Skip to main content

Advertisement

SpringerLink
Log in

Compact radial zigzag for five-axis machining of STL surfaces

  • ORIGINAL ARTICLE
  • Published:
The International Journal of Advanced Manufacturing Technology Aims and scope Submit manuscript

Abstract

The paper presents a new method to generate efficient milling toolpaths for five-axis sculptured surface machining in an important case when the vector field of preferred directions (VFPD) forms a star-like, radial configuration. To optimize the toolpath, a new modification of the radial toolpath aligned with the VFPD called the compact radial zigzag (CRZ) has been proposed, analyzed, and verified practically. The CRZ is combined with transfinite interpolation (TFI) to treat an irregular VFPD. The method is designed for the machining of industrial stereolithography (STL) part surfaces characterized by complex geometries and sharp extrema. A demo of the algorithm is at https://drive.google.com/open?id=1OM_z4cAOUqGu2RPAzkZOIBcEnfptdTq7.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30

Similar content being viewed by others

References

  1. Dragomatz D, Mann S (1997) A classified bibliography of literature on NC milling path generation. Comput Aided Des 29:239–247

    Google Scholar 

  2. Lasemi A, Xue D, Gu P (2010) Recent development in CNC machining of freeform surfaces: a state-of-the-art review. Comput Aided Des 42:641–654

    Google Scholar 

  3. Makhanov SS (2010) Adaptable geometric patterns for five-axis machining: a survey. Int J Adv Manuf Technol 47:1167–1208

    Google Scholar 

  4. Feng H-Y, Li H (2002) Constant scallop-height tool path generation for three-axis sculptured surface machining. Comput Aided Des 34:647–654

    Google Scholar 

  5. Ding S, Mannan MA, Poo AN, Yang DCH, Han Z (2003) Adaptive iso-planar tool path generation for machining of free-form surfaces. Comput Aided Des 35:141–153

    Google Scholar 

  6. Suresh K, Yang DCH (1994) Constant scallop-height machining of free-form surfaces. J Eng Ind 116:253–259

    Google Scholar 

  7. Lo C-C (1999) Efficient cutter-path planning for five-axis surface machining with a flat-end cutter. Comput Aided Des 31:557–566

    MATH  Google Scholar 

  8. Lee E (2003) Contour offset approach to spiral toolpath generation with constant scallop height. Comput Aided Des 35:511–518

    Google Scholar 

  9. Marshall S, Griffiths JG (1994) A new cutter-path topology for milling machines. Comput Aided Des 26:204–214

    MATH  Google Scholar 

  10. Chiou CJ, Lee YS (2002) A machining potential field approach to tool path generation for multi-axis sculptured surface machining. Comput Aided Des 34:357–371

    MATH  Google Scholar 

  11. Liu W, Zhou L-S, An L-L (2012) Constant scallop-height tool path generation for three-axis discrete data points machining. Int J Adv Manuf Technol 63:137–146

    Google Scholar 

  12. Hu P, Chen L, Tang K (2017) Efficiency-optimal iso-planar tool path generation for five-axis finishing machining of freeform surfaces. Comput Aided Des 83:33–50

    Google Scholar 

  13. Zou Q, Zhang J, Deng B, Zhao J (2014) Iso-level tool path planning for free-form surfaces. Comput Aided Des 53:117–125

    MathSciNet  Google Scholar 

  14. Kim T, Sarma SE (2002) Toolpath generation along directions of maximum kinematic performance; a first cut at machine-optimal paths. Comput Aided Des 34:453–468

    Google Scholar 

  15. Hu P, Tang K (2016) Five-axis tool path generation based on machine-dependent potential field. Int J Comput Integr Manuf 29:636–651

    Google Scholar 

  16. Makhanov SS (1999) An application of the grid generation techniques to optimize a tool-path of industrial milling robots. J Comput Math Phys 39:1589–1600

    MATH  Google Scholar 

  17. Makhanov SS, Ivanenko SA (2010) Grid generation as a new concept of CNC-based part optimization. In: IMACS Word Congress on Computational Mathematics and Simulations, 21-25 August, Switzerland

  18. Bieterman MB, Sandstrom DR (2003) A curvilinear tool-path method for pocket machining. J Manuf Sci Eng 125:709–715

    Google Scholar 

  19. Chen L, Hu P, Luo M, Tang K (2018) Optimal interface surface determination for multi-axis freeform surface machining with both roughing and finishing. Chin J Aeronaut 31:370–384

    Google Scholar 

  20. Moodleah S, Bohez EJ, Makhanov SS (2016) Five-axis machining of STL surfaces by adaptive curvilinear toolpaths. Int J Prod Res 54:7296–7329

    Google Scholar 

  21. Kumazawa GH, Feng HY, Barakchi Fard MJ (2015) Preferred feed direction field: a new tool path generation method for efficient sculptured surface machining. Comput Aided Des 67–68:1–12

    Google Scholar 

  22. Liu X, Li Y, Ma S, Lee CH (2015) A tool path generation method for freeform surface machining by introducing the tensor property of machining strip width. Comput Aided Des 66:1–13

    Google Scholar 

  23. Teng Z, Feng H-Y, Azeem A (2006) Generating efficient tool paths from point cloud data via machining area segmentation. Int J Adv Manuf Technol 30:254–260

    Google Scholar 

  24. Chen ZC, Dong Z, Vickers GW (2003) Automated surface subdivision and tool path generation \( 3\frac{1}{2}\frac{1}{2} \)-axis CNC machining of sculptured parts. Comput Ind 50:319–331

    Google Scholar 

  25. Tuong NV, Pokorný P (2010) A practical approach for partitioning free-form surfaces. Int J Comput Integr Manuf 23:992–1001

    Google Scholar 

  26. Wang N, Tang K (2008) Five-axis tool path generation for a flat-end tool based on iso-conic partitioning. Comput Aided Des 40:1067–1079

    Google Scholar 

  27. Gordon WJ, Hall CA (1973) Construction of curvilinear co-ordinate systems and applications to mesh generation. Int J Numer Methods Eng 7:461–477

    MathSciNet  MATH  Google Scholar 

  28. The Siemens PLM software. https://www.plm.automation.siemens.com/en/docs/nx/11.shtml. Accessed 13 Aug 2018

  29. Ozturk E, Tunc LT, Budak E (2009) Investigation of lead and tilt angle effects in 5-axis ball-end milling processes. Int J Mach Tools Manuf 49:1053–1062

    Google Scholar 

  30. Makhanov SS, Anotaipaiboon W (2007) Advanced numerical methods to optimize cutting operations of five-axis milling machines. Springer

  31. Zhang K, Tang K (2014) An efficient greedy strategy for five-axis tool path generation on dense triangular mesh. Int J Adv Manuf Technol 74:1539–1550

    Google Scholar 

  32. Moodleah S, Makhanov SS (2015) Five-axis machining using a curvilinear tool path aligned with the direction of the maximum removal rate. Int J Adv Manuf Technol 80:65–90

    Google Scholar 

  33. Flusser J, Zitova B, Suk T (2009) Moments and moment invariants in pattern recognition. John Wiley & Sons

  34. Floater MS, Hormann K (2005) Surface parameterization: a tutorial and survey. In: Advances in multiresolution for geometric modelling. Springer, Berlin, Heidelberg, pp 157–186

  35. Sheffer A, Praun E, Rose K (2007) Mesh parameterization methods and their applications. Found Trends Comput Graph Vis 2:105–171

    MATH  Google Scholar 

  36. Tutte WT (1963) How to draw a graph. Proc Lond Math Soc 3:743–767

    MathSciNet  MATH  Google Scholar 

  37. Lévy B, Petitjean S, Ray N, Maillot J (2002) Least squares conformal maps for automatic texture atlas generation. ACM Trans Graph 21:362–371

    Google Scholar 

  38. Sheffer A, Lévy B, Mogilnitsky M, Bogomyakov A (2005) ABF++: fast and robust angle based flattening. ACM Trans Graph 24:311–330

    Google Scholar 

  39. Desbrun M, Meyer M, Alliez P (2002) Intrinsic parameterizations of surface meshes. Comput Graph Forum 21:209–218

    Google Scholar 

  40. Floater MS (1997) Parametrization and smooth approximation of surface triangulations. Comput Aided Geom Des 14:231–250

    MathSciNet  MATH  Google Scholar 

  41. Xu J, Sun Y, Wang S (2013) Tool path generation by offsetting curves on polyhedral surfaces based on mesh flattening. Int J Adv Manuf Technol 64:1201–1212

    Google Scholar 

  42. Yuwen S, Dongming G, Zhenyuan J, Haixia W (2006) Iso-parametric tool path generation from triangular meshes for free-form surface machining. Int J Adv Manuf Technol 28:721–726

    Google Scholar 

  43. Ren F, Sun Y, Guo D (2009) Combined reparameterization-based spiral toolpath generation for five-axis sculptured surface machining. Int J Adv Manuf Technol 40:760–768

    Google Scholar 

  44. Sun Y, Ren F, Zhu X, Guo D (2012) Contour-parallel offset machining for trimmed surfaces based on conformal mapping with free boundary. Int J Adv Manuf Technol 60:261–271

    Google Scholar 

  45. Shu CF, Jain RC (1993) Direct estimation and error analysis for oriented patterns. CVGIP Image Underst 58:383–398

    Google Scholar 

  46. Schlemmer M, Heringer M, Morr F, Hotz I, Bertram MH, Garth C, Kollmann W, Hamann B, Hagen H (2007) Moment invariants for the analysis of 2D flow fields. IEEE Trans Vis Comput Graph 13:1743–1750

    Google Scholar 

  47. Corpetti T, Mémin E, Pérez P (2003) Extraction of singular points from dense motion fields: An analytic approach. J Math Imaging Vis 19:175–198

    MathSciNet  MATH  Google Scholar 

  48. Koch S, Kasten J, Wiebel A, Scheuermann G, Hlawitschka M (2016) 2D Vector field approximation using linear neighborhoods. Vis Comput 32:1563–1578

    Google Scholar 

  49. Liu W, Ribeiro E (2012) Detecting singular patterns in 2D vector fields using weighted Laurent polynomial. Pattern Recogn 45:3912–3925

    Google Scholar 

  50. Gonzalez RC, Woods RE (2002) Digital image processing, 2nd edn. Prentice Hall

  51. Zhang Q, Yan H (2007) Fingerprint orientation field interpolation based on the constrained Delaunay triangulation. Int J Inf Syst Sci 3:438–452

    MATH  Google Scholar 

  52. Hu MK (1962) Visual pattern recognition by moment invariants. IRE Trans Inf Theory 8:179–187

    MATH  Google Scholar 

  53. Flusser J, Suk T (1994) Affine moment invariants: a new tool for character recognition. Pattern Recogn Lett 15:433–436

    Google Scholar 

  54. Campisi P, Neri A, Panci G, Scarano G (2004) Robust rotation-invariant texture classification using a model based approach. IEEE Trans Image Process 13:782–791

    Google Scholar 

  55. Srikanth C, Deekshatulu B, Rao C, Bhagvati C (2009) Classification and identification of Telugu Aksharas using moment invariants and C4.5 algorithms. Int J Comput Intell Res 5:225–232

    Google Scholar 

  56. Shu CF, Jain RC (1994) Vector field analysis for oriented patterns. IEEE Trans Pattern Anal Mach Intell 16:946–950

    Google Scholar 

  57. Lin R-S, Koren Y (1996) Efficient tool-path planning for machining free-form surfaces. J Eng Ind 118:20–28

    Google Scholar 

  58. Petitjean S (2002) A Survey of methods for recovering quadrics in triangle meshes. ACM Comput Surv 34:211–262

    Google Scholar 

  59. Goldfeather J, Interrante V (2004) A novel cubic-order algorithm for approximating principal direction vectors. ACM Trans Graph 23:45–63

    Google Scholar 

  60. Sun Y, Xu J, Jin C, Guo D (2016) Smooth tool path generation for 5-axis machining of triangular mesh surface with nonzero genus. Comput Aided Des 79:60–74

    Google Scholar 

  61. Chen X, Schmitt F (1992) Intrinsic surface properties from surface triangulation. In: European Conference on Computer Vision. Springer, Berlin, Heidelberg, pp 739–743

    Google Scholar 

  62. Martin RR (1998) Estimation of principal curvatures from range data. Int J Shape Model 4:99–109

    Google Scholar 

  63. Cohen-Steiner D, Morvan J-M (2003) Restricted Delaunay triangulations and normal cycle. In: Proceedings of the 19th Annual Symposium on Computational Geometry. ACM, pp 312–321

  64. Alliez P, Cohen-Steiner D, Devillers O, Lévy B, Desbrun M (2003) Anisotropic polygonal remeshing. In: ACM Transactions on Graphics (TOG). ACM, pp 485–493

  65. Rusinkiewicz S (2004) Estimating curvatures and their derivatives on triangle meshes. In: Proceedings. 2nd International Symposium on 3D Data Processing, Visualization and Transmission. 3DPVT 2004. IEEE, pp 486–493

  66. Cipolla R, Giblin P (2000) Visual motion of curves and surfaces. Cambridge University Press

  67. Telea A, Van Wijk JJ (1999) Simplified representation of vector fields. In: Proceedings Visualization ’99: Celebrating Ten Years. IEEE Computer Society Press, pp 35–42

  68. Wang Q-H, Zhang X-M, Li J-R, Tang C-S (2015) Prediction of toolpath redundancy for NC machining of free-form surfaces based on automatic recognition of steep-wall features. Int J Prod Res 53:4304–4316

    Google Scholar 

  69. Knorr EM, Ng RT (1999) Finding intensional knowledge of distance-based outliers. Proceedings 25th International Conference on Very Large Data Bases (VLDB 1999). pp 211–222

  70. ISO 4278: Geometrical Product Specifications (GPS) – Surface Texture (1997) Profile method terms. Defin Surf Texture Parameters 1st ed

  71. The Matlab Library. https://www.mathworks.com/products/matlab.html. Accessed 13 Aug 2018

  72. Makhanov SS, Munlin M (2007) Optimal sequencing of rotation angles for five-axis machining. Int J Adv Manuf Technol 35:41–54

    Google Scholar 

  73. Anotaipaiboon W, Makhanov SS (2011) Minimization of the kinematics error for five-axis machining. Comput Aided Des 43:1740–1757

    Google Scholar 

  74. Sun C, Wang Y, Huang N (2015) A new plunge milling tool path generation method for radial depth control using medial axis transform. Int J Adv Manuf Technol 76:1575–1582

    Google Scholar 

  75. Tournier C, Duc E (2005) Iso-scallop tool path generation in 5-axis milling. Int J Adv Manuf Technol 25:867–875

    Google Scholar 

  76. Wang Y, Yan C, Yang J, Lee CH (2017) Tool path generation algorithm based on covariant field theory and cost functional optimization and its applications in blade machining. Int J Adv Manuf Technol 90:927–943

    Google Scholar 

  77. Lin Z, Fu J, Yao X, Sun Y (2015) Improving machined surface textures in avoiding five-axis singularities considering tool orientation angle changes. Int J Mach Tools Manuf 98:41–49

    Google Scholar 

Download references

Funding

This research is supported by the Center of Excellence in Biomedical Engineering, Thammasat University, Thailand.

Author information

Authors and Affiliations

  1. Sirindhorn International Institute of Technology, Thammasat University, Rangsit, Pathum Thani, 12120, Thailand

    L. V. Dang & S. S. Makhanov

  2. National Institute of Metrology (Thailand) (NIMT), Khlong Luang, Pathum Thani, 12120, Thailand

    K. Vacharanukul

Authors
  1. L. V. Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Vacharanukul
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. S. Makhanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. S. Makhanov.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix. The kinematic error

Appendix. The kinematic error

To compare the accuracy of the proposed method, we evaluate the kinematic errors induced by non-linear trajectories of the machine for examples 3 and 4. Note that since the surfaces are approximated by the STL mesh, the testing includes several sources of error. The kinematic error depends on the accuracy of the approximation of the desired surface by the STL mesh (quality of triangulation). Furthermore, the procedure includes barycentric interpolation, transforming the triangulated surfaces into the Cartesian system. This transformation invokes certain numerical errors. If a CC curve on the real surface is characterized by a smooth variation of the rotation angles, the CC curve on the STL surface is piecewise linear. The corresponding normals and rotation angles could change abruptly, leading to substantial kinematic errors. A certain toolpath may pass through such singularities and generate large kinematic errors, whereas a different configuration may eventually avoid these errors. However, we have included these inaccuracies in our evaluation.

The toolpaths produced by NX have not been included since they have been generated from the solid models rather than from the STL. Although our experiments show that their kinematic errors are in the same range, the Appendix compares the errors obtained only on the STL meshes.

Furthermore, we exclude the kinematic error at the pole. Our approach is not to cross the peak (singularity) which inevitably leads to large kinematic errors [72, 73]. The trajectory loops invoked by the singularity can destroy the workpiece or even cause a global collision. Therefore, when the SRZ and CRZ reach the singularity point, they turn back, following the next track (see our demo in the Abstract). The CP cuts around the singularity as in Section 7. Finally, if the ZZ or ISOP path runs near the singularity, we withdraw the tool, turn it in the air, and continue the cut from the other side (plunging [74], see also Fig. 31). Note that there exist a number of efficient methods to avoid singularities [75,76,77]; however, this experiment compares a regular kinematic error that appears when cutting a relatively smooth surface characterized by the radial VFPD. Statistically, the singularity is an outlier, which must be excluded.

Fig. 31
figure 31

Practical treatment of the singularity. a ZZ path. b SRZ and CRZ path. c CP path

Full size image

Table 6 shows εmax and ε′ obtained by the ZZ, CP, SRZ, ISOP, and CRZ toolpaths. Each toolpath has a different length. Therefore, we measure the kinematic error per unit length of the forward step (0.5 mm). The size of the forward step was selected so that h ≈ 0.1.

Table 6 Kinematic error for surfaces in examples 3 and 4
Full size table

The results show that the maximum and average kinematic errors by the CRZ are in the same range as the errors produced by competing methods; however, the machining time is consistently shorter.

The kinematic error is graphically illustrated in Fig. 32a and b. The CP trajectories substantially deviate from the desired surface (in example 4) due to the large kinematic error, whereas the CRZ tool trajectories follow the low curvature and, therefore, are lying on the surface.

Fig. 32
figure 32

Tool trajectories on the twisted surface. a CP path. b CRZ path

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dang, L.V., Vacharanukul, K. & Makhanov, S.S. Compact radial zigzag for five-axis machining of STL surfaces. Int J Adv Manuf Technol 105, 1853–1882 (2019). https://doi.org/10.1007/s00170-019-03897-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00170-019-03897-7

Keywords

Search

Navigation