Skip to main content
SpringerLink
Log in

Propagation of assembly errors in multitasking machines by the homogenous matrix method

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

Abstract

In this work, a methodology for the assessment of the geometrical accuracy of a multiaxis machine, the type usually called multitasking machine, was fully developed. For this purpose, the well-known formulation by Denavit and Hartenberg was applied. Multitasking machines are derived from lathes in which turrets are substituted by milling spindles, or they are also derived from milling centres including a turning headstock on the machine bed. In both cases, they are kinematically much more complex than lathes or milling centres, where it is not easy to previously calculate the consequence of parallelism and squareness errors between elements and joints. In this paper, a radical new ‘multitasking’ machine model was studied. Literature shows a complete lack of values for this kind of machine. In the methodology presented here, errors were introduced as additional geometric parameters in each elemental transformation matrices, resulting in the real transformation matrix for a multitasking machine. Elemental errors and the way to introduce them into the Denavit and Hartenberg matrices are fully described. Some simulations are tested, giving a useful outcome regarding the sensitivity of the machine with respect to the feasible assembly errors or errors produced by light misalignments caused by the machine tool continuous use.

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

Similar content being viewed by others

References

  1. Lamikiz A, López de Lacalle LN (2009) Machine tools: a general view, machine tools for high performance machining. Springer, Berlin

    Google Scholar 

  2. Rahman M (2004) Modelling and measurement of multi-axis machine tools to improve positioning accuracy in a software way. Ph.D. thesis, University of Oulu, Finland

  3. Ramesh R, Mannan MA, Poo AN (2000) Error compensation in machine tools—a review: part I: geometric, cutting-force induced and fixture-dependent errors. Int J Mach Tool Manuf 40(9):1235–1256

    Article  Google Scholar 

  4. Slocum A (1992) Precision machine design. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  5. Masaomi Tsutsumi S, Saito A (2004) Identification of angular and positional deviations inherent to 5-axis machining centres with a tilting-rotary table by simultaneous four-axis control movements. Int J Mach Tool Manuf 44:1333–1342

    Article  Google Scholar 

  6. Srivasta A, Veldhuis MA (1995) Modelling geometric and thermal errors in five-axes CNC machine tool. Int J Mach Tool Manuf 35:1321–1337

    Article  Google Scholar 

  7. Nawara L, Kowalski J, Sladek J (1989) The influence of kinematic errors on the profile shapes by means of CMM. Ann CIRP 38(1):511–516

    Article  Google Scholar 

  8. Soons J, Theuws F, Schllekens P (1992) Modelling the errors of multi axis machines: a general methodology. Precis Eng 14(1):5–19

    Article  Google Scholar 

  9. Suh SH, Lee JJ, Kim SK (1998) Multiaxis machining with additional-axis NC system: theory and development. Int J Adv Manuf Technol 14:865–875

    Article  Google Scholar 

  10. Ferreira P, Liu C (1989) An analytical quadratic model for the geometric errors of a machine tool. J Manuf Syst 5(1):51–63

    Article  Google Scholar 

  11. Mou J, Liu CR (1995) A method for enhancing the accuracy of CNC machine tools for on-machine inspection. J Manuf Syst 11(4):29–237

    Google Scholar 

  12. Sørby K (2007) Inverse kinematics of five-axis machines near singular configurations. Int J Mach Tool Manuf 47(2):299–306

    Article  Google Scholar 

  13. Cho J, Cho M, Kim K (1994) Volumetric error analysis of a multi-axis machine tool machining a sculptured workpiece. Int J Prod Res 32(2):345–363

    Article  MATH  Google Scholar 

  14. Jha BK, Kumar A (2003) Analysis of geometric errors associated with five-axis machining centre in improving the quality of cam profile. Int J Mach Tool Manuf 43:629–636

    Article  Google Scholar 

  15. Lei WT, Hsu YY (2002) Accuracy test of five-axis CNC machine tool with 3D probe-ball. Part II: errors estimation. Int J Mach Tool Manuf 42:1163–1170

    Article  Google Scholar 

  16. Lin PD, Ehmann KF (1993) Direct volumetric error evaluation for multi-axis machines. Int J Mach Tool Manuf 33(5):675–693

    Article  Google Scholar 

  17. Lee RS, She CH (1997) Developing a postprocessor for three types of five-axis machine tools. Int J Adv Manuf Technol 13(9):658–665

    Article  Google Scholar 

  18. Lin YJ, Shen YJ (2003) Modelling of five-axis machine tool metrology models using the matrix summation approach. Int J Adv Manuf Technol 21(4):243–248

    Article  Google Scholar 

  19. Mahbubur RMD, Heikkala J, Lappalainen K, Karjalainen JA (1997) Positioning accuracy improvement in five-axis milling by post processing. Int J Mach Tool Manuf 37(2):223–236

    Article  Google Scholar 

  20. Remus T-FO, Feng H-Y (2004) Configuration analysis of five-axis machine tools using a generic kinematic model. Int J Mach Tool Manuf 44(11):1235–1243

    Article  Google Scholar 

  21. Moon SK, Moon Y-M, Kota S (2001) Screw theory based metrology for design and error compensation of machine tools. Proceedings of DETC’01: ASME 2001 Design Engineering Technical Conferences, Pittsburgh, Pennsylvania, 9–12 September

  22. Lee E, Suh S, Shon J (1998) A comprehensive method for calibration of volumetric positioning accuracy of CNC-machines. Int J Adv Manuf Technol 14:43–49

    Article  Google Scholar 

  23. Raksiri C, Parnichkun M (2004) Geometric and force errors compensation in a 3-axis CNC milling machine. Int J Mach Tool Manuf 44(12–13):1283–1291

    Article  Google Scholar 

  24. Chen G, Yuan J, Ni J (2001) A displacement measurement approach for machine geometric error assessment. Int J Mach Tool Manuf 41(1):149–161

    Article  Google Scholar 

  25. Sartori S, Zhang GX (1995) Geometric error measurement and compensation of machines. Ann CIRP 44(2):99–609

    Article  Google Scholar 

  26. Fan K, Lin J, Lu S, (1996) Measurement and compensation of thermal error on a machining centre. In: Proceedings of the fourth International conference on automatic technology, Hsinchu, Taiwan, 8–11 July 1996, pp. 261–268

  27. Salgado MA, López de Lacalle LN, Lamikiz A, Muñoa J, Sánchez JA (2005) Evaluation of the stiffness chain on the deflection of end-mills under cutting forces. Int J Mach Tool Manuf 45:727–739

    Article  Google Scholar 

  28. Uriarte L, Herrero A, Zatarain M, Santiso G, Lopéz de Lacalle LN, Lamikiz A, Albizuri J (2007) Error budget and stiffness chain assessment in a micromilling machine equipped with tools less than 0.3 mm in diameter. Precis Eng 31(n.1):1–12

    Article  Google Scholar 

  29. Armarego EJA, Deshpande NP (1991) Computerized end-milling force predictions with cutting models allowing for eccentricity and cutter deflections. Ann CIRP 40(1):25–29

    Article  Google Scholar 

  30. López de Lacalle LN, Lamikiz A, Sánchez JA, Salgado MA (2004) Effects of tool deflection in the high speed milling of inclined surfaces. Int J Adv Manuf Technol 24(9):621–631

    Article  Google Scholar 

  31. Kim GM, Kim BH, Chu CN (2003) Estimation of cutter deflection and form error in ball-end milling process. Int J Mach Tool Manuf 43:917–924

    Article  Google Scholar 

  32. López de Lacalle LN, Lamikiz A, Sánchez JA, Salgado MA (2007) Toolpath selection based on the minimum deflection cutting forces in the programming of complex surfaces milling. Int J Mach Tool Manuf 47(2):388–400

    Article  Google Scholar 

  33. Ikua BW, Tanaka H, Obata F, Sakamoto S (2001) Prediction of cutting forces and machining error in ball end milling of curved surfaces-I theoretical analysis. Precis Eng 25:266–273

    Article  Google Scholar 

  34. Donaldson RR (1980) Error budgets. In: Hocken RJ (ed) Technology of machine tool. Machine tool accuracy, vol.5, Ch 9. Machine Tool Task Force

  35. Walter MM, Norlund B, Koning RJ, Roblee JW (2002) Error budget as a design tool for ultra-precision diamond turning machines. In: ASPE’s 17th Annual Meeting, 20–25 October 2002

  36. Treib T, Matthias E (1987) Error budgeting—applied to the calculation and optimization of the volumetric error field of multiaxis systems. Ann CIRP 36(1):365–368

    Article  Google Scholar 

  37. Lee JH, Liu Y, Yang SH (2006) Accuracy improvement of miniaturized machine tool: geometric error modeling and compensation. Int J Mach Tool Manuf 46(12–13):1508–1516

    Article  Google Scholar 

  38. Huertas Talón JL, Boria DR, Muro LB, Gómez CL, Zurdo JJM, Calvo FV, Barace JJG (2011) Functional check test for high-speed milling centres of up to five axes. Int J Adv Manuf Technol 55(1–4):39–52

    Article  Google Scholar 

  39. Zargarbashi SHH, Mayer JRR (2006) Assessment of machine tool trunnion axis motion error, using magnetic double ball bar. Int J Mach Tool Manuf 46(14):1823–1834

    Article  Google Scholar 

  40. Lee KI, Lee DM, Kweon SH, Yang SH (2010) Geometric errors estimation of a rotary table using double ball-bar. J Korean Soc Precis Eng 27(11):98–105

    Google Scholar 

  41. López de Lacalle LN, Lamikiz A, Sanchez JA, Ocerin O, Maidagan E (2008) The Denavit and Hartenberg approach applied to evaluate the consequences in the tool tip position of geometrical errors in five-axis milling centres. International Journal of Advanced Manufacturing Technology 37(1–2):122–139

  42. ISO 230-1:1996: Test code for machine tools—part 1: geometric accuracy of machines operating under no-load or quasi-static conditions

  43. ISO 10791-2:2001: Test conditions for machining centres—part 2: geometric tests for machines with vertical spindle or universal heads with vertical primary rotary axis (vertical Z-axis)

  44. ISO 10791-1:1998: Test conditions for machining centres—part 1: geometric tests for machines with horizontal spindle and with accessory heads (horizontal Z-axis)

  45. Rocha CR, Tonetto CP, Dias A (2011) A comparison between the Denavit–Hartenberg and the screw-based method used in kinematic modeling of robot manipulators. Robot Comput Integr Manuf 27:723–728

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of the Basque Country, ETSI, C/Alameda de Urquijo s/n, 48013, Bilbao, Spain

    E. Díaz-Tena, U. Ugalde, L. N. López de Lacalle, A. de la Iglesia, A. Calleja & F. J. Campa

Authors
  1. E. Díaz-Tena
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. U. Ugalde
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. L. N. López de Lacalle
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. de la Iglesia
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Calleja
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. F. J. Campa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. N. López de Lacalle.

Appendices

Appendix A. Transformation matrix of the squareness error

$$ \begin{array}{*{20}c} {{}_{\mathrm{tcp}}^1{{\mathbf{T}}_{{r\_\mathrm{squareness}}}}=\left( {\begin{array}{*{20}c} {{}_{\mathrm{tcp}}^1\mathbf{R}} & {{}_{\mathrm{tcp}}^1\mathbf{d}} \\ \mathbf{0} & 1 \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1\mathbf{R}=\left( {\begin{array}{*{20}c} {1.10196\cdot {10^{-5 }}\cos (B)\cos \left( {{C_1}} \right)-\cos \left( {{C_1}} \right)\sin (B)} & {\sin ({C_1})} & {\cos (B)\cos \left( {{C_1}} \right)+1.10196\cdot {10^{-5 }}\cos \left( {{C_1}} \right)\sin (B)} \\ {1.10196\cdot {10^{-5 }}\cos (B)\sin \left( {{C_1}} \right)-\sin (B)\sin \left( {{C_1}} \right)} & {-\cos ({C_1})} & {1.10196\cdot {10^{-5 }}\sin (B)\sin \left( {{C_1}} \right)+\cos (B)\sin \left( {{C_1}} \right)} \\ {\cos (B)+1.10196\cdot {10^{-5 }}\sin (B)} & 0 & {-1.10196\cdot {10^{-5 }}\cos (B)+\sin (B)} \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1\mathbf{d}=\left( {\begin{array}{*{20}c} {-{X_1}\cos \left( {{C_1}} \right)+1.10196\cdot {10^{-5 }}\cdot 300\cos \left( {{C_1}} \right)-1,400\cos \left( {{C_1}} \right)+{L_{\mathrm{tool}}}\left( {\cos (B)\cos \left( {{C_1}} \right)+1.10196\cdot {10^{-5 }}\cos \left( {{C_1}} \right)\sin (B)} \right)-700\sin \left( {{C_1}} \right)+{Y_1}\sin \left( {{C_1}} \right)} \\ {700\cos \left( {{C_1}} \right)-{Y_1}\cos \left( {{C_1}} \right)+{L_{\mathrm{tool}}}\left( {1.10196\cdot {10^{-}}\sin (B)\sin \left( {{C_1}} \right)+\cos (B)\sin \left( {{C_1}} \right)} \right)-{X_1}\sin ({C_1})+1.10196\cdot {10^{-5 }}\cdot 300\sin ({C_1})-1,400\sin ({C_1})} \\ {-840+1.10196\cdot {10^{-5 }}{X_1}+{Z_1}+{L_{\mathrm{tool}}}\left( {-1.10196\cdot {10^{-5 }}\cos (B)+\sin (B)} \right)} \\ \end{array}} \right)} \\ \end{array} $$

Appendix B. Transformation matrix of the tool clamping error

$$ \begin{array}{*{20}c} {{}_{\mathrm{tcp}}^1{{\mathbf{T}}_{{\mathrm{r}\_\mathrm{clamping}}}}=\left( {\begin{array}{*{20}c} {{}_{\mathrm{tcp}}^1\mathbf{R}} & {{}_{\mathrm{tcp}}^1\mathbf{d}} \\ \mathbf{0} & 1 \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1\mathbf{R}=\left( {\begin{array}{*{20}c} {-\cos \left( {{C_1}} \right)\sin (B)} & {1.36732\cdot {10^{-5 }}\cos (B)\cos \left( {{C_1}} \right)+\sin \left( {{C_1}} \right)} & {\cos (B)\cos \left( {{C_1}} \right)-1.36732\cdot {10^{-5 }}\sin \left( {{C_1}} \right)} \\ {-\sin (B)\sin \left( {{C_1}} \right)} & {-\cos \left( {{C_1}} \right)+1.36732\cdot {10^{-5 }}\cos (B)\sin \left( {{C_1}} \right)} & {1.36732\cdot {10^{-5 }}\cos \left( {{C_1}} \right)+\cos (B)\sin \left( {C{}_1} \right)} \\ {\cos (B)} & {1.36732\cdot {10^{-5 }}\sin (B)} & {\sin (B)} \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1\mathbf{d}=\left( {\begin{array}{*{20}c} {-1,400\cos \left( {{C_1}} \right)-{X_1}\cos \left( {{C_1}} \right)+{L_{\mathrm{tool}}}\left( {\cos (B)\cos \left( {{C_1}} \right)-1.36732\cdot {10^{-5 }}\sin \left( {{C_1}} \right)} \right)-700\sin \left( {{C_1}} \right)+{Y_1}\sin \left( {{C_1}} \right)} \\ {700\cos \left( {{C_1}} \right)-{Y_1}\cos \left( {{C_1}} \right)-1,400\sin \left( {{C_1}} \right)-{X_1}\sin \left( {{C_1}} \right)+{L_{\mathrm{tool}}}\left( {\cos (B)\sin \left( {{C_1}} \right)+1.36732\cdot {10^{-5 }}} \right)} \\ {-840+{Z_1}+{L_{\mathrm{tool}}}\sin (B)} \\ \end{array}} \right)} \\ \end{array} $$

Appendix C. Transformation matrix of the combined squareness and clamping error

$$ \begin{array}{*{20}c} {{}_{\mathrm{tcp}}^1{{\mathbf{T}}_{{r\_\mathrm{squa}\_\mathrm{clam}}}}=\left( {\begin{array}{*{20}c} {{}_{\mathrm{tcp}}^1\mathbf{R}} & {{}_{\mathrm{tcp}}^1\mathbf{d}} \\ \mathbf{0} & 1 \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1\mathbf{R}=\left( {\begin{array}{*{20}c} {{}_{\mathrm{tcp}}^1{{\mathbf{R}}^1}} & {{}_{\mathrm{tcp}}^1{{\mathbf{R}}^2}} & {{}_{\mathrm{tcp}}^1{{\mathbf{R}}^3}} \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1{{\mathbf{R}}^1}=\left( {\begin{array}{*{20}c} {1.10196\cdot {10^{-5 }}\cos (B)\cos \left( {{C_1}} \right)-\cos \left( {{C_1}} \right)\sin (B)} \\ {1.10196\cdot {10^{-5 }}\cos (B)\sin \left( {{C_1}} \right)-\sin (B)\sin \left( {{C_1}} \right)} \\ {\cos (B)+1.10196\cdot {10^{-5 }}\sin (B)} \\ 0 \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1{{\mathbf{R}}^2}=\left( {\begin{array}{*{20}c} {1.36732\cdot {10^{-5 }}\cos (B)\cos \left( {{C_1}} \right)+1.50673\cdot {10^{-10 }}\cos \left( {{C_1}} \right)\sin (B)+\sin \left( {{C_1}} \right)} \\ {-\cos \left( {{C_1}} \right)+1.50673\cdot {10^{-10 }}\sin (B)\sin \left( {{C_1}} \right)+1.36732\cdot {10^{-5 }}\cos (B)\sin \left( {{C_1}} \right)} \\ {-1.50673\cdot {10^{-10 }}\cos (B)+1.36732\cdot {10^{-5 }}\sin (B)} \\ 0 \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1{{\mathbf{R}}^3}=\left( {\begin{array}{*{20}c} {\cos (B)\cos \left( {{C_1}} \right)+1.10196\cdot {10^{-5 }}\cos \left( {{C_1}} \right)\sin (B)-1.36732\cdot {10^{-5 }}\sin \left( {{C_1}} \right)} \\ {1.36732\cdot {10^{-5 }}\cos \left( {{C_1}} \right)+1.10196\cdot {10^{-5 }}\sin (B)\sin \left( {{C_1}} \right)+\cos (B)\sin \left( {{C_1}} \right)} \\ {-1.10196\cdot {10^{-5 }}\cos (B)\sin (B)} \\ 0 \\ \end{array}} \right)} \\ {{}_{\mathrm{tcp}}^1{{\mathbf{d}}^T}=\left( {\begin{array}{*{20}c} {-{X_1}\cos \left( {{C_1}} \right)+3.30588\cdot {10^{-3 }}\cos \left( {{C_1}} \right)-1,400\cos \left( {{C_1}} \right)+{L_{\mathrm{tool}}}\left( {\cos (B)\cos \left( {{C_1}} \right)+1.10196\cdot {10^{-5 }}\cos \left( {{C_1}} \right)\sin (B)-1.36732\cdot {10^{-5 }}\sin \left( {{C_1}} \right)} \right)-700\sin \left( {{C_1}} \right)+{Y_1}\sin \left( {{C_1}} \right)} \\ {700\cos \left( {{C_1}} \right)-{Y_1}\cos \left( {{C_1}} \right)+{L_{\mathrm{tool}}}\left( {1.36732\cdot {10^{-5 }}\cos \left( {{C_1}} \right)+1.10196\cdot {10^{-5 }}\sin (B)\sin \left( {{C_1}} \right)+\cos (B)\sin \left( {{C_1}} \right)} \right)-{X_1}\sin \left( {{C_1}} \right)+3.30588\cdot {10^{-3 }}\sin \left( {{C_1}} \right)-1,400\sin \left( {{C_1}} \right)} \\ {-840+1.10196\cdot {10^{-5 }}{X_1}+{Z_1}+{L_{\mathrm{tool}}}\left( {-1.10196\cdot {10^{-5 }}\cos (B)+\sin (B)} \right)} \\ \end{array}} \right)} \\ \end{array} $$

Appendix D. Transformation matrix of misalignment between both spindles

$$ \begin{array}{*{20}c} {{}_4^1{{\mathbf{T}}_{{r\_\mathrm{desalignment}}}}=\left( {\begin{array}{*{20}c} {{}_4^1\mathbf{R}} & {{}_4^1\mathbf{d}} \\ \mathbf{0} & 1 \\ \end{array}} \right)} \\ {{}_4^1\mathbf{R}=\left( {\begin{array}{*{20}c} {\cos \left( {{C_2}} \right)\left( {5.24\cdot {10^{-4 }}\sin \left( {{C_1}} \right)+\cos \left( {{C_1}} \right)} \right)-\sin \left( {{C_2}} \right)\left( {-5.24\cdot {10^{-4 }}\cos ({C_1})+\sin ({C_1})} \right)} & {-\cos ({C_2})(-5.24\cdot {10^{-4 }}\cos ({C_1})+\sin ({C_1}))-\sin ({C_2})(\cos ({C_1})+5.24\cdot {10^{-4 }}\sin ({C_1}))} & 0 \\ {-\cos \left( {{C_2}} \right)\left( {5.24\cdot {10^{-4 }}\cos \left( {{C_1}} \right)-\sin \left( {{C_1}} \right)} \right)+\sin \left( {{C_2}} \right)\left( {\cos \left( {{C_1}} \right)+5.24\cdot {10^{-4 }}\sin \left( {{C_1}} \right)} \right)} & {\cos ({C_2})(\cos ({C_1})+5.24\cdot {10^{-4 }}\sin ({C_1}))+\sin ({C_2})(5.24\cdot {10^{-4 }}\cos ({C_1})+\sin ({C_1}))} & 0 \\ 0 & 0 & 1 \\ \end{array}} \right)} \\ {{}_4^1\mathbf{d}=\left( {\begin{array}{*{20}c} {-1,400\cos \left( {{C_1}} \right)+1,400\cos \left( {{C_2}} \right)\left( {\cos \left( {{C_1}} \right)+5.24\cdot {10^{-4 }}\sin \left( {{C_1}} \right)} \right)-1,400\sin \left( {{C_2}} \right)\left( {-5.24\cdot {10^{-4 }}\cos \left( {{C_1}} \right)+\sin \left( {{C_1}} \right)} \right)} \\ {-1,400\cos \left( {{C_2}} \right)\left( {5.24\cdot {10^{-4 }}\cos \left( {{C_1}} \right)-\sin \left( {{C_1}} \right)} \right)-1,400\sin \left( {{C_1}} \right)+1,400\sin \left( {{C_2}} \right)\left( {\cos \left( {{C_1}} \right)+5.24\cdot {10^{-4 }}\sin \left( {{C_1}} \right)} \right)} \\ {A-2,280} \\ \end{array}} \right)} \\ \end{array} $$

Rights and permissions

Reprints and permissions

About this article

Cite this article

Díaz-Tena, E., Ugalde, U., López de Lacalle, L.N. et al. Propagation of assembly errors in multitasking machines by the homogenous matrix method. Int J Adv Manuf Technol 68, 149–164 (2013). https://doi.org/10.1007/s00170-012-4715-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00170-012-4715-x

Keywords

Search

Navigation