工程科学与技术   2021, Vol. 53 Issue (3): 159-165

1. 中国工程物理研究院 机械制造工艺研究所，四川 绵阳 621900;
2. 国家机床产品质量监督检验中心（四川），四川 成都 610200;
3. 南京理工大学 发射动力学研究所，江苏 南京 210094

Measurement Method of Geometric Error of Coordinate Measuring Machine Using Laser Tracer
HAN Lin1,2, MI Liang1,2, LIU Xingbao1,2,3, TENG Qiang1,2, TANG Qiang1,2, XIA Yangqiu1,2
1. Inst. of Machinery Manufacturing Technol., China Academy of Eng. Physics, Mianyang 621900, China;
2. National Machine Tool Production Quality Supervision Testing Center (Sichuan), Chengdu 610200, China;
3. Inst. of Launch Dynamics, Nanjing Univ. of Sci. and Technol., Nanjing 210094, China
Abstract: Geometric error is an important error source of coordinate measuring machine (CMM). The geometric error measurement methods currently used have the problems of low efficiency and low accuracy, which seriously deteriorate the further improvement of the performance of CMM. Hence, a novel geometric error measurement and direct separation method of CMM using laser tracer was proposed. Firstly, the geometric error model was established based on the multi-body system theorem and homogeneous coordinate transformations. Furthermore, the mathematical model of geometric error measurement using laser tracer was formulated. And the geometric errors were directly separated based on the constraints of geometric error characteristics combining with the Levenberg–Marquardt method. Finally, the experiment of geometric error measurement of a CMM using a laser tracer and comparison experiments of single axis positioning error and volumetric positioning error were carried out. The results showed that the positioning errors ofX, Y and Z axis are –27.06, 43.75 and 36.76 μm, respectively, and the squareness error between X and Y axis was –82.89 μrad, which were the main error sources. The maximum volumetric error predicted of CMM was 74.64 μm, which was located in the limit area of the measurement volume. The identification results of geometric errors compared with the results measured by laser interferometer showed that, the maximum difference of three-axis positioning errors was 7.43 μm and the maximum difference of body diagonal positioning errors predicted was 10.51 μm. Compared with other laser tracking geometric error measurement method, X and Y axis volumetric positioning errors measured by this proposed method were the closest to the results measured by laser interferometer with the maximum difference of 5.3 μm which verified the effectiveness of this method. In addition, the 17 geometric errors of CMM could be measured within 2 hours by the method, which is of high speed and accuracy, and has a large application in the field of accuracy measurement of CMM and CNC machine tool.
Key words: geometric error    laser tracer    constraints    volumetric positioning error

1 三坐标测量机几何误差建模

 图1 Y轴6项几何误差项 Fig. 1 Six geometric errors of Y axis

 图2 垂直度误差分布 Fig. 2 Distribution of squareness errors

 $\left\{\!\!\!\! \begin{array}{l} {D_{{X}}}{{ = }}{E_{{{XX}}}} + {E_{{{XY}}}} + {E_{{{X}}{\textit{Z}}}} + {\textit{Z}}\left( {{E_{{{BO}}{\textit{Z}}}}{{ + }}{E_{{{{\rm{BX}}}}}}{{ + }}{E_{{{{\rm{BY}}}}}}} \right),\\ {D_{{Y}}}{{ = }}{E_{{{YX}}}} + {E_{{{YY}}}} + {E_{{{Y}}{\textit{Z}}}} - {\textit{Z}}\left( {{E_{{{AO}}{\textit{Z}}}}{{ + }}{E_{{{{\rm{AX}}}}}}{{ + }}{E_{{{{\rm{AY}}}}}}} \right) + \\ \;\;\;\;\;\;\;\;\;X\left( {{E_{{{COY}}}} + {E_{{{{\rm{CY}}}}}}} \right),\\ {D_{\textit{Z}}}{{ = }}{E_{{{YX}}}} + {E_{{{YY}}}} + {E_{{{Y}}{\textit{Z}}}} - X{E_{{{{\rm{BY}}}}}} \end{array} \right.$ (1)

 ${{E}} = {{C}} \cdot {{{P}}_{{C}}}$ (2)

 ${D_{\rm{V}}}{\rm{ = }}\sqrt {{D_{{X}}}^{\rm{2}}{{ + }}{D_{{Y}}}^{\rm{2}} + {D_{\textit{Z}}}^{\rm{2}}}$ (3)

2 跟踪仪几何误差测量与分离原理

 图3 激光跟踪仪检测基本原理 Fig. 3 Measurement principle of laser tracer

 $\left\| {{{{Q}}_i} - \left( {{{{P}}_j} + {{{E}}_j}} \right)} \right\| = {l_{ij}} + {l_i}$ (4)

 $F{\rm{ = }}{\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\left[ {\left\| {{{{Q}}_i} - \left( {{{{P}}_j} + {{{E}}_j}} \right)} \right\| - \left( {{l_{ij}} + {l_i}} \right)} \right]^2} } }$ (5)

 ${{J}} \cdot {{d}} = {{{F}}_{\rm{d}}}$ (6)

 ${{B}} \cdot {{d}} = {\bf{0}}$ (7)

 ${{M}} \cdot {{d}} = {{G}}$ (8)

 ${{d}} = {\left( {{{{M}}^{\rm{T}}}{{M}} + \mu {{I}}} \right)^{ - 1}}{{{M}}^{\rm{T}}}{{G}}$ (9)

3 试验验证 3.1 激光跟踪仪几何误差检测试验

 图4 激光跟踪仪检测几何误差示意图 Fig. 4 Schematic diagram of geometric error measurement of CMM using laser tracer

 图5 三坐标测量机空间位置误差预测分布 Fig. 5 Volumetric error distribution of coordinate measuring machine

3.2 单项几何误差对比试验验证

 图6 激光干涉仪检测定位误差 Fig. 6 Positioning error measurement using laser interferometer

 图7 X轴定位误差对比 Fig. 7 Comparison of positioning error of X axis

 图8 Y轴定位误差对比 Fig. 8 Comparison of positioning error of Y axis

 图9 Z轴定位误差对比 Fig. 9 Comparison of positioning error of Z axis

3.3 空间位置误差对比试验验证

 图10 不同分离方法的空间误差对比结果 Fig. 10 Volumetric error comparison results of different measurement methods

 图11 激光干涉仪检测PPP体对角线定位误差现场 Fig. 11 Positioning error measurement of PPP body diagonal using laser interferometer

 图12 体对角线预测与实测空间位置误差对比 Fig. 12 Volumetric positioning error comparison of measured and predicted values of body diagonals

4 结　论

1）建立了三坐标测量机的几何误差模型和激光跟踪仪检测原理模型，并利用几何误差约束条件，实现了几何误差的直接分离。该方法不需要进行激光跟踪仪位置的自标定，简化了求解步骤，提高了几何误差测量精度。

2）在某三坐标测量机上进行了几何误差检测试验和空间误差预测。与激光干涉仪检测结果相比，三轴定位误差最大相差7.43 μm，体对角线定位误差最大相差10.51 μm，验证了几何误差分离方法和空间位置误差预测模型的正确性。该方法与常用的激光跟踪仪检测几何误差方法相比有较大优势，能够更为准确地反映空间误差分布。三坐标测量机预测空间误差最大值为74.64 μm，且位于测量空间极限区域位置。

3）应用激光跟踪仪的三坐标测量机几何误差检测总耗时2 h，比传统的激光干涉仪对X轴、Y轴和Z轴定位精度和4条体对角线位置精度的检测效率高；而且便于三坐标测量机空间位置误差补偿，在三坐标测量机和数控机床精度检测领域有较大的应用空间。

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