Advertisement

A geometric constraint solver for 3-D assembly modeling

  • Xiaobo Peng
  • Kunwoo Lee
  • Liping Chen
Original Article

Abstract

In this paper, we propose a geometric constraint solver for 3-D assembly applications. First, we give a new geometry and constraint expression based on Euler parameters, which can avoid singular points during the solving process and simplify constraint types. Then we present a directed graph based constructive method to geometric constraint system solving that can handle well-, over- and under-constrained systems efficiently. The basic idea of this method is that it first simplifies the constraint graph by pruning those vertices which have only in-arcs from the graph and then reduces the size of strongly connected components (SCCs) left in the graph by DOF-based analysis. The method can solve all kinds of configurations including closed-loops. After that, we apply a hybrid numerical method of Newton–Raphson and Homotopy to solve under-constrained systems. The hybrid method makes use of the high efficiency of the Newton–Raphson method as well as the outstanding convergence of the Homotopy method. Finally, we give a practical example and conclusion.

Keywords

Assembly Constraint decomposition  Constraint graph Geometric constraint Numerical solving  

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kim J, Kim K, Choi K, Lee JY (2000) Solving 3D geometric constraints for assembly modeling. Int J Adv Manuf Technol 16:843–849CrossRefGoogle Scholar
  2. 2.
    Light R, Gossard D (1982) Modification of geometric models through variational geometry. Comput Aided Des 14(3):209–214CrossRefGoogle Scholar
  3. 3.
    Sutherland I (1963) Sketchpad, a man-machine graphical communication system. Proc Spring Joint Comp Conference, North Holland, pp 329–345Google Scholar
  4. 4.
    Gao XS, Zhou SC (1998) Solving geometric constraint systems II. A symbolic approach and decision of rc-constructibility. Comput Aided Des 30(2):115–22CrossRefGoogle Scholar
  5. 5.
    Kondo K (1992) Algebraic method for manipulation of dimensional relationships in geometric models. Comput Aided Des 24(3):141–7CrossRefzbMATHGoogle Scholar
  6. 6.
    Aldefeld B (1988) Variation of geometries based on a geometric-reasoning method. Comput Aided Des 20(3):117–126CrossRefzbMATHGoogle Scholar
  7. 7.
    Roller D (1990) A System for Interactive Variation Design. In: Wozny MJ, Turner JU, Preiss K (eds) Geometric Modeling for Product Engineering. Elsevier Science Publishers B.V. (North Holland), pp 207–210Google Scholar
  8. 8.
    Verroust A, Schonek F, Roller D (1992) Rule-oriented method for parameterized computer-aided design. Comput Aided Des 24(3):531–540CrossRefzbMATHGoogle Scholar
  9. 9.
    Sunde G (1987) A CAD system with declarative specification of shape. Eurographics workshop on intelligent CAD systems. Noorwijkerhout, The Netherlands, pp 90–104Google Scholar
  10. 10.
    Kramer G (1992) Solving geometric constraint system: A case study in kinematics. MIT Press, Cambridge, MAGoogle Scholar
  11. 11.
    Owen J C (1991) Algebraic solution for geometry from dimensional constraints. Proc ACM Symp Found of Solid Modeling, Austin, TX, pp 397–407Google Scholar
  12. 12.
    Bouma W, Fudos I, Hoffmann C, Cai J, Paige R (1995) Geometric constraint solver. CAD 27(5):487–501zbMATHGoogle Scholar
  13. 13.
    Hoffmann CM, Vermeer P (1995) Geometric constraint solving in R2 and R3. In: Computing in Euclidean geometry. World Scientific Publishing, , pp 170–195Google Scholar
  14. 14.
    Lee JY, Kim K (1998) A 2-D geometric constraint solver using DOF-based graph reduction. Comput Aided Des 30(9):883–896CrossRefzbMATHGoogle Scholar
  15. 15.
    Li YT, Hu SM, Sun JG (2002) A constructive approach to solving 3-D geometric constraint systems using dependence analysis. Comput Aided Des 30:97–108CrossRefGoogle Scholar
  16. 16.
    Peng XB, Chen LP, Zhou FL, Zhou J (2002) Singularity analysis of geometric constraint system. J Comput Sci Technol 17(3):314–323zbMATHCrossRefGoogle Scholar
  17. 17.
    Hoffmann CM, Lomonosov A, Sitharam M (1998) Geometric constraint decomposition. In: Geometric constraint solving and applications. Springer, Berlin Heidelberg New York, pp 170–195Google Scholar
  18. 18.
    Ge J-X, Chou S-C, Gao X-S (1999) Geometric constraint satisfaction using optimization methods. Comput Aided Des 31:867–879CrossRefzbMATHGoogle Scholar
  19. 19.
    Kang YS (1996) Principles and methods of variational assembly design. Dissertation, Huazhong Univ Sci & TechGoogle Scholar
  20. 20.
    Lamure H, Michelucci D (1996) Solving geometric constraints by homotopy. IEEE Trans Visual Comput Graph 2(1):28–34.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of Mechanical Design & Production EngineeringSeoul National UniversitySeoulSouth Korea
  2. 2.School of Mechanical Science & EngineeringHuazhong University of Science & TechnologyWuhanChina

Personalised recommendations