﻿ 激光跟踪仪分段积分测距改正模型研究与分析
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 大地测量与地球动力学  2019, Vol. 39 Issue (4): 377-381  DOI: 10.14075/j.jgg.2019.04.009

### 引用本文

PAN Guorong, ZHANG Chuntao, ZHOU Zhi, et al. Research and Analysis on Segmented Integral Ranging Correction Method Based on Laser Tracker[J]. Journal of Geodesy and Geodynamics, 2019, 39(4): 377-381.

### Foundation support

Fundamental Research Funds for the Central Universities.

### 第一作者简介

PAN Guorong, professor, PhD supervisor, majors in measurement data processing and 3D visualized simulation, E-mail: pgr2@163.com.

### 文章历史

1. 同济大学测绘与地理信息学院，上海市四平路1239号，200092

1 分段积分测距改正模型

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;{\left( {n - 1} \right)_s} \times {10^8} = \\ 8\;342.54 + \frac{{2\;406\;147}}{{130 - {\sigma ^2}}} + \frac{{15\;998}}{{38.9 - {\sigma ^2}}} \end{array}$ (1)

 $\begin{array}{l} \;\;\;{\left( {n - 1} \right)_{tp}} = \frac{{{{\left( {n - 1} \right)}_s} \times P}}{{96\;095.43}} \times \\ \frac{{1 + {{10}^{ - 8}} \times \left( {0.601 - 0.009\;72t} \right)}}{{1 + \alpha t}} \end{array}$ (2)

 $\begin{array}{l} {\rm{ppm}} = 271.687 - 291.580 \times \\ \frac{{1 + {{10}^{ - 8}}\left( {0.601 - 0.009\;72t} \right)}}{{1 + \alpha t}} \end{array}$ (3)

ppm为气象改正中按距离成比例的部分，单位是10-6

1.1 两点积分测距改正模型

 图 1 两点温度分布示意图 Fig. 1 Two-point temperature distribution diagram
 $t = \frac{{{t_1} - {t_0}}}{S}x + {t_0}$ (4)

 $\begin{array}{l} D = \smallint _0^S{\rm{ppm}}\left( t \right){\rm{d}}x = \smallint _0^S[271.687 - 291.580 \times \\ \;\;\;\;\;\;\frac{{1 + {{10}^{ - 8}}\left( {0.601 - 0.009\;72t} \right)}}{{1 + \alpha t}}]{\rm{d}}x \end{array}$ (5)

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\Delta D = 271.687S - \\ \frac{{291.580(1 + {{10}^{ - 8}}(0.601 - 0.009\;72{t_0}))S}}{{\alpha ({t_1} - {t_0})}} \times \\ \;\;\;\;\ln \frac{{1 + \alpha {t_1}}}{{1 + \alpha {t_0}}} + 2.831\;456\;7 \times {10^{ - 8}} \times \\ \;\;\;\;\;\;\frac{S}{\alpha }\left( {1 - \frac{{1 + \alpha {t_0}}}{{\alpha ({t_1} - {t_0})}}\ln \frac{{1 + \alpha {t_1}}}{{1 + \alpha {t_0}}}} \right) \end{array}$ (6)

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;{\rm{ppm}} = \frac{{\Delta D}}{S} = 271.687 - \\ \frac{{291.580(1 + {{10}^{ - 8}}(0.601 - 0.009\;72{t_0}))}}{{\alpha ({t_1} - {t_0})}} \times \\ \;\;\;\;\;\ln \frac{{1 + \alpha {t_1}}}{{1 + \alpha {t_0}}} + 2.831\;456\;7 \times {10^{ - 8}} \times \\ \;\;\;\;\;\;\;\frac{1}{\alpha }\left( {1 - \frac{{1 + \alpha {t_0}}}{{\alpha ({t_1} - {t_0})}}\ln \frac{{1 + \alpha {t_1}}}{{1 + \alpha {t_0}}}} \right) \end{array}$ (7)
1.2 三点等距积分测距改正模型

 图 2 均匀三点温度分布示意图 Fig. 2 Uniform three-point temperature distribution diagram
 $\left\{ \begin{array}{l} t = \frac{{2({t_1} - {t_0})}}{S}x + {t_0}\\ t = \frac{{2({t_2} - {t_1})}}{S}x + 2{t_1} - {t_2} \end{array} \right.$ (8)

 ${\rm{ppm}} = \frac{1}{2}{\rm{pp}}{{\rm{m}}_{0, 1}} + \frac{1}{2}{\rm{pp}}{{\rm{m}}_{1, 2}}$ (9)
 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;{\rm{pp}}{{\rm{m}}_{0, 1}} = 271.687 - \\ \frac{{291.580(1 + {{10}^{ - 8}}(0.601 - 0.009\;72{t_0}))}}{{\alpha ({t_1} - {t_0})}} \times \\ \;\;\;\;\ln \frac{{1 + \alpha {t_1}}}{{1 + \alpha {t_0}}} + 2.831\;456\;7 \times {10^{ - 8}} \times \\ \;\;\;\;\;\;\frac{1}{\alpha }\left( {1 - \frac{{1 + \alpha {t_0}}}{{\alpha ({t_1} - {t_0})}}\ln \frac{{1 + \alpha {t_1}}}{{1 + \alpha {t_0}}}} \right) \end{array}$ (10)
 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{pp}}{{\rm{m}}_{1, 2}} = \\ - \frac{{291.580(1 + {{10}^{ - 8}}(0.601 - 0.009\;72(2{t_1} - {t_2})))}}{{\alpha ({t_2} - {t_1})}} \times \\ \;\;\;\;\;\;\;\;\;\;\ln \frac{{1 + \alpha {t_2}}}{{1 + \alpha {t_1}}} + 2.831\;456\;7 \times {10^{ - 8}} \times \\ \;\;\;\frac{1}{\alpha }\left( {1 - \frac{{1 + \alpha (2{t_1} - {t_2})}}{{\alpha ({t_2} - {t_1})}}\ln \frac{{1 + \alpha {t_2}}}{{1 + \alpha {t_1}}}} \right) + 271.687 \end{array}$ (11)
1.3 多点分段积分测距改正模型

 图 3 多点温度分布示意图 Fig. 3 Multi-point temperature distribution diagram

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{pp}}{{\rm{m}}_{0, n}} = \\ \frac{{{S_1}}}{S}{\rm{pp}}{{\rm{m}}_{0, 1}} + \frac{{{S_2}}}{S}{\rm{pp}}{{\rm{m}}_{1, 2}} + \cdots + \frac{{{S_n}}}{S}{\rm{pp}}{{\rm{m}}_{n - 1, n}} \end{array}$ (12)

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;{\rm{pp}}{{\rm{m}}_{0, 1}} = 271.687 - \\ \frac{{291.580(1 + {{10}^{ - 8}}(0.601 - 0.009\;72{t_0}))}}{{\alpha ({t_1} - {t_0})}} \times \\ \;\;\;\ln \frac{{1 + \alpha {t_1}}}{{1 + \alpha {t_0}}} + 2.831\;456\;7 \times {10^{ - 8}} \times \\ \;\;\;\frac{1}{\alpha }\left( {1 - \frac{{1 + \alpha {t_0}}}{{\alpha ({t_1} - {t_0})}}\ln \frac{{1 + \alpha {t_1}}}{{1 + \alpha {t_0}}}} \right) \end{array}$ (13)
 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{pp}}{{\rm{m}}_{i, i + 1}} = \\ - \frac{{291.580(1 + {{10}^{ - 8}}(0.601 - 0.009\;72(2{t_i} - {t_{i + 1}})))}}{{\alpha ({t_{i + 1}} - {t_i})}}\\ \;\;\;\;\;\;\;\; \times \ln \frac{{1 + \alpha {t_{i + 1}}}}{{1 + \alpha {t_i}}} + 2.831\;456\;7 \times {10^{ - 8}} \times \\ \;\;\frac{1}{\alpha }\left( {1 - \frac{{1 + \alpha (2{t_i} - {t_{i + 1}})}}{{\alpha ({t_{i + 1}} - {t_i})}}\ln \frac{{1 + \alpha {t_{i + 1}}}}{{1 + \alpha {t_i}}}} \right) + 271.687 \end{array}$ (14)

2 实验和分析

2.1 倾斜单测线实验

 图 4 倾斜单测线实验现场配置 Fig. 4 Field configuration of tilt single line experiment

2.2 水平单测线实验

 图 5 水平单测线实验现场配置 Fig. 5 Field configuration of horizontal single line experiment

2.3 实验分析

 图 6 不同测距改正方法差值比较 Fig. 6 Comparison of different range correction methods

3 实际应用与分析

 图 7 装配现场传感器配置 Fig. 7 Sensor configuration at assembly site

 图 8 装配现场热力分布 Fig. 8 Assembly site thermal distribution

4 结语

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Research and Analysis on Segmented Integral Ranging Correction Method Based on Laser Tracker
PAN Guorong1     ZHANG Chuntao1     ZHOU Zhi1     WANG Suihui1
1. College of Surveying and Geo-Informatics, Tongji University, 1239 Siping Road, Shanghai 200092, China
Abstract: We are concerned with the difficulty of constant temperature measurement in the assembly site, which has a great influence on distance measurement. In this paper, a correction method for laser tracker segmented integral distance measurement is advanced, and the correction model of segmented integral distance measurement is established. Through the single line temperature ranging correction experiment, based on radian laser tracker, the experimental results show that the method can effectively reduce the influence of temperature gradient, provide theoretical support for the temperature gradient correction of the assembly site, and help to improve the accuracy of aircraft and other assembly and manufacturing.
Key words: laser tracker; assembly site; temperature gradient; correction model of segmented integral ranging