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

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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.

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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
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  2. U. Ugalde
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  3. L. N. López de Lacalle
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  4. A. de la Iglesia
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  5. A. Calleja
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  6. F. J. Campa
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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

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

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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

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