﻿ 基于全站仪交会测量的空间距离校准装置测量方法
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 大地测量与地球动力学  2020, Vol. 40 Issue (9): 985-990  DOI: 10.14075/j.jgg.2020.09.021

### 引用本文

PENG Youzhi, ZHANG Xin, HE Haopeng. A Measuring Method for Space Distance Calibration Field Based on Total Station Intersection Survey[J]. Journal of Geodesy and Geodynamics, 2020, 40(9): 985-990.

### Foundation support

Scientific Research Fund of Institute of Seismology and Institute of Crustal Dynamics, CEA, No.IS20176253.

### 第一作者简介

PENG Youzhi, engineer, majors in metrological calibration of geodetic instruments, E-mail: 8452590@qq.com.

### 文章历史

1. 中国地震局地震研究所地震大地测量重点实验室, 武汉市洪山侧路40号，430071;
2. 武汉地震计量检定与测量工程研究院有限公司，武汉市洪山侧路40号，430071

1 测量方法与数据处理 1.1 整体思路

 图 1 测量及数据处理流程 Fig. 1 Flow chart of measurement and data processing

1.2 测量方法

 图 2 平面式空间距离标准装置测量方法布置示意图 Fig. 2 Schematic layout of measurement method for plane space distance standard device

 $\left\{ \begin{array}{l} R = \frac{{{\rm{sin}}\alpha /2}}{{1 - {\rm{sin}}\alpha /2}}L{\rm{sin}}V\\ x = {x_i} + (R + L){\rm{sin}}V{\rm{cos}}{H_z}\\ y = {y_i} + (R + L){\rm{sin}}V{\rm{cos}}{H_z}\\ z = {z_i} + (R + L){\rm{cos}}V \end{array} \right.$ (1)

1.3 整体平差与精度估计

 ${x_P} = {X_P} - X_P^0, {y_P} = {Y_P} - Y_P^0, {z_P} = {Z_P} - Z_P^0$

 $\begin{array}{l} {\rm{tan}}{A_{mn}} = \frac{{{X_n} - {X_m}}}{{{Y_n} - {Y_m}}} \Rightarrow {\rm{tan}}{A_{mn}}\left( {{y_n} - {y_m}} \right) - \\ \;\;\;\;\;\;\;\left( {{x_n} - {x_m}} \right) = \Delta X_{mn}^0 - \Delta Y_{mn}^0{\rm{tan}}{A_{mn}} \end{array}$

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;{\rm{tan}}\left( {\frac{\pi }{2} - {A_{mn}}} \right) = \frac{{{Y_n} - {Y_m}}}{{{X_n} - {X_m}}} \Rightarrow \\ {\rm{tan}}\left( {\frac{\pi }{2} - {A_{mn}}} \right)\left( {{x_n} - {x_m}} \right) - \left( {{y_n} - {y_m}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Delta Y_{mn}^0 - \Delta X_{mn}^0{\rm{tan}}\left( {\frac{\pi }{2} - {A_{mn}}} \right) \end{array}$

1) 水平方向观测值：

 ${\widehat{L}}_{mn}={\rm arctan}\left(\frac{{\widehat{X}}_{n}-{\widehat{X}}_{m}}{{\widehat{Y}}_{n}-{\widehat{Y}}_{m}}\right)-{\widehat{\rm{SA}}}_{k}$ (2a)

2) 天顶距观测值：

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\hat T}_{mn}} = \frac{\pi }{2} - \\ {\rm{arctan}}\left( {\frac{{{{\hat Z}_n} - {{\hat Z}_m}}}{{\sqrt {{{\left( {{{\hat X}_n} - {{\hat X}_m}} \right)}^2} + {{\left( {{{\hat Y}_n} - {{\hat Y}_m}} \right)}^2}} }}} \right) \end{array}$ (2b)

3) 起始方位角差观测值：

 ${\widehat {{\rm{DA}}}_{kl}} = {\widehat {{\rm{SA}}}_k} - {\widehat {{\rm{SA}}}_l}$ (2c)

4) 距离观测值：

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\hat S}_{{{mn}}}} = \\ \sqrt {{{\left( {{{\hat X}_n} - {{\hat X}_m}} \right)}^2} + {{\left( {{{\hat Y}_n} - {{\hat Y}_m}} \right)}^2} + {{\left( {{{\hat Z}_n} - {{\hat Z}_m}} \right)}^2}} \end{array}$ (2d)

 $\left\{ \begin{array}{l} {V_{L_{mn}}} = \frac{{\Delta X_{mn}^0}}{{{{\left( {{\rm{SH}}_{mn}^0} \right)}^2}}}{y_m} - \frac{{\Delta Y_{mn}^0}}{{{{\left( {{\rm{SH}}_{mn}^0} \right)}^2}}}{x_m} - \\ \;\;\;\frac{{\Delta X_{mn}^0}}{{{{\left( {{\rm{SH}}_{mn}^0} \right)}^2}}}{y_n} + \frac{{\Delta Y_{mn}^0}}{{{{\left( {{\rm{SH}}_{mn}^0} \right)}^2}}}{x_n} - {a_k} - {l_{mn}} + \\ \;\;\;{\rm{arctan}}\left( {\frac{{\Delta X_{mn}^0}}{{\Delta Y_{mn}^0}}} \right) - SA_k^0\\ {V_{T_{mn}}} = \frac{{\Delta Z_{mn}^0\Delta X_{mn}^0}}{{{{\left( {S_{mn}^0} \right)}^2}{\rm{SH}}_{mn}^0}}{x_n} - \frac{{\Delta Z_{mn}^0\Delta X_{mn}^0}}{{{{\left( {S_{mn}^0} \right)}^2}{\rm{SH}}_{mn}^0}}{x_m} + \\ \;\;\;\frac{{\Delta Z_{mn}^0\Delta Y_{mn}^0}}{{{{\left( {S_{mn}^0} \right)}^2}{\rm{SH}}_{mn}^0}}{y_n} - \frac{{\Delta Z_{mn}^0\Delta Y_{mn}^0}}{{{{\left( {S_{mn}^0} \right)}^2}{\rm{SH}}_{mn}^0}}{y_m} - \\ \;\;\;\frac{{{\rm{SH}}_{mn}^0}}{{{{\left( {S_{mn}^0} \right)}^2}}}{z_n} + \frac{{{\rm{SH}}_{mn}^0}}{{{{\left( {S_{mn}^0} \right)}^2}}}{z_m} + \frac{\pi }{2} - \\ \;\;\;\;{\rm{arctan}}\left( {\frac{{\Delta Z_{mn}^0}}{{{\rm{SH}}_{mn}^0}}} \right) - {t_{mn}}\\ {V_{{\rm{d}}{{\rm{a}}_{kl}}}} = {a_k} - {a_l} + A_k^0 - A_l^0 - {\rm{d}}{{\rm{a}}_{kl}}\\ {V_{{S_{mn}}}} = \frac{{\Delta X_{mn}^0}}{{S_{mn}^0}}{x_n} - \frac{{\Delta X_{mn}^0}}{{S_{mn}^0}}{x_m} + \frac{{\Delta Y_{mn}^0}}{{S_{mn}^0}}{y_n} - \\ \;\;\;\;\;\;\;\frac{{\Delta Y_{mn}^0}}{{S_{mn}^0}}{y_m} + \frac{{\Delta Z_{mn}^0}}{{S_{mn}^0}}{z_n} - \frac{{\Delta Z_{mn}^0}}{{S_{mn}^0}}{z_m} + S_{mn}^0 - {{\rm{s}}_{mn}} \end{array} \right.$ (3)

 $\mathit{\boldsymbol{V}} = \mathit{\boldsymbol{Bx}} + \mathit{\boldsymbol{l}}$ (4)

 $\begin{array}{l} \mathit{\boldsymbol{x}} = ({x_1}\;\;{y_1}\;\;{z_1}\;\;{x_2}\;\;{y_2}\;\;{z_2}\;\; \cdots \\ \;\;\;\;\;\;\;\;\;{x_N}\;\;{y_N}\;\;{z_N}\;\;{a_{k1}}\;\;{a_{k2}}\; \cdots \;\;{a_{kM}}{)^{\rm{T}}} \end{array}$

 $\mathit{\boldsymbol{Gx}} = \mathit{\boldsymbol{t}}$

G作奇异值分解(SVD)：

 $\mathit{\boldsymbol{G}} = \mathit{\boldsymbol{U \boldsymbol{\varLambda} }}{\mathit{\boldsymbol{H}}^{\rm{T}}} = \left( {{\mathit{\boldsymbol{U}}_{{r}}}} \right)\left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_r}}&{{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_0}} \end{array}} \right){\left( {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{H}}_r}}&{{\mathit{\boldsymbol{H}}_0}} \end{array}} \right)^{\rm{T}}}$

 $\mathit{\boldsymbol{x}} = {\mathit{\boldsymbol{H}}_0}\mathit{\boldsymbol{y}} + \mathit{\boldsymbol{w}}$

 $\mathit{\boldsymbol{w}} = {\mathit{\boldsymbol{H}}_r}\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_r^{ - 1}\mathit{\boldsymbol{U}}_r^{\rm{T}}\mathit{\boldsymbol{t}}$

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\boldsymbol{V}} = \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{H}}_0}\mathit{\boldsymbol{y}} + \mathit{\boldsymbol{Bw}} + \mathit{\boldsymbol{l}}\\ \mathop {{\rm{min}}}\limits_\mathit{\boldsymbol{y}} {\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}} = - {\left( {{\mathit{\boldsymbol{H}}_0}^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{H}}_0}} \right)^{ - 1}}{\mathit{\boldsymbol{H}}_0}^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{Bw}} + \mathit{\boldsymbol{l}}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathop {{\rm{min}}}\limits_\mathit{\boldsymbol{x}} {\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}} = \\ \;\;\;\mathit{\boldsymbol{w}} - {\mathit{\boldsymbol{H}}_0}{\left( {{\mathit{\boldsymbol{H}}_0}^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{H}}_0}} \right)^{ - 1}}{\mathit{\boldsymbol{H}}_0}^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{Bw}} + \mathit{\boldsymbol{l}}} \right)\\ \mathit{\boldsymbol{x}}\left| {{\rm{min}}{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}}}) \right. = {\mathit{\boldsymbol{H}}_r}\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_r^{ - 1}\mathit{\boldsymbol{U}}_r^{\rm{T}}\mathit{\boldsymbol{t}} - {\mathit{\boldsymbol{H}}_0}{\left( {{\mathit{\boldsymbol{H}}_0}^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}}{\mathit{\boldsymbol{H}}_0}} \right)^{ - 1}} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{H}}_0}^{\rm{T}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P}}\left( {\mathit{\boldsymbol{B}}{\mathit{\boldsymbol{H}}_r}\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_r^{ - 1}\mathit{\boldsymbol{U}}_r^{\rm{T}}\mathit{\boldsymbol{t}} + \mathit{\boldsymbol{l}}} \right) \end{array}$ (5)

 ${p_s} = {\left( {\frac{S}{{\rho ''{\sigma _s}}}} \right)^2}$ (6)

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{S_{{{mn}}}} = \\ \sqrt {{{\left( {{X_m} - {X_n}} \right)}^2} + {{\left( {{Y_m} - {Y_n}} \right)}^2} + {{\left( {{Z_m} - {Z_n}} \right)}^2}} \end{array}$

 $\begin{array}{l} {S_{{{mn}}}} = \frac{{\Delta X_{mn}^0}}{{S_{mn}^0}}{x_n} - \frac{{\Delta X_{mn}^0}}{{S_{mn}^0}}{x_m} + \frac{{\Delta Y_{mn}^0}}{{S_{mn}^0}}{y_n} - \\ \;\;\;\;\;\;\frac{{\Delta Y_{mn}^0}}{{S_{mn}^0}}{y_m} + \frac{{\Delta Z_{mn}^0}}{{S_{mn}^0}}{z_n} - \frac{{\Delta Z_{mn}^0}}{{S_{mn}^0}}{z_m} + S_{mn}^0 \end{array}$

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\mathit{\boldsymbol{D}}_{SS}}{\rm{ = }}\\ \left( {\frac{{\Delta X_{mn}^0}}{{S_{mn}^0}}, - \frac{{\Delta X_{mn}^0}}{{S_{mn}^0}}, \frac{{\Delta Y_{mn}^0}}{{S_{mn}^0}}, - \frac{{\Delta Y_{mn}^0}}{{S_{mn}^0}}, \frac{{\Delta Z_{mn}^0}}{{S_{mn}^0}}, - \frac{{\Delta Z_{mn}^0}}{{S_{mn}^0}}} \right) \cdot \\ {\mathit{\boldsymbol{D}}_{XX}}{\left( {\frac{{\Delta X_{mn}^0}}{{S_{mn}^0}}, \frac{{\Delta X_{mn}^0}}{{S_{mn}^0}}, \frac{{\Delta Y_{mn}^0}}{{S_{mn}^0}}, \frac{{\Delta Y_{mn}^0}}{{S_{mn}^0}}, \frac{{\Delta Z_{mn}^0}}{{S_{mn}^0}}, \frac{{\Delta Z_{mn}^0}}{{S_{mn}^0}}} \right)^{\rm{T}}} \end{array}$ (7)

 ${\mathit{\boldsymbol{D}}_{XX}} = {\hat \sigma _0}^2{\mathit{\boldsymbol{Q}}_{xx}}$

2 算例

 图 3 点位概略坐标示意图 Fig. 3 A schematic diagram of the outline coordinate

 图 4 点位坐标精度统计 Fig. 4 Statistics of coordinate accuracy

 图 5 空间距离精度统计 Fig. 5 Statistics for spatial distance accuracy

3 结语

1) 用全站仪交会测量后整体平差的方法可以用于地面激光扫描仪空间距离标准装置的校准，内符合精度优于0.1 mm，精度和激光跟踪仪相当。由于交会测量属于非接触测量，对于空间范围很大的空间距离标准装置校准来说，比激光跟踪仪更加安全、更方便操作。若使用激光跟踪仪测量，要充分考虑测量点的稳定性和分布。

2) 和工业经纬仪测量系统相比，全站仪使用广泛、价格便宜，使得建立工业测量系统更加便捷。用全站仪测量大尺度的空间标准距离，可以利用全站仪的免棱镜、后方交会等功能来直接测量概略坐标，更加直观、便捷，还可避免坐标推算造成的系统误差积累。

3) 交会测量后用整体平差法可以更好地利用全站仪的技术特点，直接使用补偿后的天顶方向为坐标纵轴，直接互瞄后(不需要内外觇标)得到水平角观测值，方法更加简便，精度一致性、可靠性都能满足工业测量的要求，可以用于更多室内精密测量的场景。

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A Measuring Method for Space Distance Calibration Field Based on Total Station Intersection Survey
PENG Youzhi1     ZHANG Xin2     HE Haopeng1
1. Key Laboratory of Earthquake Geodesy, Institute of Seismology, CEA, 40 Hongshance Road, Wuhan 430071, China;
2. Wuhan Seismic Metrological Verification and Surveying Engineering Institute Co Ltd, 40 Hongshance Road, Wuhan 430071, China
Abstract: Space distance standard device is an important metrological standard for the calibrating terrestrial laser scanner. The measurement of the space distance of the target ball has always been a difficult problem. At present, most are measured by laser tracker. In this paper, we propose the coordinate measurement of target sphere by total station intersection survey. Firstly, the outline coordinates of feature points can be obtained quickly and conveniently by rear rendezvous and non-prism measurement. Then, the orientation datum can be established by mutual aiming. Finally, the high-precision coordinates of feature points can be obtained by overall adjustment. The example proves that the accuracy of this system is better than 0.1 mm, and it can meet the calibration accuracy requirement of space distance standard device.
Key words: laser scanner; calibration device; total station; intersection measurement; overall adjustment