-
通过蛙跳技术实现激光跟踪仪对组合式测量网络的全局控制,将不同站位下柔性关节坐标测量臂的测量数据统一到同一个激光跟踪仪坐标系下。在激光跟踪仪与测量臂的公共测量范围内布置n(n≥4)个临时基准点,分别用激光跟踪仪和测量臂测量这些临时基准点,可以得到激光跟踪仪坐标系下的坐标矩阵:
$$A = \left[ {\begin{array}{*{20}{c}} {{x_1}}&{{x_2}}&{...}&{{x_i}} \\ {{y_1}}&{{y_2}}&{...}&{{y_i}} \\ {{{\textit{z}}_1}}&{{{\textit{z}}_2}}&{...}&{{{\textit{z}}_i}} \\ 1&1&{...}&1 \end{array}} \right]$$ 以及柔性关节坐标测量臂坐标系下的坐标矩阵:
$$B = \left[ {\begin{array}{*{20}{c}} {{{x'}_1}}&{{{x'}_2}}&{...}&{{{x'}_i}} \\ {{{y'}_1}}&{{{y'}_2}}&{...}&{{{y'}_i}} \\ {{{{\textit{z}}'}_1}}&{{{{\textit{z}}'}_2}}&{...}&{{{{\textit{z}}'}_i}} \\ 1&1&{...}&1 \end{array}} \right]$$ 通过坐标变换关系可以得到如下关系式:
$$A = T \cdot B$$ (1) 式中:矩阵
$T$ 是一个4×4的坐标变换矩阵。再通过最小二乘法求解该坐标变换矩阵,令:
$${\left\| {A - T \cdot B} \right\|^2} \to \min $$ (2) 可以得到:
$$T = A{B^{\rm T}}{\left( {B{B^{\rm T}}} \right)^{ - 1}}$$ (3) -
激光跟踪仪测量的是跟踪头到靶镜球心的距离,柔性关节坐标测量臂测量的是测头球心的空间坐标。在使用激光跟踪仪与测量臂测量临时基准点的过程中,靶镜与测头的半径不同很难保证二者测量中心重合。为此,在空间配准的过程中用靶镜代替临时基准点,使用测量臂测量靶镜球面上的多个点,并且保证测量臂测头与靶镜相切,这样就实现了测头球心与靶镜球心的距离为定值,即为两球半径之和,如图3所示。
将测量臂测量得到不同的坐标值Pi(xi,yi,zi)代入球心坐标拟合方程:
$$ ({x}_{i}-{x}_{0}{)}^{2}+{({y}_{i}-{y}_{0})}^{2}+{({{\textit{z}}}_{i}-{{\textit{z}}}_{0})}^{2}={(R+r)}^{2} $$ (4) 式中:(x0,y0,z0)为靶镜球心的坐标值;R为靶球半径;r为测头半径。通过求解非线性方程组(4)可以得到测量臂坐标系下具有更高精度的临时基准点坐标矩阵
$B'$ :$$ B' = \left[ {\begin{array}{*{20}{c}} {{{x{''}}_1}}&{{{x{''}}_2}}&{...}&{{{x{''}}_i}} \\ {{{y{''}}_1}}&{{{y{''}}_2}}&{...}&{{{y{''}}_i}} \\ {{{{\textit{z}}{''}}_1}}&{{{{\textit{z}}{''}}_2}}&{...}&{{{{\textit{z}}{''}}_i}} \\ 1&1&{...}&1 \end{array}} \right] $$ 再代入到坐标转换公式(3)中,可以解算得到具有更高精度控制效果的坐标转换矩阵
$T'$ 。 -
为提高激光跟踪仪对组合式测量网络的全局控制精度,摒弃其角度测量模块,采用四路激光跟踪并结合冗余数据的测量方法[13-14]。布置激光跟踪仪的位置如图4所示。
多边测量方法需要在四个不同站位下分别测量到临时基准点的距离,考虑到激光跟踪仪的成本问题,实际测量时使用单台激光跟踪仪分时多站位测量的方法。设激光跟踪仪四个不同站位的编号分别为Si(i=1,2,3,4),建立激光跟踪多边测量坐标系,选取其中一台激光跟踪仪的跟踪头位置为坐标系原点S1(0,0,0),因为测量过程中使用同一台激光跟踪仪,可以保证四个站位下激光跟踪仪跟踪头的高度相等,则站位2、站位3、站位4的坐标分别为S2(X2,Y2,0),S3(X3,Y3,0),S4(X4,Y4,0)。将激光跟踪仪位置参数X2、Y2、X3、Y3、X4、Y4作为迭代初值,通过Levenberg-Marquardt优化算法得到较高精度的临时基准点坐标。
根据空间多边理论[15],已知任意三个站位的位置坐标及其干涉测量数据,就可以解算出临时基准点的空间坐标,再计算出该基准点到第四个站位处跟踪头的理论计算距离lij,根据干涉测量数据可以得到残差方程:
$$f = {d_{ij}} - {l_{ij}}$$ (5) 式中:i=1,2,3,···,n表示临时基准点的编号;j=1,2,3,4,表示激光跟踪仪的站位编号。
设lij=F(p)为优化迭代的目标函数,p=(X2,Y2,X3,Y3,X4,Y4)T为迭代参数,并在p处对F(p)进行一阶泰勒展开,可以得到:
$$F\left( {p + \Delta p} \right) = F\left( p \right) + J\Delta p$$ (6) 式中:
$J = \dfrac{{\partial F\left( p \right)}}{{\partial p}}$ 为Jacobi矩阵;$\Delta p$ 为迭代步长,每一次迭代有:$${p_{k + 1}} = {p_k} + \Delta p$$ (7) 将上述雅可比矩阵与残差方程的计算结果代入到增量正规化方程[16-17]中:
$$\bar N\Delta p = {J^{\rm T}}f$$ (8) 可以求出其最小二乘解为:
$$\Delta p = {\bar N^{ - 1}}{J^{\rm T}}f$$ (9) 式中:
$\bar N = \left( {{J^\rm T}J + \mu I} \right)$ ,μ为阻尼因子。解算时,将迭代步长
$\Delta p$ 与预先设置好的阈值$\varepsilon $ 进行比较,若$\left\| {\Delta p} \right\| < \varepsilon $ 则结束迭代过程,得到的pk即为最优解。反之,将迭代步长$\Delta p$ 不断地代入到公式(7)中得到新的坐标迭代参数,再用改进的坐标参数重新求解残差矩阵、雅可比矩阵以及增量正规化方程,代入下一轮迭代,最终使得:$$\left\| {f - J\Delta p} \right\| \to \min $$ (10) -
激光跟踪仪多边测量方法需要确定不同站位之间的相对位置关系。采用IFM干涉测距,通过距离约束建立激光跟踪位置参数标定模型,如图5所示。
选用一块标准靶镜标定板,标定板上均匀排布着靶镜槽,每个靶镜槽之间的距离为精确值,且能保证靶镜中心的高度为定值,标定板如图6所示。在标定板上放置三个靶镜,其中一个靶镜放置在靶镜槽1的位置,另外两个靶镜则随机放置,由此,建立以靶镜槽1处靶镜中心为坐标系原点的靶镜坐标系,通过标定板参数可以确定另外两个靶镜中心在坐标系中的精确坐标,分别设为(x2,y2,0)、(x3,y3,0)。
固定好靶镜后,在靶镜坐标系的第一象限内放置激光跟踪仪,并设此时的位置为站位1,激光跟踪仪的坐标为S1(X1,Y1,Z1)。用站位1处的激光跟踪仪分别测量标定板上的三个靶镜,得到干涉测距值分别为d1、d2、d3,建立球面坐标方程组:
$$\left\{ \begin{array}{l} X_1^2 + Y_1^2 + Z_1^2 = d_1^2 \\ {\left( {{X_1} - {x_2}} \right)^2} + {\left( {{Y_1} - {y_2}} \right)^2} + Z_1^2 = d_2^2 \\ {\left( {{X_1} - {x_3}} \right)^2} + {\left( {{Y_1} - {y_3}} \right)^2} + Z_1^2 = d_3^2 \\ \end{array} \right.$$ (11) 通过最小二乘法求解该恰定非线性方程组可以得到靶镜坐标系下站位1处跟踪头的坐标值。保持标定板与靶镜不动,改变激光跟踪仪的站位,分别测量并计算得到激光跟踪仪跟踪头在站位2、站位3、站位4处的坐标值。
以站位1处跟踪头的位置为整个多边测量坐标系的原点,并以此建立X轴、Y轴和Z轴。通过上述标定方法得到四个站位下跟踪头的坐标值后,可以相应地计算出站位2、站位3、站位4相对于站位1也就是坐标系原点处X轴和Y轴的相对坐标值,从而完成多边测量方法下激光跟踪仪的位置参数标定。采用“先布置,后定位”的标定思路,能够有效地避免在实际测量过程中反复调整激光跟踪仪,大大缩短了测量时间,配合冗余数据能够有效降低测量现场环境因素的干扰,实现激光跟踪仪的高效、精准定位。
-
为验证组合式测量网络的正确性与精度提升能力,对建立的组合式测量网络进行了实验验证。根据现场测量实际条件选定FARO VantageE6型号激光跟踪仪与FARO QuantumS V2型号测量臂作为测量设备。激光跟踪仪的有效测量范围为70 m,全局测量精度为(10+2.5) μm/m,干涉测距精度为0.5 μm/m;测量臂单点测量精度高达10 μm。
所选测量仪器如图7所示。
-
建立基于激光跟踪多边测量方法的组合式测量网络,首先要对激光跟踪仪不同站位下的位置参数进行测量标定,并验证其精度控制效果。设计仿真实验,将所选测量设备的标称精度作为最大误差限引入到仿真验证模型中,每次测量随机生成含有误差的测量数据。根据特大齿轮的尺寸参数,分别布置临时基准点位于激光跟踪多边测量坐标系原点的436.353、1276.001、2072.722、3425.855 mm距离处,每组完成20次含有随机误差的激光跟踪仪干涉测距。通过激光跟踪多边测量位置参数标定模型以及L-M优化算法求解出激光跟踪仪的位置参数测量值以及临时基准点坐标的测量值。四组实验激光跟踪仪位置参数的测量平均值与标准值如表1所示。
表 1 激光跟踪仪不同站位坐标值(单位:mm)
Table 1. Coordinate values of different stations of laser tracker (Unit: mm)
Position Group 1 Group 2 Group 3 Group 4 P2 X Standard value 278.293 350.237 408.318 465.041 Measured value 278.294 350.237 408.319 465.040 Y Standard value 19.608 91.833 86.587 181.150 Measured value 19.609 91.832 86.587 181.149 P3 X Standard value 108.947 211.714 144.500 201.476 Measured value 108.947 211.714 144.499 201.476 Y Standard value 207.212 259.048 256.227 443.156 Measured value 207.212 259.048 256.227 443.156 P4 X Standard value 297.151 517.629 502.104 706.057 Measured value 297.151 517.629 502.103 706.056 Y Standard value 192.448 388.820 305.389 710.938 Measured value 192.448 388.820 305.390 710.939 将四组实验临时基准点坐标值的测量结果与标准坐标值比较,分别计算出X方向和Y方向的测量误差平均值如表2所示。
表 2 参数标定误差平均值(单位:mm)
Table 2. Average value of parameter calibration error (Unit: mm)
Direction Group 1 Group 2 Group 3 Group 4 X 0.001 0.005 0.005 0.012 Y 0.001 0.007 0.011 0.014 为更加直观地展示测量误差平均值,绘制其并列柱状图如图8所示。
对比表1中的测量数据与标准值可以得出,激光跟踪位置参数标定模型具有较高的精度控制效果,四组测量实验中位置参数的最大测量误差值为0.001 mm。对于临时基准点的测量,从图8中可以看出,临时基准点的测量误差随着测量距离的增加而增大。其中,第二组与第三组的临时基准点布置在常规测量范围内,第四组实验结果为特大齿轮极端测量范围下的测量误差,结果为X方向误差平均值为0.012 mm,Y方向误平均值为0.014 mm。由此可以证明,建立的激光跟踪多边测量位置标定模型具有较高的测量精度与较强的可行性。
-
为进一步验证组合式测量网络的精度提升能力,根据两点间距离不变原理,完成组合式测量网络的测量精度提升验证仿真实验。选取标称长度为100 mm的标准件块规,使用柔性关节坐标测量臂分别测量不同位置下块规的左右端点坐标值。根据齿形测量测点数量的选取原则,在合理的测量范围内随机测量45组不同位置下块规的左、右端点坐标值。通过测量臂坐标转换方程实现空间配准,将测量得到的块规端点坐标统一到激光跟踪多边测量坐标系下,计算块规的测量长度并与其标称值相比较,得到不同位置下的块规测量误差,如表3所示。
表 3 组合式测量网络测量块规长度测量误差(单位:mm)
Table 3. Measurement error of block gauge length measured by combined measuring network(Unit: mm)
Group Error value Group Error value Group Error value Group Error value Group Error value 1 0.012 10 −0.003 19 0.014 28 −0.004 37 0.009 2 −0.014 11 0.002 20 0.008 29 0.015 38 0.009 3 −0.005 12 −0.004 21 −0.005 30 0.005 39 0.010 4 −0.007 13 −0.003 22 0.007 31 0.006 40 0.004 5 0.003 14 0.008 23 0.003 32 −0.002 41 0.007 6 0.008 15 −0.002 24 −0.001 33 −0.004 42 −0.011 7 −0.013 16 0.005 25 0.004 34 −0.003 43 0.013 8 0.008 17 0.005 26 −0.007 35 0.014 44 −0.014 9 0.008 18 0.012 27 0.008 36 −0.005 45 0.013 为验证组合式测量网络的精度提升效果,在块规位置相同的条件下,使用激光跟踪仪与柔性关节坐标测量臂直接测量的方法得到块规的长度误差值如表4所示,绘制两组测量误差折线图如图9所示。
表 4 激光跟踪仪直接测量块规长度测量误差(单位:mm)
Table 4. Measurement error of the block gauge length measured directly by laser tracker(Unit: mm)
Group Error value Group Error value Group Error value Group Error value Group Error value 1 0.081 10 −0.058 19 0.078 28 −0.031 37 0.029 2 −0.075 11 −0.010 20 0.029 29 0.062 38 0.030 3 −0.055 12 −0.028 21 −0.047 30 0.017 39 0.042 4 −0.036 13 −0.092 22 0.047 31 0.017 40 −0.011 5 −0.056 14 0.021 23 0.043 32 −0.015 41 −0.008 6 0.025 15 0.003 24 −0.039 33 −0.025 42 −0.094 7 −0.069 16 0.049 25 −0.060 34 −0.031 43 −0102 8 −0.006 17 −0.067 26 0.007 35 0.053 44 −0.144 9 0.022 18 0.041 27 0.022 36 −0.040 45 −0.091 分析、对比两组测量结果可以得出,基于多边测量方法的组合式测量网络测量误差平均值为0.007 mm,标准差为0.004 mm,而包含角度测量模块的激光跟踪仪直接测量方法得到的测量误差平均值为0.044 mm,标准差为0.028 mm。为使结果具有普遍性,设计多组不同位置参数下的实验,测量误差对比折线图见图10。
对上述图中测量结果进行统计分析,得到两种测量方法的误差平均值与标准差如表5所示。
表 5 测量误差结果分析表(单位:mm)
Table 5. Analysis table of measurement error results(Unit: mm)
Position Optimized Direct Avg. Std. Avg. Std. P1 0.007 0.004 0.044 0.028 P2 0.005 0.003 0.027 0.020 P3 0.007 0.004 0.036 0.023 P4 0.007 0.004 0.044 0.031 P5 0.007 0.005 0.049 0.037 从表5中可以看出,使用组合式测量网络方法得到的误差远小于直接测量方法得到的误差,且离散程度较小,相对稳定。由于缺少特大齿轮样板,因此无法进行实际测量实验。根据国标《GB 10095—1988》,分度圆直径为3000 mm的特大齿轮齿形公差为100 μm,当组合式测量网络的测量误差小于公差带的20%~30%时,即视为测量精度满足要求。可见,基于激光跟踪多边测量方法的组合式测量网络不仅能够满足特大齿轮测量精度的需求,而且具有较高的精度提升效果。
Laser tracking measurement accuracy improvement method for mega gear
-
摘要: 为实现特大齿轮激光跟踪测量精度的提升,采用激光跟踪仪与柔性关节坐标测量臂相结合的测量方式,建立了基于激光跟踪多边测量方法的特大齿轮组合式测量网络。采用柔性关节坐标测量臂蛙跳技术确定激光跟踪仪全局坐标系与柔性关节坐标测量臂坐标系之间的坐标转换关系,实现不同站位下测量臂测量数据的空间配准。引入激光跟踪仪多边测量方法,摒弃其角度测量模块,建立激光跟踪多边测量位置参数标定模型,通过测量冗余数据并对其进行L-M优化迭代,以提高激光跟踪仪的全局控制精度。对建立的组合式测量网络进行仿真实验,分析对比测量数据,组合式测量网络的测量误差平均值为0.007 mm,误差标准差为0.004 mm,相同条件下,使用激光跟踪仪直接测量方法的测量误差平均值为0.044 mm。仿真实验分析表明,该方法显著提升了测量精度,满足了特大齿轮现场齿形测量的要求,具有较好的理论与工程应用价值。Abstract: In order to improve the precision of laser tracking measurement for mega gear, a combined measuring network for mega gear was established by combining laser tracking device and flexible joint coordinate measuring arm. The coordinate transformation relationship between the global coordinate system of laser tracker and the coordinate system of flexible joint coordinate measuring arm was determined by using the frog jump technology to realize the spatial registration of measurement data at different stations. A multi-dimensional measurement method of laser tracker was introduced, which had abandoned the angle measurement module, a multi-dimensional measurement position parameter calibration model of laser tracker was established, redundant data were measured and L-M optimization iteration was carried out in this method in order to improve the global control accuracy of laser tracker. The simulation experiment of combined measurement network was carried out, and the measurement data were analyzed and compared. The average error of combined measurement network was 0.007 mm and the error standard deviation was 0.004 mm. Under the same conditions, the average error of direct measurement method using laser tracker was 0.044 mm. The simulation results show that this method can improve the measuring accuracy obviously and meet the requirements of the measurement of tooth shape of mega gear. It has a good theoretical and engineering application value.
-
表 1 激光跟踪仪不同站位坐标值(单位:mm)
Table 1. Coordinate values of different stations of laser tracker (Unit: mm)
Position Group 1 Group 2 Group 3 Group 4 P2 X Standard value 278.293 350.237 408.318 465.041 Measured value 278.294 350.237 408.319 465.040 Y Standard value 19.608 91.833 86.587 181.150 Measured value 19.609 91.832 86.587 181.149 P3 X Standard value 108.947 211.714 144.500 201.476 Measured value 108.947 211.714 144.499 201.476 Y Standard value 207.212 259.048 256.227 443.156 Measured value 207.212 259.048 256.227 443.156 P4 X Standard value 297.151 517.629 502.104 706.057 Measured value 297.151 517.629 502.103 706.056 Y Standard value 192.448 388.820 305.389 710.938 Measured value 192.448 388.820 305.390 710.939 表 2 参数标定误差平均值(单位:mm)
Table 2. Average value of parameter calibration error (Unit: mm)
Direction Group 1 Group 2 Group 3 Group 4 X 0.001 0.005 0.005 0.012 Y 0.001 0.007 0.011 0.014 表 3 组合式测量网络测量块规长度测量误差(单位:mm)
Table 3. Measurement error of block gauge length measured by combined measuring network(Unit: mm)
Group Error value Group Error value Group Error value Group Error value Group Error value 1 0.012 10 −0.003 19 0.014 28 −0.004 37 0.009 2 −0.014 11 0.002 20 0.008 29 0.015 38 0.009 3 −0.005 12 −0.004 21 −0.005 30 0.005 39 0.010 4 −0.007 13 −0.003 22 0.007 31 0.006 40 0.004 5 0.003 14 0.008 23 0.003 32 −0.002 41 0.007 6 0.008 15 −0.002 24 −0.001 33 −0.004 42 −0.011 7 −0.013 16 0.005 25 0.004 34 −0.003 43 0.013 8 0.008 17 0.005 26 −0.007 35 0.014 44 −0.014 9 0.008 18 0.012 27 0.008 36 −0.005 45 0.013 表 4 激光跟踪仪直接测量块规长度测量误差(单位:mm)
Table 4. Measurement error of the block gauge length measured directly by laser tracker(Unit: mm)
Group Error value Group Error value Group Error value Group Error value Group Error value 1 0.081 10 −0.058 19 0.078 28 −0.031 37 0.029 2 −0.075 11 −0.010 20 0.029 29 0.062 38 0.030 3 −0.055 12 −0.028 21 −0.047 30 0.017 39 0.042 4 −0.036 13 −0.092 22 0.047 31 0.017 40 −0.011 5 −0.056 14 0.021 23 0.043 32 −0.015 41 −0.008 6 0.025 15 0.003 24 −0.039 33 −0.025 42 −0.094 7 −0.069 16 0.049 25 −0.060 34 −0.031 43 −0102 8 −0.006 17 −0.067 26 0.007 35 0.053 44 −0.144 9 0.022 18 0.041 27 0.022 36 −0.040 45 −0.091 表 5 测量误差结果分析表(单位:mm)
Table 5. Analysis table of measurement error results(Unit: mm)
Position Optimized Direct Avg. Std. Avg. Std. P1 0.007 0.004 0.044 0.028 P2 0.005 0.003 0.027 0.020 P3 0.007 0.004 0.036 0.023 P4 0.007 0.004 0.044 0.031 P5 0.007 0.005 0.049 0.037 -
[1] 中国齿轮专业协会. 中国齿轮工业年鉴[M]. 北京理工大学出版社, 2010. China Gear Professional Association. China Gear Industry Yearbook[M]. Beijing : Beijing Institute of Technology Press, 2010. (in Chinese) [2] Peng Peng, Xiao Tiyi, Liu Tao, et al. Reliability of dynamic kinematic accuracy for gear transmission [J]. Journal of Mechanical Transmission, 2019, 43(5): 96-100, 125. (in Chinese) [3] 郑中鹏. 大齿轮测量系统坐标系的建立技术研究[D]. 西安工业大学, 2017. Zheng Zhongpeng. Research on the establishment of coordinate system for large gear geasurement system[D]. Xi'an: Xi'an University of Technology, 2017. (in Chinese) [4] Shi Zhaoyao, Zhang Bai, Lin Jiachun, et al. Principle and key technology of laser tracking in-situ measurement for extra-large gears [J]. Optics and Precision Engineering, 2013, 21(9): 2340-2347. (in Chinese) doi: 10.3788/OPE.20132109.2340 [5] Zhang Bai, Lin Jiachun. New method of tooth pitch deviation measurement based on laser tracker for mega spur gear [J]. Journal of Mechanical Transmission, 2019, 43(10): 146-150. (in Chinese) [6] Chen Hongfang, Yan Hao, Shi Zhaoyao. Laser tracking multi-station positioning for extra-large gears [J]. Optics and Precision Engineering, 2014, 22(9): 2375-2380. (in Chinese) [7] 许友. 激光跟踪绝对测长多边法坐标测量系统研究[D]. 天津大学, 2018. Xu You. Research on 3D coordinate measuring system based on laser tracking absolute length measurement multilateral method[D]. Tianjin: Tianjin University, 2018. (in Chinese) [8] Ma Guoqing, Liu Li, Yu Zhenglin, et al. Application and development of three-dimensional profile measurement for large and complex surface [J]. Chinese Optics, 2019, 12(2): 214-228. (in Chinese) doi: 10.3788/co.20191202.0214 [9] Li Yuyang, Ma Yingbao, Zheng Yilong, et al. Error compensation of coordinate transformation method for combined large-scale measuring system [J]. Tool Engineering, 2016, 50(12): 100-103. (in Chinese) doi: 10.3969/j.issn.1000-7008.2016.12.024 [10] 顾永奇. 组合式超大尺寸精密测量技术在EAST超导托卡马克装置中的应用[D]. 合肥工业大学, 2012. Gu Yongqi. Research on combined large-scale precision measur-ing techniques and application to the EAST superconducting tokamak device[D]. Hefei: Hefei University of Technology, 2012. (in Chinese) [11] Xu Yaming, Zheng Qi, Guan Xiao. Measurement accuracy analysis of Leica AT960 laser tracker [J]. Journal of Geomatics, 2020, 45(1): 8-12. (in Chinese) [12] Zhang Shuai, Miao Dongjing, Li Jishuang, et al. Influence of tracking mode on measurement accuracy in multi-purpose pose measurement system [J]. Acta Metrology Sinica, 2020, 41(9): 1055-1061. (in Chinese) doi: 10.3969/j.issn.1000-1158.2020.09.04 [13] Galliana Flavio, Lanzillotti Marco. Report of a multilateral accurate measurement comparison on a high-precision multimeter to evaluate the traceability transfer from INRIM in the field of low-frequency electrical quantities [J]. Journal of Metrology Society of India, 2019, 34(1): 473-478. doi: 10.1007/s12647-019-00317-9 [14] 沈睿. 激光跟踪仪多站测量的精度提升方法研究[D]. 电子科技大学, 2020. Shen Rui. Research on the method of improving the accuracy of multi Station measurement of laser tracker[D]. Chengdu: University of Electronic Science and Technology of China, 2020. (in Chinese) [15] Bachevskii S V, Martem’yanov I S. Cooperative processing of measurements in multilateral radial-range difference system of passive radiolocation [J]. Russian Aeronautics (Iz VUZ), 2015, 58(4): 478-483. doi: 10.3103/S1068799815040200 [16] Wu Bin, Xu You, Yang Fengting, et al. 3D coordinate measuring system based on laser tracking absolute length measurement multilateral method [J]. Infrared and Laser Engineering, 2008, 37(8): 140-145. (in Chinese) [17] 闫昊. 面向特大零件激光跟踪测量精度的提升方法研究[D]. 北京工业大学, 2014.
Yan Hao. Method to improve the laser tracking measuring accuracy for large-scale parts[D]. Beijing: Beijing University of Technology, 2014. (in Chinese)