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采用光刻工艺制做一块狭缝目标板,目标板形状为圆形,直径为30 mm,基底材料为K9玻璃,具体布局见图4,狭缝目标板中心制作有十字线,十字竖线一侧平行地刻划有三条狭缝,十字线和狭缝透光,其余部分均不透光,
$ {N_1} $ 、$ {N_2} $ 、$ {N_3} $ 分别为狭缝1、狭缝2、狭缝3与十字横线的交点,$ O $ 为十字线交点。三条狭缝距离十字竖线的距离分别用$ O{N_1} $ 、$ O{N_2} $ 、$ O{N_3} $ 表示,使用德国马尔公司制造的10106 HA型三坐标测量仪对狭缝到十字竖线的距离进行测量,测量结果由小到大依次为2.9245、6.5432、9.4979 mm。将制做好的目标板安装到检测架上平行光管的焦面位置处,在地基环上架设瑞士徕卡公司制造的TC2003型全站仪,如图5所示,使用全站仪首先瞄准目标板经平行光管出射的十字线交叉点
$ O $ ,记方位角为$ \angle {A_o} $ ,俯仰角为$ \angle {E_o} $ ,其次瞄准十字横线与狭缝的交点,记方位角为$ \angle A $ ,俯仰角为$ \angle E $ 。转动目标板并通过全站仪目镜进行观察,使得线段
$ {N_1}{N_3} $ 经平行光管出射后分别成像于全站仪分划板的第一象限、第二象限、第三象限、第四象限及与分划板的竖丝和横丝平行,按照上述测试流程依次瞄准交点$ O $ 、$ {N_1} $ 、$ {N_2} $ 、$ {N_3} $ 并采集对应的方位角和俯仰角,过程数据如表1所示。Points to be measured Imaging area of ${N_1},{N_3}$ First
quadrantSecond
quadrantThird
quadrantFourth
quadrantCoincidence with the
horizontal wireCoincidence with the
vertical wire$ {N_1} $ $ \angle {A_o} $ 0 0 0 0 0 0 $ \angle {E_o} $ 1.044524 1.044598 1.044626 1.044598 1.044621 1.044580 $ \angle A $ 0.000625 0.000568 0.002066 0.002547 0.002935 0 $ \angle E $ 1.045966 1.046047 1.043576 1.043868 1.044611 1.046057 $ {N_2} $ $ \angle {A_o} $ 0 0 0 0 0 0 $ \angle {E_o} $ 1.044524 1.044598 1.044626 1.044598 1.044621 1.044580 $ \angle A $ 0.001348 0.001277 0.004519 0.005625 0.006489 0 $ \angle E $ 1.047709 1.047788 1.042299 1.042988 1.044615 1.047836 $ {N_3} $ $ \angle {A_o} $ 0 0 0 0 0 0 $ \angle {E_o} $ 1.044524 1.044598 1.044626 1.044598 1.044621 1.044580 $ \angle A $ 0.002003 0.001749 0.006530 0.008002 0.009276 0 $ \angle E $ 1.049068 1.049175 1.041322 1.042263 1.044620 1.049236 Table 1. Process data when the N1, N3 is located in different quadrants and parallel to the horizontal and vertical wire of the reticle (Unit: rad)
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将线段
${N_1}、{N_3}$ 分别位于不同象限及与分划板横丝平行时所采集的$ \angle {E_o} $ ,$ \angle A $ (另$ \angle {A_o} = 0 $ ,则$ \angle A - \angle {A_o} = \angle A $ ),$ \angle E $ 代入公式(5)可解得修正投影误差后的平行光管焦距,最终解算的焦距值如表2所示。由表2可知,经投影误差修正后由同名测试点解算的焦距最大差值为4.41 mm,测试点$ {N_1} $ 的焦距平均值为1982.19 mm,波动值为4.40 mm,测试点$ {N_2} $ 的焦距平均值为2009.94 mm,波动值为3.39 mm,测试点$ {N_3} $ 的焦距平均值为2039.15 mm,波动值为3.79 mm。各测试点焦距平均值出现较大差异是由畸变引入的原理性误差所致。平行光管半视场为15 mm,则$ {N_1} $ 点位于0.19视场位置处,$ {N_2} $ 点位于0.44视场位置处,$ {N_3} $ 点位于0.63视场位置处,由平行光管的光学设计结果可知0.19视场处系统畸变为0.02%,该值可忽略,0.44及0.63视场处系统畸变分别为−1.3%和−2.8%,根据畸变设计结果对$ O{N_2} $ 、$ O{N_3} $ 的长度进行修正重新代入公式(5)计算焦距,如表3所示,测试点$ {N_2} $ 的焦距平均值变为1983.81 mm、测试点$ {N_3} $ 的焦距平均值变为1982.06 mm,修正畸变后,测试点$ {N_2} $ 和$ {N_3} $ 的焦距平均值与测试点$ {N_1} $ 的焦距平均差值分别为1.62 mm和−0.13 mm。Points to be measured Imaging area of ${N_1}, {N_3}$ First quadrant Second quadrant Third quadrant Fourth quadrant Coincidence with the horizontal wire $ {N_1} $ 1981.77 1980.37 1980.24 1984.65 1983.93 $ {N_2} $ 2009.70 2011.13 2011.01 2010.13 2007.73 $ {N_3} $ 2041.14 2038.21 2037.35 2040.33 2038.74 Table 2. Focal length when the N1, N3 is located in different quadrants and parallel to the horizontal wire of the reticle after correcting the projection error (Unit: mm)
Points to be measured The imaging area of ${N_1}, {N_3}$ First quadrant Second quadrant Third quadrant Fourth quadrant Coincidence with the horizontal wire $ {N_1} $ 1981.77 1980.37 1980.24 1984.65 1983.93 $ {N_2} $ 1983.58 1984.98 1984.86 1984.00 1981.63 $ {N_3} $ 1983.99 1981.14 1980.31 1983.20 1981.66 Table 3. Focal length of different testing points after correcting the projection error and distortion (Unit: mm)
为了解耦投影误差并获得焦距的真值,根据全站仪工作原理,转动水平轴轴系进行俯仰方向的角度测量时不存在投影误差,因此可以利用表1中线段
${N_1},{N_3}$ 与分划板竖丝平行时的过程数据解算焦距并作为真值来验证模型的正确性,需要注意的是,公式中的$ \omega $ 需以$ 2\left| {\angle {E_o} - \angle E} \right| $ 代替,最终解算的焦距值如表4所示。Points to be measured Coincidence with the vertical wire $ {N_1} $ 1980.03 $ {N_2} $ 2009.58 $ {N_3} $ 2039.91 Table 4. Focal length when the N1, N3 is parallel to the vertical wire of the reticle (Unit: mm)
同样由于畸变的存在,各测试点解算的焦距值差异较大,测试点
$ {N_1} $ 、$ {N_2} $ 、$ {N_3} $ 的焦距值分别为1980.03、2009.58、2039.91 mm,进行畸变修正后对应的焦距值分别为1980.03、1983.45、1982.79 mm,波动值为3.43 mm,各测试点焦距平均值即真值为1982.09 mm。线段${N_1}、{N_3}$ 位于不同象限及与分划板横丝平行时的各测试点在修正投影误差和畸变后焦距平均值为1982.69 mm,此值与真值差值为0.6 mm,验证了模型的正确性及鲁棒性。修正畸变后将线段
${N_1}、{N_3}$ 与分划板横丝平行时所采集的$ \angle {A_o} $ 、$ \angle {E_o} $ 、$ \angle A $ 及$ \angle E $ 代入公式(2)中可解得传统意义下的焦距,需要注意的是公式中的$ \omega $ 需以$ 2\left| {\angle A - \angle {A_o}} \right| $ 代替,最终解算的焦距值如表5所示。Points to be measured Coincidence with the horizontal wire $ {N_1} $ 996.42 $ {N_2} $ 995.23 $ {N_3} $ 995.22 Table 5. Focal length when the N1, N3 is parallel to the horizontal wire of the reticle with correcting the distortion (Unit: mm)
由表5可知,修正畸变但未修正投影误差的情况下由测试点
$ {N_1} $ 、$ {N_2} $ 、$ {N_3} $ 解算的焦距结果分别为996.42、995.23、995.22 mm,焦距平均值与真值之差为986.47 mm,相对误差为50.2%。 -
令公式(5)中的
则焦距f的表达式可变换为:
$\angle A$ 、$\angle E$ 、$\angle {E_\textit{o}}$ 及$O{N_1}$ 对焦距的灵敏度系数分别如下式所示:设
$\angle A$ 、$\angle E$ 、$\angle {E_o}$ 及$O{N_1}$ 的测量标准不确定度分别为$ {\mu _A} $ 、${\;\mu _E}$ 、$ {\;\mu _{{E_o}}} $ 、$ {\;\mu _{O{N_1}}} $ 且相互独立。则线段${N_1}、{N_3}$ 位于不同象限及与分划板横丝平行时焦距测量结果的不确定度应是所有不确定度分量的合成,用合成标准不确定度${\;\mu _{{c}}}$ 来表示,如下式所示:线段
${N_1}、{N_3}$ 与分划板竖丝平行时焦距的解算公式可表示为:合成标准不确定度用
${\;\mu _{{c}}}^\prime$ 来表示,如下式所示:通过查验全站仪和三坐标测量仪计量证书可知
$ \;{\mu }_{A}\text{=0}\text{.3}″ $ ,$ \;{\mu }_{E}\text{=}{\mu }_{{E}_{o}}\text{=}0.4″ $ ,$\; {\mu _{O{N_1}}}{\text{ = }}1.0\;{\text{μ}} {\rm{m}}$ 。修正畸变后将采集的过程数据及相应的测量标准不确定度代入公式(11)和公式(13)中,可得各测试点对应焦距的合成标准不确定度,取包含因子k=2,则展伸不确定度U=2${\;\mu _{{c}}}$ (2$\;{\mu _{{c}}}'$ ),如表6所示。Points to be measured Extended uncertainty First quadrant Second quadrant Third quadrant Fourth quadrant Coincidence with the
horizontal wireCoincidence with the
vertical wire$ {N_1} $ 7.33 7.35 5.58 4.26 2.39 1.36 $ {N_2} $ 3.33 3.34 2.55 1.93 1.08 0.60 $ {N_3} $ 2.33 2.33 1.76 1.36 0.76 0.42 Table 6. Extended uncertainty of focal length (Unit: mm)
由表6可知:线段
${N_1}、{N_3}$ 位于不同象限及与分划板横丝平行时,各测试点的焦距测量结果展伸不确定度均大于与竖丝平行时的焦距测量结果展伸不确定度,测量结果的展伸不确定度与焦距真值的最大相对误差为0.36%,该值远小于GB/T 9917.1—2002 照相镜头中实测焦距对名义焦距的相对误差不超过±5%的规定。
Research on focal length calibration method of oblique installation collimator
doi: 10.3788/IRLA20220124
- Received Date: 2022-02-08
- Rev Recd Date: 2022-04-20
- Publish Date: 2022-11-30
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Key words:
- focal length /
- distortion /
- projection error /
- in-situ calibration /
- random phase
Abstract: When the collimator is placed horizontally and installed obliquely, its optical parameters will be greatly different due to different stress states. In order to accurately evaluate the focal length of collimator, according to the mapping relationship between the point on the focal plane of the collimator and the angle of the total station, an accurate mathematical model of the relationship between the focal length and the angle of the total station under the condition of oblique installation is established, the principle projection error caused by the rotation of the vertical axis of the total station is corrected. Several groups of data are collected by total station and experimental verification is carried out. After correcting the distortion, the focal length calculated by each testing point when the line segment is parallel to the vertical wire are 1 980.03 mm, 1 983.45 mm, 1 982.79 mm, the average focal length, i.e. the true value, is 1 982.09 mm. When the distortion is corrected but the projection error is not corrected, the focal length calculated from each testing point when the line segment is parallel to the horizontal wire of the reticle is 996.42 mm, 995.23 mm, 995.22 mm, the relative error of the average focal length is 50.2%. The range of focal length calculated by each testing point when the line segment is located in different quadrants and parallel to the horizontal wire of the reticle is 4.74 mm after correcting the projection error and distortion, the average focal length of all testing points is 1982.69 mm, the difference between the average value and the true value is 0.6 mm. The maximum relative error between the extended uncertainty of the focal length calculated by different testing point and the true value of the focal length is 0.36%. This value is far less than the stipulation in GB/T 9917.1-2002 that the relative error between the measured focal length and the nominal focal length in the photographic lens does not exceed ±5%. The experimental results show that the model has universality and high accuracy, the phase of the target slit in the reticle is allowed to be a random value, there is no need to adjust the slit to be strictly parallel to the vertical wire of the total station, the model has great engineering application value for the in-situ detection of the focal length of the collimator under the condition of oblique installation.