Calculation of Effect Ratio of 21 Geometric Errors and Detection of Surface Figure Error

Abstract

Traditionally, a machine tool is regarded as a rigid-multi body system, and it is studied by integrated geometric error modeling. Considering a three axis-machine, this study introduced the ratio and effects of the geometric errors on the surface figure error. First, based on synchronous iteration location, a surface matching model was employed to assess the coordinate measurement result. Subsequently, considering all the geometric error functions having the same parameters, the effect ratio and the surface figure error of a single geometric error were obtained. Concurrently, based on the effect ratio and surface figure error results, the main geometric errors were obtained, such as EZX, EZY, EAX, and COY. Moreover, the proposed method and the main errors were experimentally verified. By compensating the main errors, the sphere accuracy was improved by 53.4% and the flat accuracy by 70.0%. Additionally, the method proposed in this paper could be utilized to detect the surface figure error.

Appendix

For a three-axis machine, four relative moving parts and four coordinate frames are built, as shown in Fig. 1. At the initial position, the four origins were coincident. To obtain P_error (x’, y’, z’), first, the position of the stylus tip, P, in coordinate frame O₃X₃Y₃Z₃ is given by.

$${\varvec{O}}_{3} {\varvec{P}} = \left( {x_{p} ,y_{p} ,z_{p} } \right).$$

(16)

Subsequently, there were two steps to obtain the position of stylus tip P in the coordinate frame, O₂X₂Y₂Z₂. In the first step, the three axes of O₃X₃Y₃Z₃ were rotated parallel to the axes of O₂X₂Y₂Z₂, and the second step was that origin O₃ moved to coincide with origin O₂. Therefore, the position of P in the coordinate frame, O₂X₂Y₂Z_2, was successively.

$$\begin{aligned} &{\varvec{O}}_{2} {\varvec{P}} = {\varvec{O}}_{2} {\varvec{O}}_{3} + {\varvec{R}}^{ - 1} \left( {\varvec{z}} \right) \times {\varvec{O}}_{3} {\varvec{P}}, \hfill \\ &where \begin{array}{*{20}c} {\begin{array}{*{20}c} {{\varvec{O}}_{2} {\varvec{O}}_{3} = Z = \left[ {\begin{array}{*{20}c} {EXZ - zBOZ} \\ {EYZ - zAOZ} \\ {z + EZZ} \\ \end{array} } \right]} \\ &{R\left( z \right) = \left[ {\begin{array}{*{20}c} 1 & {ECZ} & { - EBZ} \\ { - ECZ} & 1 & {EAZ} \\ {EBZ} & { - EAZ} & 1 \\ \end{array} } \right],} \\ \end{array} } & \\ \end{array} \hfill \\ \end{aligned}$$

(17)

where matrix ${\mathrm{O}}_{2}{\mathrm{O}}_{3}$ ${\mathrm{O}}_{2}{\mathrm{O}}_{3}$ O₂O₃ is the linear movement matrix along the Z axis, and matrix R(z)$\mathrm{R}(\mathrm{z})$ $\mathrm{R}(\mathrm{z})$ is the rotation matrix.

Similarly, to obtain the position of P in the coordinate frame, O₁X₁Y₁Z₁, the three axes of O₂X₂Y₂Z₂ were rotated parallel to the axes of O₁X₁Y₁Z₁, and the origin, O₂, was moved to coincide with origin O₁ along with the Y axis. Therefore, the position of P in the coordinate frame, O₁X₁Y₁Z_1, was successively

$$\begin{aligned} & {\varvec{O}}_{1} {\varvec{P}} = {\varvec{O}}_{1} {\varvec{O}}_{2} + {\varvec{R}}^{{\left( { - 1} \right)}} \left( {\varvec{y}} \right) \times {\varvec{O}}_{2} \user2{P,} \\ & where\, O_{1} O_{2} = Y = \left[ {\begin{array}{*{20}c} {EXY - yCOY} \\ {y + EYY} \\ {EZY} \\ \end{array} } \right] \;and \\ & R\left( y \right) = \left[ {\begin{array}{*{20}c} 1 & {ECY} & { - EBY} \\ { - ECY} & 1 & {EAY} \\ {EBY} & { - EAY} & 1 \\ \end{array} } \right]. \\ \end{aligned}$$

(18)

The position of P in the coordinate frame, OXYZ, was successively.

$$\begin{aligned} & {\varvec{OP}} = {\varvec{OO}}_{1} + {\varvec{R}}^{{\left( { - 1} \right)}} \left( {\varvec{x}} \right) \times {\varvec{O}}_{1} {\varvec{P}}, \\ & where\; OO_{1} = X = \left[ {\begin{array}{*{20}c} {x + EXX} \\ {EYX} \\ {EZX} \\ \end{array} } \right]and \\ & {\varvec{R}}\left( x \right) = \left[ {\begin{array}{*{20}c} 1 & {ECX} & { - EBX} \\ { - ECX} & 1 & {EAX} \\ {EBX} & { - EAX} & 1 \\ \end{array} } \right]. \\ \end{aligned}$$

(19)

In fact, the position of P in the coordinate frame OXYZ is P_error (x’, y’, z’). The summation of (16)–(19) yields Eq. (1).