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长度标准器以两基准点球心连线为长度标准。如图1所示,设长度标准两端基准点为P1, P2,激光跟踪仪原点位置为O,原点O与P1, P2连线夹角为
$ \beta $ ,由余弦定理可得长度标准$ L $ 的测量模型如下式:$$ L = \sqrt {l_1^2 + l_2^2 - 2{l_1}{l_2}\cos \beta } $$ (1) 式中:
$ {l_1},{l_2} $ 为基准点P1, P2由激光跟踪仪干涉测长测得的观测值。根据测量不确定度传播律(GUM法) [12],
$ L $ 的合成标准不确定度为:$$ {u_L} = \sqrt {{{\left( {\frac{{\partial L}}{{\partial {l_1}}}} \right)}^2} \cdot u_{{l_1}}^2 + {{\left( {\frac{{\partial L}}{{\partial {l_2}}}} \right)}^2} \cdot u_{{l_2}}^2 + {{\left( {\frac{{\partial L}}{{\partial \beta }}} \right)}^2} \cdot u_\beta ^2} $$ (2) 式中:
$ {u_{{l_1}}},{u_{{l_2}}},{u_\beta } $ 为输入量$ {l_1},{l_2},\beta $ 的标准不确定度;$\dfrac{\partial L}{\partial {l}_{1}}, $ $ \dfrac{\partial L}{\partial {l}_{2}}, \dfrac{\partial L}{\partial \beta }$ 为输入量$ {l_1},{l_2},\beta $ 对$ L $ 的传递系数。$$ \begin{gathered} \frac{{\partial L}}{{\partial {l_1}}} = \frac{{{l_1} - {l_2} \cdot \cos \beta }}{{{{\left( {l_1^2 - 2{l_1} \cdot {l_2} \cdot \cos \beta + l_2^2} \right)}^{1/2}}}} \hfill \\ \frac{{\partial L}}{{\partial {l_2}}} = \frac{{{l_2} - {l_1} \cdot \cos \beta }}{{{{\left( {l_1^2 - 2{l_1} \cdot {l_2} \cdot \cos \beta + l_2^2} \right)}^{1/2}}}} \hfill \\ \frac{{\partial L}}{{\partial \beta }} = \frac{{{l_1} \cdot {l_2} \cdot \sin \beta }}{{{{\left( {l_1^2 - 2{l_1} \cdot {l_2} \cdot \cos \beta + l_2^2} \right)}^{1/2}}}} \hfill \\ \end{gathered} $$ (3) 激光跟踪仪的测角不确定度远大于测距不确定度[13-14],为了获得最小的长度测量不确定度
$ {u_L} $ ,应使$ \partial L/\partial \beta $ 尽可能小。当$ \beta $ 为0时,即O,P1, P2三点共线,$ {u_L} $ 最小。此时,待测长度即为激光干涉测长模式测得的长度$ {L_0} $ ,达到了最佳测量状态,$ {l_{{P_1}}},{l_{{P_2}}} $ 为共线时跟踪仪干涉测长测得P1, P2的观测值。$$ {L_0} = {l_{{P_1}}} - {l_{{P_2}}} $$ (4) 为了解决激光跟踪仪较为笨重,位置调整难度大的问题,镜面反射式激光跟踪干涉测长方法将平面镜置于基准点P1前侧,调节平面镜的俯仰和偏转旋钮,实时监控测量软件中两基准点的夹角,直至夹角尽可能接近于0,实现O', P1, P2三点共线的最佳测量状态,即仅用激光跟踪干涉测长对两基准点P1, P2进行不断光连续测量。测量原理如图2所示,O’表示激光跟踪仪的镜像原点。
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由于平面镜存在调节分辨率、平面度等误差因素,激光束经镜面反射无法完全达到O,P1',P2'三点共线的理想测量状态,产生原点与两基准点连线的夹角
$ \beta $ 、光程差$ \delta $ ,从而引起长度测量误差。图3为平面镜角度偏差示意图,P1',P2'表示两基准点P1,P2的虚像。(1)按典型布局,O到平面镜中心距离约为3000 mm,平面镜中心到P1距离约为100 mm,则平面镜角度偏差
$ \alpha $ 与夹角$ \beta $ 的关系如图4所示。如图4所示,
$ \alpha $ 与$\;\beta$ 近似呈线性关系,随着长度变长,斜率增加趋于平缓,取长度为8000 mm的斜率作为趋近值$ k{\text{ = 1}}{\text{.4}} $ ,则:$$ \beta = k \times \alpha = 1.4\alpha $$ (5) 通常经过精细调整后,调节分辨率引起的角度偏差
$ \alpha $ 可控制在0.005°以内,则由此引入的标准不确定度分量$ {u_1} $ 为:$$ {u_1} = \frac{{\partial L}}{{\partial \beta }} \times {u_\beta } $$ (6) 式中:
${u_\beta } = k{u_\alpha } = 1.4 \times \dfrac{1}{2} \times 0.005 \times \dfrac{1}{{\sqrt 3 }} \approx {0.002^ \circ }$ 。(2)图5为平面度引起的光程差示意图,虚线部分表示增加的光程差
$ \delta $ ,$ \sigma $ 表示平面镜的平面度。设激光在镜面的入射角为
$ \gamma $ ,根据光程差$ \delta $ 、平面度$ \sigma $ 和入射角$ \gamma $ 三者的几何关系,增加的光程差可由下式计算:$$ \delta = 2 \times \frac{\sigma }{{\cos \gamma }} $$ (7) 由此引入的标准不确定度分量
$ {u_2} $ 为:$$ {u_2} = \frac{\delta }{{\sqrt 3 }} = \frac{{2\sigma }}{{\sqrt 3 \cos \gamma }} $$ (8) 由于平面镜调整机构的限制,入射角范围约为
$ \left( {0,{{45}^ \circ }} \right) $ 。为保证较高的测量精度且标准不确定度分量不被低估,应选择较大的入射角,较小的平面度进行计算。当$ \gamma = {45^ \circ } $ ,$ \sigma = \lambda /4 $ 时,标准不确定度分量约为${u_2} = 0.259~{\text{μm}}$ 。 -
(1)激光跟踪仪干涉测长的仪器误差是长度测量的主要误差来源,以Leica AT960跟踪仪为例[14],其干涉测长的最大允许误差为
$\pm 0.5~{\text{μm/m}}$ ,则由此引入的标准不确定度分量为:$$ {u_3} = {u_{{\text{IFM}}}} = \frac{{0.5 \times {{10}^{ - 6}} \times {L_0}}}{{\sqrt 3 }} $$ (9) 式中:
$ {L_0} $ 为激光干涉测长模式测得的长度。(2)实际测量中,激光跟踪仪测角的仪器误差是长度测量的误差来源之一,以Leica AT960跟踪仪为例[14],其角度测量最大允许误差为
$\pm ( 15\;{\text{μm}} + 6\;{\text{μm/m}}$ ),则角度测量的标准不确定度为:$$ {u_\theta } = {u_\varphi } = \arcsin \left( {\frac{{15 + 6l}}{l} \times {{10}^{ - 6}}} \right) $$ (10) 设两基准点的球坐标值分别为
${{{P}}_1}\left( {{l_1},{\theta _1},{\varphi _1}} \right)$ 和${{{P}}_2} $ $ \left( {{l_2},{\theta _2},{\varphi _2}} \right)$ ,则两基准点距离为[15-16]:$$ \begin{split} &{L^2} = l_1^2 + l_2^2 \hfill \\& - 2{l_1}{l_2}\left( {\sin {\varphi _1}\sin {\varphi _2}\cos \left( {\left| {{\theta _1} - {\theta _2}} \right|} \right) + \cos {\varphi _1}\cos {\varphi _2}} \right) \hfill \end{split}$$ (11) 则由此引入的标准不确定度分量
$ {u_4} $ 为:$$ {u_4} = \sqrt {{{\left( {\frac{{\partial L}}{{\partial {\varphi _1}}}} \right)}^2}u_{_\varphi }^2 + {{\left( {\frac{{\partial L}}{{\partial {\varphi _2}}}} \right)}^2}u_{_\varphi }^2 + {{\left( {\frac{{\partial L}}{{\partial {\theta _1}}}} \right)}^2}u_{_\theta }^2 + {{\left( {\frac{{\partial L}}{{\partial {\theta _2}}}} \right)}^2}u_{_\theta }^2} $$ (12) (3)温度、湿度和压力等外部环境干扰造成大气折射率变化,会引起跟踪仪产生测长误差[12,17]。考虑到跟踪仪带有环境补偿单元,且实验室环境温度为(20±0.5) ℃,一次测量时间较短,环境参数近似恒定,因此由环境干扰引入的标准不确定度忽略不计。
(4)由长度
$ L $ 的$ n $ 次测量值求得单次测量的标准差为:$$ {\sigma _L} = \sqrt {\frac{{\displaystyle\sum\limits_{i = 1}^n {{{\left( {{L_i} - \bar L} \right)}^2}} }}{{n - 1}}} $$ (13) 式中:
$ {L_i} $ 为第$ i $ 次测量的长度值;$ \bar L $ 为$ n $ 次测量的平均长度。则长度测量重复性引起的标准不确定度分量$ {u_5} = {\sigma _L} $ 。由上述各不确定度分量合成长度测量不确定度$ U $ (k=2)。 -
由2.1节可知,平面镜角度调节偏差导致了激光跟踪仪测得两基准点产生夹角
$\;\beta$ ,进而产生长度测量误差,因此合理地调节平面镜角度从而限制夹角$\;\beta$ ,有利于减小测量不确定度$ U $ 并提高测量效率。(1)固定长度
$ L $ ,$ U $ 随$\;\beta$ 变化的仿真实验以典型布局下测量
$L = 1\;{\text{m}}$ 为例,以如图2所示的最佳测量状态为初始状态。令$\;\beta$ 分别沿水平和垂直两个方向以0.005°逐渐偏离初始状态,进行了20次平面镜角度的调节仿真。随着$\;\beta$ 变化,引起的长度测量不确定度$ U $ (k=2)变化规律如图6所示。由图6可知,在水平和垂直两个方向上,
$\;\beta$ 对$ U $ 的影响程度相同,均近似呈对称的二次函数关系,$\;\beta$ 为0时$ U $ 最小,为1.1 µm,$ U $ 随$\;\beta$ 绝对值增大而增大;根据大尺寸测量现场几何量溯源需求,$ U $ 阈值设为3 µm,只需调整$\;\beta$ 在(−0.01°~0.01°)范围即可满足。(2)固定夹角
$\;\beta$ ,$ U $ 随$ L $ 变化的仿真实验保持激光跟踪仪、平面镜和基准点1的位置不变,移动基准点2,使
$ L $ 以间隔0.25 m从1 m增加至6 m,仿真$\;\beta = {0^ {\circ} }\sim{0.05^ {\circ} }$ 下长度测量不确定度的变化规律,结果如图7所示。由图7可知,随长度变长,
$\;\beta$ 对$ U $ 的影响逐渐减小,对于1~2.5 m的长度,$ U $ 主要由$\;\beta$ 引入,为使$ U $ 不大于3 µm,须使$ \left| \beta \right| \leqslant {0.01^ \circ } $ ;对于2.5 m及以上的长度,$ U $ 主要由激光跟踪干涉测长引入,且随长度变长,$ U $ 趋近于最佳测量状态$ \left( {\beta {\text{ = }}{0^ \circ }} \right) $ 的测量不确定度,为使$ \beta $ 对$ U $ 的影响可忽略(占比不超过1/5),须使$ \left| \beta \right| \leqslant {0.03^ \circ } $ 。综上,给出了以下针对不同长度的平面镜角度调节允许范围:
(1)针对1~2.5 m的长度,调整平面镜使得激光跟踪仪测得两基准点的夹角在(−0.01°~0.01°)范围即可;
(2)针对2.5 m以上的长度,调整平面镜使得激光跟踪仪测得两基准点的夹角在(−0.03°~0.03°)范围即可。
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通过两个实验验证文中的长度测量不确定度和仿真实验结果。实验1验证文中测量不确定度的准确性;实验2验证平面镜角度调节的准确性、高效性。
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坐标测量机长度测量比对实验如图8所示,调整平面镜至近似最佳测量状态(
$ {\theta _1}{\text{ = }}{\theta _2},{\varphi _1} = {\varphi _2} $ ),采用Leica AT960跟踪仪不断光连续测量P1、P2,计算两点间的长度,重复测量60次。按文中方法评定长度测量不确定度,计算60次测量的平均值及偏差,并与测量不确定度比较,结果如图9所示。图9中,60次测量的长度与平均值的偏差均包含在长度测量不确定度以内。
相同状态下,采用坐标测量机对P1、P2进行采点测量,给出长度
${L_{{{{P}}_{\text{1}}}{{{P}}_{\text{2}}}}}$ 及其测量不确定度$ {u_{{L_0}}} $ ,计算长度$ L $ 与${L_{{{{P}}_{\text{1}}}{{{P}}_{\text{2}}}}}$ 的En:$$ {E_{\text{n}}} = \frac{{\left| {L - {L_{{{{P}}_{\text{1}}}{{{P}}_{\text{2}}}}}} \right|}}{{\sqrt {u_L^2 + u_{{L_0}}^2} }} $$ (14) 结果如表1所示。
表 1 En验证结果
Table 1. En verification result
Interferometric length measurement Coordinate measuring machine $ L/{\text{mm}} $ $ U/{\text{μm}} $ $ {L_{ { { {P} }_{\text{1} } }{ { {P} }_{\text{2} } } } }/{\text{mm} } $ $ {U_0}/{\text{μm}} $ En 1000.2833 1.1 1000.2825 2.6 0.3 表1中,长度测量对比的En小于1,即两种测量方法的测量结果的不确定度均在各自评定的不确定度范围内,比对结果满意。
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以测量1 m和2.5 m的长度为例,第一步仅肉眼观察,粗调平面镜,使激光在平面镜上的反射点与长度标准器的P1、P2近似共线,记录此时表2中相关数据;第二步按照2.3节的角度调节,由测量软件读取两基准点的夹角
$\;\beta$ ,微调平面镜的偏转与俯仰旋钮,直到$\;\beta$ 分别小于0.01°(L=1 m)和0.03°(L=2.5 m),记录数据;第三步精调平面镜,继续多次反复交替微调平面镜的偏转和俯仰,直至$\;\beta$ 接近于0°,达到如图2所示的最佳测量状态,记录数据。重新放置平面镜,重复三次上述步骤,计算平均调整次数$ \overline M $ 和平均用时$ \overline t $ 。将每一步测得的长度与最佳测量状态的长度计算差值$ \Delta L $ ,采用文中方法评定三个状态下的长度测量不确定度(k=2),结果如表2所示。表 2 角度调节验证结果
Table 2. Angle adjustment verification result
$ L/{\text{mm}} $ Adjustment strategy Angle $ \;\beta /({^ \circ } )$ $ \overline M $/times $ \overline t /\min $ $ \Delta L/{\text{μm}} $ $ U/{\text{μm}} $
1000Rough adjustment 0.0421 1 4 9.2 11.7 This paper strategy 0.0096 9 10 1.2 3.0 Optimal state 0.0003 18 23 — 1.1 Rough adjustment 0.2746 1 5 7.6 8.1 2500 This paper strategy 0.0293 10 13 2.6 3.7 Optimal state 0.0005 21 28 — 3.1 采用文中给出的平面镜角度调节允许范围,得到的长度测量误差均小于3
${\text{μm}}$ ,调整次数比最佳测量状态时减少了约50%,用时减少了约56%。而仅粗调平面镜得到的长度偏差近10${\text{μm}}$ ,不确定度无法满足溯源需求。综上,文中给出的平面镜角度调节允许范围可同时兼顾测量精度和测量效率,提高了镜面反射式激光跟踪干涉测长方法的测量效率,具有准确性、高效性。
Study on measurement method of mirror reflection laser tracking interferometric length measurement
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摘要: 为了解决镜面反射式激光跟踪干涉测长方法在不同尺寸长度标准溯源中缺乏有效的误差分析,测量效率低等问题,文中研究了平面镜调节分辨率、平面度引入的长度测量误差,并给出了针对不同长度的平面镜角度调节允许范围。建立了镜面反射式激光跟踪干涉测长模型,分析了该模型的测量不确定度来源,模拟仿真了平面镜角度调节的影响,并进行了坐标测量机长度测量比对实验和平面镜角度调节实验。实验表明:长度测量比对的En验证结果小于1,证明了长度测量不确定度的准确性。通过针对不同长度的平面镜角度调节实验,得到了测量不确定度为3
$ {\text{μm}} $ (长度1 m)和3.7$ {\text{μm}} $ (长度2.5 m)的测量结果;同时,测量效率提高了56%,验证了此方法的准确性、高效性。文中方法实现了长度的高精度测量与测量不确定度分析。Abstract: In the traceability of length standards of different sizes, in order to solve the problems of lack of effective error analysis and low measurement efficiency of mirror reflection laser tracking interferometric length measurement, the length measurement error caused by the plane mirror adjustment resolution and flatness was studied. The allowable range of plane mirror angle adjustment for different lengths was obtained. The model of mirror reflection laser tracking interferometry was established, and the source of measurement uncertainty of the model was analyzed. The influence of plane mirror angle adjustment was simulated. The length measurement comparison experiment of coordinate measuring machine and the plane mirror angle adjustment experiment were carried out. The experiment shows that the En verification result of length measurement comparison is less than 1, which proves the accuracy of length measurement uncertainty. Through the plane mirror angle adjustment experiment for different lengths, the measurement results with measurement uncertainty of 3$ {\text{μm}} $ (length 1 m) and 3.7$ {\text{μm}} $ (length 2.5 m) were obtained. The measurement efficiency was improved by 56%. Experiments verify the accuracy and efficiency of this method. This method realizes the high-precision measurement of length and the analysis of measurement uncertainty.-
Key words:
- interferometry /
- mirror reflection /
- uncertainty evaluation /
- length standard /
- laser tracker
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表 1 En验证结果
Table 1. En verification result
Interferometric length measurement Coordinate measuring machine $ L/{\text{mm}} $ $ U/{\text{μm}} $ $ {L_{ { { {P} }_{\text{1} } }{ { {P} }_{\text{2} } } } }/{\text{mm} } $ $ {U_0}/{\text{μm}} $ En 1000.2833 1.1 1000.2825 2.6 0.3 表 2 角度调节验证结果
Table 2. Angle adjustment verification result
$ L/{\text{mm}} $ Adjustment strategy Angle $ \;\beta /({^ \circ } )$ $ \overline M $ /times$ \overline t /\min $ $ \Delta L/{\text{μm}} $ $ U/{\text{μm}} $
1000Rough adjustment 0.0421 1 4 9.2 11.7 This paper strategy 0.0096 9 10 1.2 3.0 Optimal state 0.0003 18 23 — 1.1 Rough adjustment 0.2746 1 5 7.6 8.1 2500 This paper strategy 0.0293 10 13 2.6 3.7 Optimal state 0.0005 21 28 — 3.1 -
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