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Production Engineering

, Volume 8, Issue 1–2, pp 81–89 | Cite as

A procedure for the evaluation and compensation of form errors by means of global isometric registration with subsequent local reoptimization

  • Laura Klein
  • Tobias Wagner
  • Christoph Buchheim
  • Dirk Biermann
Computer Aided Engineering

Abstract

Stresses remaining in the component after sheet metal forming processes can result in complex form errors, such as springback and torsions. In order to compensate these process-induced deformations, the local and global deformations have to be analyzed. Hence, an appropriate comparison between the actually manufactured and the target design is required. For this purpose, the surface of the actual workpiece is scanned and the so-obtained scan points have to be assigned to corresponding points of the target shape defined by the workpiece model. From these correspondences, a field of deformation vectors can be computed which represents the basis for the compensation strategy. The task of finding appropriate correspondences is called registration. It is usually solved using rigid transformations, i.e., translation and rotation. Due to the locality, strength and complexity of the deformations, rigid transformations are usually not sufficient. As a more flexible alternative, a procedure for non-rigid registration is presented in this paper. Therein, isometry, i.e., the conservation of distances between corresponding points within an appropriate neighborhood structure, is defined as the objective function. The procedure consists of three steps: definition of the neighborhood structure, global registration, and local reoptimization. The main focus of the paper is set to the latter, where an adapted gradient descent method also allowing projections into the triangles of the target shape is presented and experimentally validated. With these three steps, an assignment between both shapes can be calculated, even for strong local deformations and coarse triangular meshes representing the workpiece model.

Keywords

Forming Springback analysis Non-rigid isometric registration Local optimization Quadratic assignment problem 

Notes

Acknowledgments

This work is funded as subproject C4 of the Collaborative Research Center “3D-Surface Engineering” (SFB 708) by the German Research Foundation (DFG). Elias Kuthe and Cesaire Fondjo are acknowledged for their participation in the implementation and experimental evaluation of the presented local reoptimization algorithm.

References

  1. 1.
    Amberg B, Romdhani S, Vetter T (2007) Optimal step nonrigid ICP algorithms for surface registration. In: IEEE conference on computer vision and pattern recognition, pp 1–8. doi: 10.1109/CVPR.2007.383165
  2. 2.
    Anstreicher K (2003) Recent advances in the solution of quadratic assignment problems. Math Program B 97:27–42MATHMathSciNetGoogle Scholar
  3. 3.
    Besl P, McKay N (1992) A method for registration of 3-d shapes. IEEE Trans Pattern Anal Mach Intell 14(2):239–256CrossRefGoogle Scholar
  4. 4.
    Biermann D, Sacharow A, Surmann T, Wagner T (2010) Direct free-form deformation of NC programs for surface reconstruction and form-error compensation. Prod Eng Res Dev 4(5):501–507CrossRefGoogle Scholar
  5. 5.
    Biermann D, Surmann T, Sacharow A, Skutella M, Theile M (2008) Automated analysis of the form error caused by springback in metal sheet forming. In: Proceedings of the 3rd international conference on manufacturing engineering, pp 737–746Google Scholar
  6. 6.
    Bronstein A, Bronstein M, Kimmel R (2006) Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc Natl Acade Sci 103(5):1168–1172CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Çela E (1998) The quadratic assignment problem—theory and algorithms. Kluwer Academic, Dordrecht, The NetherlandsCrossRefMATHGoogle Scholar
  8. 8.
    Chen Y, Medioni G (1991) Object modeling by registration of multiple range images. In: Proceedings of the IEEE international conference on robotics and automation, vol 3, pp 2724–2729Google Scholar
  9. 9.
    Elad A, Kimmel R (2003) On bending invariant signatures for surfaces. IEEE Trans Pattern Anal Mach Intell 25(10):1285–1295CrossRefGoogle Scholar
  10. 10.
    Gan W, Wagoner R (2004) Die design method for sheet springback. Int J Mech Sci 46(7):1097–1113CrossRefGoogle Scholar
  11. 11.
    Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman, New YorkMATHGoogle Scholar
  12. 12.
    Horn BKP (1987) Closed-form solution of absolute orientation using unit quaternions. J Opt Soc Am A 4(4):629–642CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hussain M (2013) Volume and normal field based simplification of polygonal models. J Inf Sci Eng 29(2):267–279Google Scholar
  14. 14.
    Kleiner M, Tekkaya A, Chatti S, Hermes M, Weinrich A, Ben-Khalifa N, Dirksen U (2009) New incremental methods for springback compensation by stress superposition. Prod Eng Res Dev 3(2):137–144CrossRefGoogle Scholar
  15. 15.
    Lawler EL (1963) The quadratic assignment problem. Manag Sci 9(4):586–599. doi: 10.1287/mnsc.9.4.586 CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Mémoli F, Sapiro G (2005) A theoretical and computational framework for isometry invariant recognition of point cloud data. Found Comput Math 5(3):313–347CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Nocedal J, Wright S (2006) Numerical optimization, 2nd edn. Springer, BerlinGoogle Scholar
  18. 18.
    Sacharow A, Balzer J, Biermann D, Surmann T (2011) Non-rigid isometric ICP: a practical registration method for the analysis and compensation of form errors in production engineering. Comput Aided Des 43(12):1758–1768. doi: 10.1016/j.cad.2011.07.007 CrossRefGoogle Scholar
  19. 19.
    Selimovic I (2006) Improved algorithms for the projection of points on NURBS curves and surfaces. Comput Aided Geom Des 23:439–445CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Tam GKL, Cheng ZQ, Lai YK, Langbein FC, Liu Y, Marshall D, Martin RR, Sun XF, Rosin PL (2013) Registration of 3d point clouds and meshes: a survey from rigid to non-rigid. IEEE Trans Vis Comput Graph. doi: 10.1109/TVCG.2012.310
  21. 21.
    Wagner T, Michelitsch T, Sacharow A (2007) On the design of optimisers for surface reconstruction. In: Thierens D, et al (eds) Proceedings of 9th annual genetic and evolutionary computation conference (GECCO 2007). ACM, New York, NJ, London, UK, pp 2195–2202. doi: 10.1145/1276958.1277379
  22. 22.
    Weiher J, Rietman B, Kose K, Ohnimus S, Petzoldt M (2004) Controlling springback with compensation strategies. In: AIP conference proceedings, pp 1011–1015Google Scholar

Copyright information

© German Academic Society for Production Engineering (WGP) 2013

Authors and Affiliations

  • Laura Klein
    • 1
  • Tobias Wagner
    • 2
  • Christoph Buchheim
    • 1
  • Dirk Biermann
    • 2
  1. 1.Fakultät für MathematikTU DortmundDortmundGermany
  2. 2.Institut für Spanende FertigungTU DortmundDortmundGermany

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