced feature. This standard result of the torsor formulation is applied to the unified model. The effect of this is to significantly reduce the unified model size. A example application is also presented.
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List of references
A.A.G. REQUICHA. «Towards a theory of geometric tolerancing», the International Journal of Robotics Research, Vol. 2, n4 , 1983, pp. 45–60.
E. BALLOT, P. BOURDET, F. THIÉBAUT. «Determination of relative situations of parts for tolerance computation», 7th CIRP International Seminar on Computer Aided Tolerancing, ENS de Cachan, France, 2001, April 24–25.
L. RIVEST, C. FORTIN, C. MOREL. «Tolerancing a solid model with a kinematic formulation », Computer-Aided Design, v 26, n6, 1994, pp. 465–485.
A. DESROCHERS, A. CLÉMENT, A. RIVIÈRE. «Theory and practice of 3D tolerancing for assembly», CIRP International Working Seminar on Tolerancing, Penn. State Univ., 1991, pp. 25–55.
A. WIRTZ. «Vectoriel tolerancing», International Conference on CAD/CAM and AMT in Israël, CIRP session on tolerancing for function in a CAD/CAM environment, Vol. 2, dec. 11–14, 1989.
E. BALLOT.«Lois de comportement géométrique des mécanismes pour le tolérancement », Thèse Ph. D., École Normale Supérieure de Cachan, 1995.
D. GAUNET. «Vectoriel tolerancing model », 3rd CIRP Seminar on Computer Aided Tolerancing, April 27–28, 1993.
A. DESROCHERS. «Modèle conceptuel du dimensionnement et du tolérancement des mécanismes : Représentation dans les systèmes CFAO», Thèse Ph.D., École Centrale de Paris, 1991.
D.E WHITNEY. , J.D. ADAMS. «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 1999, pp. 75–80
D.E WHITNEY., O.L. GILBERT, M. JASTRZEBSKI. «Representation of geometric variations using matrix transformation for statistical tolerance analysis in assembly», Research in engineering, Vol. 6, 1994, pp.191–210.
K. W CHASE, J. GAO, S. P. MAGLEBY. «Generalized 2-D Tolerance Analysis of Mechanical Assemblies with Small Kinematic Adjustments», Journal of Design and Manufacturing, Vol. 5, 1995, No. 2.
K. W CHASE, J. GAO, S. P. MAGLEBY. «General 3-D Tolerance Analysis of Mechanical Assemblies with Small Kinematics Adjustments», IIE Transactions, 1998, Vol. 30, , pp. 367–377.
A. DESROCHERS, L. RIVEST, C. FORTIN. « Tolerance modeling for 3D analysis: Presenting a kinematic formulation», 3rd CIRP Seminar on Computer Aided Tolerancing, April 27–28,1993.
L. RIVEST, C. MOREL, C. FORTIN. «Tolerancing a solid model with a kinematic formulation», Computer Aided design, Vol. 26, 1994, N 6, pp. 465–485,.
L. LAPERRIERE, P. LAFOND. «Identification of dispersions affecting pre-defined functional requirements of mechanical assemblies » Proc. Of IDMME Conference, Compiège , France, May27–29,1998, pp. 721–728.
L. LAPERRIERE, P. LAFOND. «Tolerances Analysis and synthesis using virtuals joints », 6th CIRP international seminar on computer aided tolerancing, Enschede, Netherlands, March 22–24, 1999, pp 405–414.
A. DESROCHERS. «Determination of Part Position Uncertainty Within Mechanical Assembly Using Screw Parameters», CD-ROM Proc. 25th ASME Design Automation Conference, Las Vegas, 1997, DAC-8587.
P. BOURDET, E. BALLOT. «Équations formelles et tridimensionnelles des chaînes de dimensions dans les mécanismes », Séminaire Tolérancement et chaînes de cotes, fév., 1995, pp. 135–146.
J. TURNER, S. SRIKANTH. «Constraint Representation and Reduction in Assembly Modeling and Analysis», Rensselear Design Research Centre, Tech Report , 1990, No. 90027.
G. ZHANG, M. PORCHET. «Some new developments in tolerance design in CAD », Proceedings of the 19th ASME Annual International Design Automation Conference And Exposition, Albuquerque, Vol. 2, 1993, pp.175–185.
G. ZHANG. « Simultaneous tolerancing for design and manufacturing », Int. J. Prod. Res., Vol. 34, N° 12, 1996, pp. 3361–3382.
W. GHIE, L. LAPERRIÈRE. «Automatic generation of tolerance chains around functional requirements of mechanical assemblies», CD-ROM Proceeding of 3rd IDMME conference, Montreal, Canada,2000.
E. HANSEN. «Global optimization using interval analysis», Marcel Dekker, ISBN 0–8247 – 8696–3, 1992.
L. LAPERRIÈRE, T. KABORÉ. «Monte Carlo simulation of tolerance synthesis equations », Int. J. Prod. Res., Vol. 39, N° 11, 2001, pp. 2395–2406.
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Ghie, W., Laperrière, L., Desrochers, A. (2003). A Unified Jacobian-Torsor Model for Analysis in Computer Aided Tolerancing. In: Gogu, G., Coutellier, D., Chedmail, P., Ray, P. (eds) Recent Advances in Integrated Design and Manufacturing in Mechanical Engineering. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0161-7_7
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DOI: https://doi.org/10.1007/978-94-017-0161-7_7
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