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Volume 43 Issue 1
Jan.  2018
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Article Contents

FAN Baixing, LI Guangyun, ZHOU Weihu, YI Wangmin, YANG Zhen, YANG Zaihua. Precision Analysis of the Unified Spatial Metrology Network Adjustment Model[J]. Geomatics and Information Science of Wuhan University, 2018, 43(1): 120-126. doi: 10.13203/j.whugis20130536
Citation: FAN Baixing, LI Guangyun, ZHOU Weihu, YI Wangmin, YANG Zhen, YANG Zaihua. Precision Analysis of the Unified Spatial Metrology Network Adjustment Model[J]. Geomatics and Information Science of Wuhan University, 2018, 43(1): 120-126. doi: 10.13203/j.whugis20130536

Precision Analysis of the Unified Spatial Metrology Network Adjustment Model

doi: 10.13203/j.whugis20130536
Funds:

The National Natural Science Foundation of China 41101446

the Open Research Fund Program of Spacecraft Precision Measurement Laboratory 201402

More Information
  • Author Bio:

    FAN Baixing, PhD, associate professor, specializes in the theories and methods of precise Industry and engineering survey. E-mail: fbxhrhr@sina.com

  • Received Date: 2017-02-22
  • Publish Date: 2018-01-05
  • In industrial metrology, a multi-type instrument is only combined to completing all measurement task, and measurement data type and accuracy of each type instrument is different, so valid adjustment model is need to setting up and solving spatial position and attitude parameter of multi-type instrument. Based on observation equation with condition equations, the article established generalized USMN adjustment model and combined adjust for different type instruments, such as laser tracker, total station and theodolite. Furthermore, because of different data type, efficient weight matric model is build up to get accuracy weight, and this model can get precision position without abundant redundant observation. Based on angular resection and coordinate transformation, spatial network preliminary computation model can solve approximate parameter without theodolite each aiming angle and scalebar measurement. At last, measurement datum is solved with generalized USMN adjustment model, the solution result shows that the model can solve different instrument and raise the point accuracy.
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Precision Analysis of the Unified Spatial Metrology Network Adjustment Model

doi: 10.13203/j.whugis20130536
Funds:

The National Natural Science Foundation of China 41101446

the Open Research Fund Program of Spacecraft Precision Measurement Laboratory 201402

  • Author Bio:

Abstract: In industrial metrology, a multi-type instrument is only combined to completing all measurement task, and measurement data type and accuracy of each type instrument is different, so valid adjustment model is need to setting up and solving spatial position and attitude parameter of multi-type instrument. Based on observation equation with condition equations, the article established generalized USMN adjustment model and combined adjust for different type instruments, such as laser tracker, total station and theodolite. Furthermore, because of different data type, efficient weight matric model is build up to get accuracy weight, and this model can get precision position without abundant redundant observation. Based on angular resection and coordinate transformation, spatial network preliminary computation model can solve approximate parameter without theodolite each aiming angle and scalebar measurement. At last, measurement datum is solved with generalized USMN adjustment model, the solution result shows that the model can solve different instrument and raise the point accuracy.

FAN Baixing, LI Guangyun, ZHOU Weihu, YI Wangmin, YANG Zhen, YANG Zaihua. Precision Analysis of the Unified Spatial Metrology Network Adjustment Model[J]. Geomatics and Information Science of Wuhan University, 2018, 43(1): 120-126. doi: 10.13203/j.whugis20130536
Citation: FAN Baixing, LI Guangyun, ZHOU Weihu, YI Wangmin, YANG Zhen, YANG Zaihua. Precision Analysis of the Unified Spatial Metrology Network Adjustment Model[J]. Geomatics and Information Science of Wuhan University, 2018, 43(1): 120-126. doi: 10.13203/j.whugis20130536
  • 工业测量的对象呈现尺寸越来越大、结构越来越复杂、测量精度越来越高的趋势,通常需要多种工业测量仪器联合测量,充分发挥各种测量仪器的优势,才能完成全部测量任务[1-2]

    国际上发展了空间联合精密控制网(unified spatial metrology network,USMN)技术,主要用于解决多种球坐标测量系统(如激光跟踪仪、全站仪、激光雷达等)的空间联合定位和定向平差问题,为多种测量仪器的空间坐标测量的统一提供位置和方位基准[3]

    中国的航空航天等精密测量领域广泛使用经纬仪进行空间点三维交会测量和空间姿态准直测量,通常需要将经纬仪非球坐标测量系统和跟踪仪球坐标测量系统进行联合平差解算,建立空间三维测量控制网。与球坐标测量系统相比,经纬仪测量系统缺少长度观测基准,需要通过基准尺引入长度基准,并通过互瞄观测值提高各个经纬仪测站的定向精度[2]。参与USMN平差的测量数据可在非整平甚至倒置状态下获取,与经典三维控制网平差模型具有较大的不同。

    本文在球坐标测量系统联合平差原理的基础上,针对经纬仪测量系统缺少长度基准而具有互瞄基准和基准尺约束条件等特点,采用附有条件的参数平差模型,构建了包含激光跟踪仪和经纬仪测量系统的USMN平差模型,在经纬仪不存在互瞄和基准观测值的情况下也可以实现整网平差解算,克服了USMN模型无法解算缺乏经纬仪互瞄值的整网平差的缺点,解决了经纬仪和跟踪仪联合平差解算及其优化问题。

  • m(m≥1)台激光跟踪仪对q(q≥3)个定向点进行观测,n台经纬仪两两互瞄并测量了p(p≥1)个位置的基准尺,每台经纬仪依次测量q个定向点,如图 1所示,则整个控制网的未知参数的个数t为:

    Figure 1.  Chart of Combined Measurement

    则整个控制网的观测方程总数N为:

    在平差解算中,通常以第一测站为测量坐标系,此时,第一测站的3个平移和3个旋转参数均为零,整个控制网需要满足:

    此时,即可按照最小二乘原理求解。在整个控制网中,误差方程可以分为水平角误差方程、垂直角误差方程、斜距误差方程和基准尺长度约束条件方程四类。设第i(i=1,2,…,m)个测站相对于测量坐标系的平移和旋转参数为(X0iY0iZ0iRxiRyiRzi),第k(k=1,2,…,q)个定向点在第i个测站坐标系下的坐标为(XikYikZik),其在测量坐标系下的坐标为(XkYkZk)。在利用角度、距离和互瞄值列误差方程时,需要将第k个定向点的观测值从第i个测站坐标系转换到测量坐标系下[1, 4-5]

    式中,ai1ai2,…,ci3为第i个测站旋转参数(Rxi, Ryi, Rzi)的函数。

  • 激光跟踪仪和激光雷达等仪器可以在不整平状态下进行测量,而经纬仪和全站仪都必须在整平状态下才能测量,但整平后的全站仪和经纬仪也存在微小倾斜,因此其平差模型相同,而基准尺测量点的误差方程模型和定向点相同。

    设第i测站对定向点k的角度和距离观测值为(HikVikSik),若测站为经纬仪则Sik=0。此时,Hik是以仪器水平度盘平面为基准水平角,如图 2所示。

    Figure 2.  Relation of Coordinate and Observed Value

    点坐标与角度观测值的函数关系为[2]

    顾及式(4),可将定向点坐标从测站坐标系转换到测量坐标系下,则式(5)变为:

    此外,点坐标与距离观测值的函数关系如下式[5-6]

    分别对式(6)和式(7)线性化即可得到定向点和基准尺端点的观测值误差方程:

    式中,d1,…,d9e1,…,e9f1,…,f9分别为斜距、水平角和垂直角对未知参数的一阶偏导;likSlikHlikV为常数项;δX0iδY0iδZ0iδRxiδRyiδRzi为第i个测站参数的近似改正数;δXkδYkδZk为第k个定向点的近似坐标改正数。

    i台经纬仪瞄向第i+1台经纬仪时,角度互瞄值为(Hi(i+1)Vi(i+1)),设两台经纬仪的测站参数为(X0iY0iZ0iRxiRyiRzi)和(X0(i+1)Y0(i+1)Z0(i+1)Rxi+1Ryi+1Rzi+1),则角度互瞄值的误差方程为[1, 4]

    式中,g1,…,g12h1,…,h12分别经纬仪互瞄的水平角和垂直角对测站参数的一阶偏导;liH(i+1),liV(i+1)为常数项;δX0iδY0iδZ0iδRxiδRyiδRzi为第i个测站参数的近似改正数;δX0(i+1)δY0(i+1)δZ0(i+1)δRx0(i+1)δRy0(i+1)δRz0(i+1)为第i+1个测站参数的近似改正数。则式(8)和式(9)联合即可组成定向点和互瞄观测误差方程的矩阵形式:

    式中,V为改正数向量;A为系数矩阵;X包含测站点、定向点、基准尺端点等未知参数矩阵;l为常数项矩阵。

  • 设基准尺两个端点为第kk+1个定向点,基准尺参考长度为S0,坐标值参数为(XkYkZk)和(Xk+1Yk+1Zk+1),则利用基准尺的参考长度即可构成约束条件方程[5]为:

    将式(11)线性化即可得到条件方程的形式为:

    式中,ckdkek分别为线性化后的系数;wk为自由项;δXkδYkδZk为基准尺第k个端点的近似坐标改正数;δXk+1δYk+1δZk+1为基准尺第k+1个端点的近似坐标改正数。若多个基准尺在多个位置观测,则式(12)可以写成矩阵形式:

    式(13)中,BX为条件方程的系数矩阵;X包含未知参数的矩阵;W为常数项矩阵。式(10)和式(13)按照具有约束条件的参数平差模型,即可解算得到[6-10]

    式中,P为观测值权阵; N = ATPA; M = BXN-1BXT

  • 与传统经纬仪和全站仪相比,激光跟踪仪是通过目标跟踪系统精密照准目标后,再测量水平和垂直度盘的角度读数,类似于智能全站仪的自动目标识别(automatic target recognization,ATR)功能。激光跟踪仪的测角误差常采用±(a+bS)的形式给出[11-14],其中a为固定误差,以μm为单位,b为比例误差系数,采用μm/m即10-6S为斜距观测值,以m为单位。激光跟踪仪的水平角权值PH、垂直角权值PV和斜距权值PS为:

    以徕卡AT901型激光跟踪仪为例,其标称测角误差为±(15μm+6×10-6·S),激光干涉(interferometer,IFM)测距误差为±0.5μm/m[15-17]。需要指出的是,与经纬仪和全站仪一样,激光跟踪仪的水平和垂直测角误差相同,但是实际测试表明,激光跟踪仪的1个测回水平方向中误差要小于1个测回垂直方向中误差,因此在平差解算时,垂直角度的权要小于水平角度权,二者的系数K一般取0.8~0.9。经纬仪和全站仪的角度和距离权的确定与传统方法相同。

  • i台电子经纬仪和第j台激光跟踪仪对第k个定向点的观测值分别为(HikVik)和(HikVikSik),以第j台激光跟踪仪作为当前测站,测站的3个平移参数和3个旋转参数均为0。则各个定向点的坐标(Xjk0, Yjk0, Zjk0)为:

    图 3所示,第i个测站对3个定向点的观测值分别为(Hi1Vi1Si1)、(Hi2Vi2Si2)、(Hi3Vi3Si3),由式(16)可以得到三个定向点坐标,分别为P1(Xj10, Yj10, Zj10)、P2(Xj20, Yj20, Zj20)、P3(Xj30, Yj30, Zj30)[18-20]

    Figure 3.  Chart of Station Preliminary

    设第i个测站未知参数为(Xi0, Yi0, Zi0, 0, 0, Rzi0),按角度后方交会原理即可解算近似坐标值为:

    式中, $ \tan \gamma = \frac{{\left( {Y_{j1}^0 - Y_{j2}^0} \right)\cot \alpha + \left( {Y_{j3}^0 - Y_{j2}^0} \right)\cot \beta + X_{j3}^0 - X_{j1}^0}}{{\left( {Y_{j1}^0 - X_{j2}^0} \right)\cot \alpha + \left( {X_{j3}^0 - X_{j2}^0} \right)\cot \beta - Y_{j3}^0 + Y_{j1}^0}} $; $ {D_{i1}} = \sqrt {{{\left( {X_i^0 - X_{j1}^0} \right)}^2} + {{\left( {Y_i^0 - Y_{j1}^0} \right)}^2} + {{\left( {Z_i^0 - Z_{j1}^0} \right)}^2}} $; $ \alpha = {H_{i2}} - {H_{i3}};\beta = {H_{i1}} - {H_{i2}} $。

    进一步,可计算第i个测站到P1点的在第j个坐标系下的方位角αi10

    式(18)中方位角的值需要根据XY坐标的差值进行判断解算,解算模型和坐标方位角判断方法一致,则第i个测站相对于第j个测站绕Z轴的旋转角度Rzi0:

    至此,完成了第i个测站的未知参数(Xi0, Yi0, Zi0, 0, 0, Rzi0)的概算。需要注意的是,此时的概算坐标系为第j个激光跟踪仪测站坐标系,在工业测量中,通常以测站1坐标系为全局测量坐标系,通常设第1测站为强制水平,可以利用坐标系转换原理,将测站坐标系参数和定向点坐标参数都转换到测站1坐标系下,并以转换后的概算值作为平差近似值参与平差解算。

  • 在实验室内布设了5个定向点P1~P5,采用2台TM5100A、2台T3000A电子经纬仪和1台AT901-B激光跟踪仪激光跟踪仪两次设站,和4台经纬仪共组成6个测站,分别对5个定向点进行了观测。其中4台经纬仪两两互瞄,6条互瞄边均测量0.9 m的基准尺,共6个位置。经纬仪的1个测回水平和垂直方向中误差的标称值为±0.5″,激光跟踪仪的IFM测距误差为±0.5 μm/m,测角误差为±(15 μm+6 μm×10-6·S)。

    该控制网中的观测值包括经纬仪对定向点的角度观测值、经纬仪互瞄角度观测值、经纬仪对基准尺的角度观测值、跟踪仪对定向点的角度和距离观测值等四类,再顾及到基准尺的长度约束条件,即可按照USMN平差模型进行解算,解算结果如表 1所示。

    未知参数 误差方程数 多余观测数 均方根误差/mm 单位权中误差/mm 备注
    条件1 87 154 67 0.062 1.007 有互瞄和基准尺
    条件2 51 100 49 0.080 1.121 有互瞄无基准尺
    条件3 51 76 25 0.056 0.731 无互瞄无基准尺

    Table 1.  Adjusted Result of Generalized USMN

    表 1中可以看出,3种条件的整网平差精度基本相当,但是条件3的多余观测数远远低于条件1和条件2,即条件3的测量效率最高。整网用USMN平差后,3种条件下测站点误差和定向点误差分别如图 4图 5所示,由于以测站1为测量坐标系,所以图 4中只显示测站2~测站6的点位误差值。

    Figure 4.  Chart of Station Error

    Figure 5.  Chart of Orientation Point Error

    图 4中可以看出,由于条件1的测站多余观测数最多,因此其测站点位误差最小。而条件3中经纬仪测站缺少测站互瞄值和基准尺测量值,即经纬仪测站的多余观测较少,导致经纬仪测站的误差远远大于激光跟踪仪测站的误差,并且远远大于经纬仪观测量较多的条件2和条件3的经纬仪测站点误差。由于跟踪仪测站的第5站和第6站观测数据没有减少,因此,条件3的跟踪仪测站点位误差与条件1和条件2中基本相当。

    图 5中,在3种条件下,由于定向点的观测量没有减少,因此定向点位误差基本相当。

    为了进一步评定定向点数量及其分布对广义USMN平差结果的影响,在周围均匀的增加了3个定向点,参与系统的平差解算,解算后,测站点误差和定向点误差的分布如图 6图 7所示。

    Figure 6.  Chart of Station Error(8 Orientation Points)

    Figure 7.  Chart of Orientation Point Error (8 Orientation Points)

    图 6中可以发现,增加3个定向点后,在3种条件下,测站中心坐标的平差精度接近一致且变化较均匀。比较图 4图 6可以发现,由于增加了3个定向点,经纬仪测站的平差精度得到了大幅度的提高,因此,增加定向点个数并优化其空间分布,可以弥补广义USMN平差模型中无经纬仪互瞄和无基准尺观测值的情况下,经纬仪测站的定向精度。而图 5图 7的趋势保持一致,即在3种条件下,定向点的平差精度基本一致。

  • USMN平差模型很好的解决了多台电子经纬仪交会测量系统、多台激光跟踪仪全站仪等球坐标测量系统的控制网整体平差,实现了多台、多类型测量仪器之间的定位和定向解算。USMN平差模型在经纬仪没有互瞄值和基准尺观测值的条件下,也可以实现经纬仪和激光跟踪仪的定位定向解算,极大地减少了多余观测量,提高了多类型测量仪器联合测量的效率,满足了航天器准直测量领域的特殊测量要求。由于激光跟踪仪的空间点位误差小于经纬仪交会测量系统,并且采用了激光跟踪仪角度和距离的合理定权模型,因此在经纬仪有无互瞄和基准尺观测值对定向点的点位误差影响不显著。经纬仪测站之间缺少互瞄值时,导致经纬仪测站的多余观测量减少,使经纬仪测站点位误差增大,而跟踪仪测站和定向点的点位误差几乎不受影响。增加定向点个数并优化其空间分布,可以提高无互瞄、无基准尺观测值条件下的经纬仪测站定向精度。

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