Spatial Expression of Assembly Geometric Errors for Multi-axis Machine Tool Based on Kinematic Jacobian-Torsor Model

Tian, Ang; Liu, Shun; Chen, Kun; Mo, Wei; Jin, Sun

doi:10.1186/s10033-023-00870-0

Chinese Journal of Mechanical Engineering

Table 4 Jacobian matrixes and deviation torsors

From: Spatial Expression of Assembly Geometric Errors for Multi-axis Machine Tool Based on Kinematic Jacobian-Torsor Model

No.	FEs	Jacobian matrixes	Tolerance torsors
1	IFE1	\(\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & {548} \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & {{ - }548} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\)	\(\left[ \begin{gathered} 0 \\ 0 \\ \left( {0,0.01} \right) \\ \left( {0,0.01/800} \right) \\ \left( {0,0.01/500} \right) \\ 0 \\ \end{gathered} \right]\)
2	CFE1	\(\left[ {\begin{array}{{20}c} {\left[ {\begin{array}{{20}c} {\cos \alpha } & 0 & {\sin \alpha } \\ 0 & 1 & 0 \\ { - \sin \alpha } & 0 & {\cos \alpha } \\ \end{array} } \right]} & {\begin{array}{{20}c} {\left[ {\begin{array}{{20}c} 0 & {220} & {292} \\ { - 220} & 0 & 0 \\ { - 292} & 0 & 0 \\ \end{array} } \right]} & \times & {\left[ {\begin{array}{{20}c} {\cos \alpha } & 0 & {\sin \alpha } \\ 0 & 1 & 0 \\ { - \sin \alpha } & 0 & {\cos \alpha } \\ \end{array} } \right]} \\ \end{array} } \\ {\left[ {\begin{array}{{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \right]} & {\left[ {\begin{array}{*{20}c} {\cos \alpha } & 0 & {\sin \alpha } \\ 0 & 1 & 0 \\ { - \sin \alpha } & 0 & {\cos \alpha } \\ \end{array} } \right]} \\ \end{array} } \right]\)	\(\left[ \begin{gathered} \left( {0,0.002} \right) \\ 0 \\ \left( {0,0.002} \right) \\ \left( {0,0.002/340} \right) \\ 0 \\ \left( {0,0.002/340} \right) \\ \end{gathered} \right]\)
3	IFE2	\(\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & {{ - }416} & {{ - }40} \\ 0 & 1 & 0 & {416} & 0 & 0 \\ 0 & 0 & 1 & {40} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\)	\(\left[ \begin{gathered} 0 \\ \left( {0,0.011} \right) \\ 0 \\ \left( {0,0.011/86} \right) \\ 0 \\ \left( {0,0.011/154} \right) \\ \end{gathered} \right]\)
4	CFE2	\(\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & { - 161} & 0 \\ 0 & 1 & 0 & {161} & 0 & { - z + 300} \\ 0 & 0 & 1 & 0 & {z - 300} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\)	\(\left[ \begin{gathered} \left( {0,2T_{stz} + \frac{2z}{{L_{Z} }}\left( {T_{z} - T_{stz} } \right)} \right) \\ 0 \\ 0 \\ 0 \\ \left( {0,\left[ {T_{stz} + \frac{z}{{L_{Z} }}\left( {T_{z} - T_{stz} } \right)} \right]/43} \right) \\ \left( {0,\left[ {T_{stz} + \frac{z}{{L_{Z} }}\left( {T_{z} - T_{stz} } \right)} \right]/77} \right) \\ \end{gathered} \right]\)
5	IFE3	\(\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & { - 161} & { - x - 280} \\ 0 & 1 & 0 & {161} & 0 & 0 \\ 0 & 0 & 1 & {x + 280} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\)	\(\left[ \begin{gathered} \left( {0,2\left[ {T_{stx} + \frac{x}{{L_{x} }}\left( {T_{x} - T_{stx} } \right)} \right]} \right) \\ 0 \\ 0 \\ 0 \\ \left( {0,\left[ {T_{stx} + \frac{x}{{L_{x} }}\left( {T_{x} - T_{stx} } \right)} \right]/43} \right) \\ \left( {0,\left[ {T_{stx} + \frac{x}{{L_{x} }}\left( {T_{x} - T_{stx} } \right)} \right]/77} \right) \\ \end{gathered} \right]\)
6	IFE4	\(\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & {y + 283} & 0 \\ 0 & 0 & 0 & { - y - 283} & 0 & {276} \\ 0 & 0 & 1 & 0 & { - 276} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\)	\(\left[ \begin{gathered} \left( {0,\left[ {T_{sty} + \frac{y}{{L_{y} }}\left( {T_{y} - T_{sty} } \right)} \right]} \right) \\ 0 \\ 0 \\ \left( {0,\left[ {T_{sty} + \frac{y}{{L_{y} }}\left( {T_{y} - T_{sty} } \right)} \right]/43} \right) \\ 0 \\ \left( {0,\left[ {T_{sty} + \frac{y}{{L_{y} }}\left( {T_{y} - T_{sty} } \right)} \right]/77} \right) \\ \end{gathered} \right]\)
7	IFE5	\(\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & {250} \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & {{ - }250} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\)	\(\left[ \begin{gathered} 0 \\ \left( {0,0.002} \right) \\ \left( {0,0.002} \right) \\ 0 \\ \left( {0,0.002/153} \right) \\ \left( {0,0.002/153} \right) \\ \end{gathered} \right]\)
8	IFE6	\(\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & {580} \\ 0 & 0 & 1 & 0 & {{ - }580} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right]\)	\(\left[ \begin{gathered} 0 \\ \left( {0,0.002} \right) \\ \left( {0,0.002} \right) \\ 0 \\ \left( {0,0.002/411} \right) \\ \left( {0,0.002/411} \right) \\ \end{gathered} \right]\)

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