CN106052556A - Airspace coordinate correction method for three-coordinate measuring machine based on multi-station measurement of laser tracking instrument - Google Patents

Abstract

The invention discloses an airspace coordinate correction method for a three-coordinate measuring machine based on multi-station measurement of a laser tracking instrument. The method comprises the steps of firstly, dividing measuring point grids within the measuring space range of the three-coordinate measuring machine, determining measuring point coordinates, moving a target mirror to each measuring point during measuring, and performing, by the laser tracking instrument, station transfer measurement beyond the grid space range to acquire a relative interference length measurement value from each measuring point to a first measuring point under different stations; secondly, solving the coordinates of each station and the distance between the corresponding station and the first measuring point by using a two-point distance formula and the principle of least square method; thirdly, solving the correction value of each measuring point via an interference length measurement error equation by using the coordinates of each station, the measuring point coordinates and the distance between each station and the first measuring point; fourthly, acquiring more accurate measuring point correction values by adopting an iteration method for improving the station coordinate precision and the distance precision from the stations to the first measuring point; and finally, acquiring the correction value of any measuring point within the grid space by using a trilinear interpolation method, thereby improving the measuring precision of the three-coordinate measuring machine.

Description

A kind of three coordinate measuring machine spatial domain coordinate measured based on laser traces instrument multi-court position is repaiied Correction method

Technical field

The present invention relates to a kind of raising three coordinate measuring machine (Coordinate Measuring Machine is called for short CMM) The analysis method of certainty of measurement, is based particularly on the three coordinate measuring machine spatial domain coordinate modification side that laser traces instrument multi-court position is measured Method, belongs to Technology of Precision Measurement and coordinate measuring technology field.

Background technology

Three coordinate measuring machine is as precision measurement system high efficiency in coordinate measuring technology, high with its certainty of measurement, fast The features such as degree is fast, flexibility is strong, play the most important effect in the fields such as the manufacturing modernized and Aeronautics and Astronautics, It is the key foundation measurement equipment in advanced manufacture field, is also quality testing and the crucial test of control in civilian industry production Equipment, it is possible to the 3 d space coordinate completing the geometric element of various part, curve and curved surface is measured, and can realize on-line checking And automatic measurement.Along with progress and the development of Ultraprecision Machining of science and technology, to three coordinate measuring engine measurement precision Requirement more and more higher.And fast and accurately CMM is demarcated, detect every error of CMM and carry out error benefit Repay, be one of important channel improving CMM certainty of measurement, be a kind of elder generation increasing substantially CMM certainty of measurement at lower cost Enter technological means.

The approach and the measure that improve coordinate measuring machine accuracy have a variety of, such as, improve frame for movement precision, reduce power change Shape, thermal deformation, raising scale precision and use suitable sampling policy etc..Owing to coordinate measuring machine structure is complicated, from raising The means of frame for movement precision ensure its precision, and not only cost is high, and the precision improved is extremely limited.Therefore in high precision, efficiently The coordinate measuring machine calibration technique of rate becomes the advanced technology means improving coordinate measuring machine certainty of measurement, and Error Compensation Technology exists Coordinate measuring machine is widely applied.What coordinate measuring machine scaling method was the more commonly used at present is laser interferometer, autocollimatic The straight high precision instrument such as instrument, optical sqaure is directly separated 21 errors, utilizes bat, ball row, ball plate etc. indirectly to separate coordinate 21 errors of measuring machine.

It is to grow up on the basis of robot meterological that laser follows the tracks of three-dimensional coordinate measurement technology the eighties in last century A kind of novel coordinate measuring technology.Since laser tracking measurement system is developed out for the first time, towards on-the-spot portable Formula coordinate system laser tracker solves coordinate measuring machine and demarcates efficiency and a difficult problem for precision raising.Follow the tracks of based on laser The measuring principle of instrument, by the location of the global positioning system (Global Positioning System, GPS) under many base stations Method can realize the demarcation of CMM.

The follower of traditional laser tracking system is the rotating shaft precise rotation device that can intersect around space two vertical axis, Two fixed joint are used to control the rotation direction of tracking lens or interferometer beam respectively, if two revolved when rotating shaft rotates The intersection point of shaft axis is unstable, will cause measurement error, laser interferometer even can be caused to break light, cause whole system to be measured Interrupt.The rigging error of tracking mirror, structural instability and thermal deformations etc. simultaneously also bring along error.It can be seen that compare In linear measure longimetry, angular surveying more can affect the uncertainty of measurement of laser tracking system.Three are carried out using laser tracker In the demarcation of coordinate measuring machine, although only used laser tracker precise interference and surveyed the long measuring method combining multi-court position, swashed The three-dimensional ball measurement of coordinates of optical tracker system is not used, but due to the restriction of conventional commercial laser tracker frame for movement, Make conventional laser follow the tracks of system accuracy to be difficult to improve.

German National quantitative study institute (PTB) and United Kingdom National physics laboratory (NPL) joint research and development are specifically designed to school The laser traces instrument of quasi-Digit Control Machine Tool and coordinate measuring machine, using the teaching of the invention it is possible to provide the range measurement of full accuracy.Interferometer is contained in universal In regulating device, the fixing spheroid around the reference mirror being only used as interferometer moves.Due to this principle, the radial direction of rotating machinery shaft Can't appreciable impact certainty of measurement with lateral deviation.The precision that laser is followed the tracks of is not affected by frame for movement, depends primarily on Quality and the change of locus with reference to sphere.Certainty of measurement can be increased substantially, the whole detection cycle can be shortened again.

It is necessary for this to design a kind of three coordinate measuring machine spatial domain coordinate modification measured based on laser traces instrument multi-court position Method, it is possible to high-precision calibrating three coordinate measuring machine, to improve the certainty of measurement of three coordinate measuring machine space measurement point.

Summary of the invention

It is an object of the invention to propose a kind of three coordinate measuring machine spatial domain coordinates compensation method, it is therefore an objective to provide a kind of base Measure in laser traces instrument multi-court position and utilize Tri linear interpolation (Trilinear interpolation) to space any point The error compensating method being modified, enables to improve the certainty of measurement of three coordinate measuring machine in reality is measured.Compare existing Some analysis methods, this method has the features such as certainty of measurement is high, measurement cost is relatively low and simple to operate.

For reaching object above, the present invention adopts the following technical scheme that:

A kind of three coordinate measuring machine spatial domain coordinates compensation method measured based on laser traces instrument multi-court position, the method includes Following step:

Step one: build laser traces instrument multi-court position measurement model.Under CMM coordinate system, if tested point in CMM measurement space For A_i(x_i,y_i,z_i), wherein i=1,2,3 ..., n；The centre of sphere of laser traces instrument internal standard ball is O；The station of laser traces instrument Position coordinate is P_j(X_j,Y_j,Z_j), wherein j=1,2,3 ..., m；P_jTo A₁The distance of point is d_j；Laser traces instrument during measurement Measurement data be l_ij, such as Fig. 1.Following relationship is set up by 2 range formulas of three dimensions:

$\sqrt{{(x_i-X_j)}^2+{(y_i-Y_j)}^2+{(z_i-Z_j)}^2}=d_j+l_{ij}---\left(1\right)$

Equation number is m × n, and unknown number number is 4m+3n.For making equation group solution should meet:

m×n≥4m+3n (2)

Then m and n meets m >=4, n >=16.

Step 2: dividing measurement space, such as Fig. 2, the summit in small cubes space is tested point, determines tested point A_i? Coordinate figure (x in the range of CMM measurement space_i,y_i,z_i), the mobile route of object of planning target mirror is as shown in Figure 3.Laser traces instrument Erect-position be P₁, control CMM moves target mirror and moves to tested point A according to the path planned_i, and measure laser now Measurement data l of tracker_i1.Move laser traces instrument successively to each erect-position P_j, wherein j=1,2,3 ..., m, and by planning Path running target mirror completes all tested point measurement data l_ijMeasurement.

Step 3: simultaneously square and transposition obtains equation by formula (1) equal sign both sides:

$x_i^2+y_i^2+z_i^2-2x_i{X}_j-2y_i{Y}_j-2z_i{Z}_j+X_j^2+Y_j^2+Z_j^2-d_j^2-2d_j{l}_{ij}-l_{ij}^2=0---\left(3\right)$

OrderThen formula (3) is converted into:

$x_i^2+y_i^2+z_i^2-2x_i{X}_j-2y_i{Y}_j-2z_i{Z}_j+k-2d_j{l}_{ij}-l_{ij}^2=0---\left(4\right)$

According to method of least square, object function is defined as:

$F(X_j,Y_j,Z_j,k)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(x_i^2+y_i^2+z_i^2-2x_i{X}_j-2y_i{Y}_j-2z_i{Z}_j+k-2d_j{l}_{ij}-l_{ij}^2)}^2---\left(5\right)$

Make F (X_j,Y_j,Z_j, k) minimum, (5) formula should meet following condition:

$\frac{\∂F}{\∂X_j}=0,\frac{\∂F}{\∂Y_j}=0,\frac{\∂F}{\∂Z_j}=0,\frac{\∂F}{\∂d_j}=0,\frac{\∂F}{\∂k}=0---\left(6\right)$

Meet simultaneously:

$\begin{array}{c}\frac{{\∂}^{2}F}{\∂X_j^2}=8\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i^2>0,\\ \frac{{\∂}^{2}F}{\∂Y_j^2}=8\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i^2>0,\\ \frac{{\∂}^{2}F}{\∂Z_j^2}=8\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i^2>0,\\ \frac{{\∂}^{2}F}{\∂d_j^2}=8\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}^2>0,\\ \frac{{\∂}^{2}F}{\∂k^2}=2>0\end{array}---\left(7\right)$

Formula (6) is write as matrix form:

$\left[\begin{array}{ccccc}2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i^2& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{y}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{z}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{l}_{ij}& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i\\ 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{y}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i^2& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i{z}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i{l}_{ij}& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i\\ 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{z}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i{z}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i^2& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i{l}_{ij}& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i\\ 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{l}_{ij}& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i{l}_{ij}& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i{l}_{ij}& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}^2& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}\\ -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}& \frac{n}{2}\end{array}\right]\left[\begin{array}{c}{X}_j\\ Y_j\\ Z_j\\ d_j\\ k\end{array}\right]=\left[\begin{array}{c}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i(x_i^2+y_i^2+z_i^2-l_{ij}^2)\\ \underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i(x_i^2+y_i^2+z_i^2-l_{ij}^2)\\ \underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i(x_i^2+y_i^2+z_i^2-l_{ij}^2)\\ \underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}(x_i^2+y_i^2+z_i^2-l_{ij}^2)\\ -\frac{1}{2}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}(x_i^2+y_i^2+z_i^2-l_{ij}^2)\end{array}\right]---\left(8\right)$

Solution formula (8) can get erect-position coordinate P_i(X_j,Y_j,Z_j) and d_j。

Step 4: formula (1) is write as error equation:

$v_{ij}=\sqrt{{(x_i-X_j)}^2+{(y_i-Y_j)}^2+{(z_i-Z_j)}^2}-d_j-l_{ij}---\left(9\right)$

The error sum of squares utilizing Least Square in Processing formula (9) to obtain is:

$E(x_1,y_1,z_1,...x_n,y_n,z_n,X_1,Y_1,Z_1,...,X_m,Y_m,Z_m)=\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{m}{\Σ}}{v}_{ij}^2---\left(10\right)$

Formula (10) is a nonlinear equation, solves the following calculating process of using for convenience:

Order

$L_{ij}=\sqrt{{(x_i-X_j)}^2+{(y_i-Y_j)}^2+{(z_i-Z_j)}^2}---\left(11\right)$

Utilize Taylor series expansion that formula (11) is carried out Taylor series expansion, obtain equation below:

$\begin{array}{c}{L}_{ij}\≈L_{ij}{|}_0+\frac{\∂L_{ij}}{\∂x_i}{|}_0\·{\mathrm{dx}}_i+\frac{\∂L_{ij}}{\∂y_i}{|}_0\·{\mathrm{dy}}_i+\frac{\∂L_{ij}}{\∂z_i}{|}_0\·{\mathrm{dz}}_i\\ +\frac{\∂L_{ij}}{\∂X_j}{|}_0\·{\mathrm{dX}}_j+\frac{\∂L_{ij}}{\∂Y_j}{|}_0\·{\mathrm{dY}}_j+\frac{\∂L_{ij}}{\∂Z_j}{|}_0\·{\mathrm{dZ}}_j\end{array}---\left(12\right)$

Formula (12) is substituted into formula (9), and abbreviation has after arranging:

$\begin{array}{c}{v}_{ij}=L_{ij}{|}_0+\frac{x_i{|}_0-X_j{|}_0}{L_{ij}{|}_0}\·({\mathrm{dx}}_i-{\mathrm{dX}}_j)+\frac{y_i{|}_0-Y_j{|}_0}{L_{ij}{|}_0}\·({\mathrm{dy}}_i-{\mathrm{dY}}_j)\\ +\frac{z_i{|}_0-Z_j{|}_0}{L_{ij}{|}_0}\·({\mathrm{dz}}_i-{\mathrm{dZ}}_j)-d_j-l_{ij}\end{array}---\left(13\right)$

Wherein:After equation (13) is optimization Solving model.In formula (12), (13), it is designated as |₀The approximation for this numerical value, x_i|₀、y_i|₀、z_i|₀Thered is provided by CMM, X_j|₀、Y_j |₀、Z_j|₀、d_jObtained by solving equation group formula (8).

Make v_ij=0, formula (13) is write as the form of matrix:

Ax=B (14)

Wherein:

$x={\[{\mathrm{dx}}_1,{\mathrm{dy}}_1,{\mathrm{dz}}_1,...,{\mathrm{dx}}_n,{\mathrm{dy}}_n,{\mathrm{dz}}_n,{\mathrm{dX}}_1,{\mathrm{dY}}_1,{\mathrm{dZ}}_1,...,{\mathrm{dX}}_m,{\mathrm{dY}}_m,{\mathrm{dZ}}_m\]}_{1\×(3n+3m)}^T---\left(15\right)$

$b={\[d_1+l_{11}-L_{11}{|}_0,...,d_j+l_{ij}-L_{ij}{|}_0,...,d_m+l_{nm}-L_{nm}{|}_0\]}_{1\×nm}^T---\left(16\right)$

Wherein dx_i、dy_i、dz_iAnd dX_j、dY_j、dZ_jCorrection value and erect-position P for tested point corresponding coordinate_jCorresponding coordinate Correction value.Solving equations (14) can get tested point A_iCorrection value (dx_i,dy_i,dz_i)。

Step 5: according to formula (12), affect tested point A_iCorrection value (dx_i,dy_i,dz_i) variable of precision has x_i|₀、y_i|₀、 z_i|₀And X_j|₀、Y_j|₀、Z_j|₀、d_j, wherein x_i|₀、y_i|₀、z_i|₀I.e. it is defined as constant after completing step one, therefore X_j|₀、Y_j|₀、 Z_j|₀、d_jThe precision solved directly affects (dx_i,dy_i,dz_i) solving precision.Recursion alternative manner is used to improve X_j|₀、Y_j|₀、Z_j |₀、d_jThe precision solved.

The step of alternative manner is as follows:

1. by tested point coordinate (x_i|₀,y_i|₀,z_i|₀) with the correction value (dx obtained_i,dy_i,dz_i) be added, obtain (x_i|₀', y_i|₀',z_i|₀')；

2. by (x_i|₀',y_i|₀',z_i|₀') and step 2 in the l that obtains_ijSubstitution formula (8), solves and obtains P_j'(X_j',Y_j', Z_j') and d_j'；

3. by (x_i|₀',y_i|₀',z_i|₀')、P_j'(X_j',Y_j',Z_j') and d_j' substitute into formula (14), solve and obtain in iteration Between correction value (dx_i',dy_i',dz'_i).When iterations is equal to 1, by (dx_i',dy_i',dz'_i) obtain with step 4 (dx_i,dy_i,dz_i) compare；When iterations is more than 1, by middle correction value and the centre of last iteration of current iteration Correction value compares, and sees whether the order of magnitude of numerical value is reducing, if there being reduction trend, follow-up need proceed iteration, as Do not reduce and then terminate iteration；

4. by (x_i|₀,y_i|₀,z_i|₀)、P_j'(X_j',Y_j',Z_j') and d_j' substituting into formula (14), solving equation group obtains high accuracy Correction value；

5. repeat step and 1. arrive process 4., the most 1. in and (x_i|₀,y_i|₀,z_i|₀) carry out additive operation correction value total Up-to-date solving obtains, until iteration ends.

Step 6: assuming that Q is any point that CMM records in the range of measurement space, coordinate is (x_Q,y_Q,z_Q).Determine Q Point small cubes space belonging in grid division space.If cubical 8 summits are A, B, C, D, E, F, G, H, wherein Plane ADHE is perpendicular to the x-axis of CMM, and the distance of Q point to plane ADHE is L_x；Plane ABFE is perpendicular to the y-axis of CMM, and Q point is to flat The distance of face ABFE is L_y；Plane ABCD is perpendicular to the z-axis of CMM, and the distance of Q point to plane ABCD is L_z。

The measuring point correction value of A, B, C, D, E, F, G, H is obtained by step 5.The correction value utilizing 8 summits passes through formula (17) method of Tri linear interpolation obtains the error correction values of this measuring point.

$\begin{array}{c}{\mathrm{\Δ}}_Q=(1-k_3)\[(1-k_1)(1-k_2){\mathrm{\Δ}}_A+k_1(1-k_2){\mathrm{\Δ}}_B+k_1{k}_2{\mathrm{\Δ}}_C+(1-k_1)k_2{\mathrm{\Δ}}_D\]\\ +k_3)\[(1-k_1)(1-k_2){\mathrm{\Δ}}_E+k_1(1-k_2){\mathrm{\Δ}}_F+k_1{k}_2{\mathrm{\Δ}}_G+(1-k_1)k_2{\mathrm{\Δ}}_H\]\end{array}---\left(17\right)$

WhereinΔ_A、Δ_B、Δ_C、Δ_D、Δ_E、Δ_F、Δ_G、Δ_H、Δ_Q, be respectively The coordinate modification value of A, B, C, D, E, F, G, H, Q.

Utilize the Δ that formula (17) is tried to achieve_QIt is the spatial correction values of Q point, the measuring point that this correction value provides plus CMM is sat Mark (x_Q,y_Q,z_Q), it is and finally optimizes the high-precision coordinate value obtained.

In sum, based on laser traces instrument multi-court position measurement technology, interfere with the high accuracy of laser traces instrument and survey Long value is constraints, is iterated correction value processing, finally by the modification method of Tri linear interpolation, it is possible to effectively carry The precision that high CMM measuring point is measured.

Accompanying drawing explanation

Fig. 1 is laser traces instrument interference length-measuring schematic diagram；

Fig. 2 is that measurement space divides schematic diagram；

Fig. 3 is the mobile route of target mirror；

Fig. 4 is CMM laser traces instrument multistation level measuring system；

Fig. 5 a is the curve chart of x-axis adjustment in direction value；

Fig. 5 b is the curve chart of y-axis adjustment in direction value；

Fig. 5 c is the curve chart of z-axis adjustment in direction value；

Fig. 6 is measurement space stress and strain model and Tri linear interpolation schematic diagram；

Fig. 7 a is the space coordinates correction value in x-axis direction；

Fig. 7 b is the space coordinates correction value in y-axis direction；

Fig. 7 c is the space coordinates correction value in z-axis direction.

Detailed description of the invention

The present invention is described in further detail below in conjunction with the accompanying drawings, to make those skilled in the art with reference to description literary composition Word can be implemented according to this.

Experiment use CMM laser traces instrument multistation level measuring system as shown in Figure 4 survey long value to carry out relative interference Measurement, be therefore analyzed by following step:

Step one: build laser traces instrument multi-court position measurement model.Consider certainty of measurement and experiment required time, determine sharp The number of light tracker erect-position is that 5,5 erect-positions are not in approximately the same plane.Tested point number under CMM coordinate system in space is 4 × 4 × 4=64, the measuring point number under sustained height is 4 × 4=16.

Step 2: divide measurement space, determine the coordinate of 64 tested points, corresponding coordinate such as table 1,

The coordinate of table 1 tested point

Laser traces instrument is positioned over default erect-position P₁, as shown in Figure 4, by the preset path running target mirror of Fig. 3, recorded Data l that when reaching tested point, laser traces instrument records_i1, until completing the measurement of whole 64 measuring points.Carry out turning station subsequently, successively Mobile laser traces instrument is to erect-position P₂, erect-position P₃, erect-position P₄, erect-position P₅, complete all tested points by path planning running target mirror Measure and record measurement data l_ij, record 5 × 64=320 numerical value altogether.

Step 3: by 64 measuring point coordinates and 320 measurement data l recorded_ijSubstitution formula (8) solving equation group, i.e. The erect-position coordinate P of laser traces instrument can be solved_j(X_j,Y_j,Y_j)、P_jTo A₁The distance of point.

Step 4: by the coordinate of 64 measuring points, 320 measurement data l_ij, laser traces instrument erect-position coordinate P_j(X_j,Y_j,Y_j) And P_jTo A₁Distance d of point_jSubstitution formula (14), solving equation group i.e. can get the space coordinates correction value of 64 measuring points.

Step 5: use iterative algorithm to improve the precision of measuring point coordinate modification value.Specifically comprise the following steps that

1. by 64 tested point coordinate (x_i|₀,y_i|₀,z_i|₀) with the correction value (dx obtained_i,dy_i,dz_i) be added, obtain 64 Individual coordinate (x_i|₀',y_i|₀',z_i|₀')；

2. by 64 coordinate (x_i|₀',y_i|₀',z_i|₀') and step 2 in 320 l obtaining_ijSubstitution formula (8), solves To P_j'(X_j',Y_j',Z_j') and d_j'；

3. by (x_i|₀',y_i|₀',z_i|₀')、P_j'(X_j',Y_j',Z_j') and d_j' substitute into formula (14), solve and obtain in iteration Between correction value (dx_i',dy_i',dz'_i).When iterations is equal to 1, by (dx_i',dy_i',dz_i') obtain with step 4 (dx_i,dy_i,dz_i) compare；When iterations is more than 1, by middle correction value and the centre of last iteration of current iteration Correction value compares, and sees whether the order of magnitude of numerical value is reducing, if there being reduction trend, follow-up need proceed iteration, as Do not reduce and then terminate iteration；

4. by (x_i|₀,y_i|₀,z_i|₀)、P_j'(X_j',Y_j',Z_j') and d_j' substituting into formula (14), solving equation group obtains precision relatively High correction value；

5. repeat step and 1. arrive process 4., the most 1. in and (x_i|₀,y_i|₀,z_i|₀) carry out additive operation correction value total Up-to-date solving obtains, until iteration ends.

For determining that iterative algorithm is effective in the present invention, also carry out an emulation experiment at this and verified. Emulation uses 18 measuring points, 4 erect-positions, the error of introducing and the result such as table 2 tried to achieve.

The error of table 2x direction of principal axis introducing, non-iteration correction value, 1 correction value of iteration, 3 correction values of iteration (unit: mm)

Respectively not iteration correction value, 1 correction value of iteration, 3 correction values of iteration are made comparisons with the error of introducing, it is possible to To be clear that, the correction value the tried to achieve ratio after adding iteration is not added with the correction value error closer to introducing of iteration, Therefore iterative algorithm is effective, and the algorithm adding iteration can obtain correction value the most accurately than method before.

The data obtained previous experiments by algorithm are calculated, the x-axis adjustment in direction of 64 measuring points finally tried to achieve Value is such as Fig. 5 a, y-axis adjustment in direction value such as Fig. 5 b, z-axis adjustment in direction value such as Fig. 5 c.It can be seen that CMM is without compensation X-axis measurement error is between-0.0026mm to 0.0035mm；Y-axis measurement error is between-0.0025mm to 0.0019mm；Z-axis Measurement error is between-0.0056mm to 0.0060mm.

The coordinate of the Q point that step 6: CMM records in the range of measurement space is (x_Q,y_Q,z_Q), determine that Q point is surveyed at grid Small cubes space belonging in quantity space, these cubical eight summits are respectively A, B, C, D, E, F, G, H, such as Fig. 6, and really Fixed corresponding coordinate, the space coordinates correction value of these 8 points is calculated by step 5, respectively Δ_A、Δ_B、Δ_C、Δ_D、Δ_E、 Δ_F、Δ_G、Δ_H、Δ_Q, calculate corresponding k₁、k₂、k₃, substitute into formula (17), the space coordinates correction value of P point can be obtained, this Sample just completes the space coordinates correction to pre-set space, such as Fig. 7 a, Fig. 7 b, Fig. 7 c.

By Fig. 7 a, Fig. 7 b, Fig. 7 c, it is clear that the x-axis of CMM, y-axis precision are preferable, and the precision of z-axis is relatively Difference；(270～340mm) × (700～800mm) × (-500～-450mm), (200～240mm) × (800～850mm) × (- 540～-400mm) it is that this grid measures the optimum measurement space divided.

Claims (1)

1. the three coordinate measuring machine spatial domain coordinates compensation method measured based on laser traces instrument multi-court position, it is characterised in that: The method comprises the steps:

Step one: build laser traces instrument multi-court position measurement model；Under CMM coordinate system, if tested point is A in CMM measurement space_i (x_i,y_i,z_i), wherein i=1,2,3 ..., n；The centre of sphere of laser traces instrument internal standard ball is O；The erect-position of laser traces instrument is sat It is designated as P_j(X_j,Y_j,Z_j), wherein j=1,2,3 ..., m；P_jTo A₁The distance of point is d_j；The survey of laser traces instrument during measurement Amount data are l_ij, set up following relationship by 2 range formulas of three dimensions:

$\sqrt{{(x_i-X_j)}^2+{(y_i-Y_j)}^2+{(z_i-Z_j)}^2}=d_j+l_{ij}---\left(1\right)$

Equation number is m × n, and unknown number number is 4m+3n；For making equation group solution should meet:

m×n≥4m+3n (2)

Then m and n meets m >=4, n >=16；

Step 2: dividing measurement space, the summit in small cubes space is tested point, determines tested point A_iAt CMM measurement space model Enclose interior coordinate figure (x_i,y_i,z_i)；The erect-position of laser traces instrument is P₁, control CMM and move target mirror according to the road planned Footpath is moved to tested point A_i, and measurement data l of the laser traces instrument that measurement is now_i1；Move laser traces instrument successively and arrive each Erect-position P_j, wherein j=1,2,3 ..., m, and complete all tested point measurement data l by path planning running target mirror_ijMeasurement；

Step 3: simultaneously square and transposition obtains equation by formula (1) equal sign both sides:

$x_i^2+y_i^2+z_i^2-2x_i{X}_j-2y_i{Y}_j-2z_i{Z}_j+X_j^2+Y_j^2+Z_j^2-d_j^2-2d_j{l}_{ij}-l_{ij}^2=0---\left(3\right)$

OrderThen formula (3) is converted into:

$x_i^2+y_i^2+z_i^2-2x_i{X}_j-2y_i{Y}_j-2z_i{Z}_j+k-2d_j{l}_{ij}-l_{ij}^2=0---\left(4\right)$

According to method of least square, object function is defined as:

$F(X_j,Y_j,Z_j,k)=\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(x_i^2+y_i^2+z_i^2-2x_i{X}_j-2y_i{Y}_j-2z_i{Z}_j+k-2d_j{l}_{ij}-l_{ij}^2)}^2---\left(5\right)$

Make F (X_j,Y_j,Z_j, k) minimum, (5) formula should meet following condition:

$\frac{\∂F}{\∂X_j}=0,\frac{\∂F}{\∂Y_j}=0,\frac{\∂F}{\∂Z_j}=0,\frac{\∂F}{\∂d_j}=0,\frac{\∂F}{\∂k}=0---\left(6\right)$

Meet simultaneously:

$\frac{{\∂}^{2}F}{\∂X_j^2}=8\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i^2>0,$

$\frac{{\∂}^{2}F}{\∂Y_j^2}=8\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i^2>0,$

$\frac{{\∂}^{2}F}{\∂Z_j^2}=8\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i^2>0,---\left(7\right)$

$\frac{{\∂}^{2}F}{\∂d_j^2}=8\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}^2>0,$

$\frac{{\∂}^{2}F}{\∂k^2}=2>0$

Formula (6) is write as matrix form:

$\left[\begin{array}{ccccc}2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i^2& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{y}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{z}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{l}_{ij}& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i\\ 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{y}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i^2& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i{z}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i{l}_{ij}& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i\\ 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{z}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i{z}_i& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i^2& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i{l}_{ij}& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i\\ 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i{l}_{ij}& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i{l}_{ij}& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i{l}_{ij}& 2\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}^2& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}\\ -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i& -\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}& \frac{n}{2}\end{array}\right]\left[\begin{array}{c}{X}_j\\ Y_j\\ Z_j\\ d_j\\ k\end{array}\right]=\left[\begin{array}{c}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{x}_i(x_i^2+y_i^2+z_i^2-l_{ij}^2)\\ \underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i(x_i^2+y_i^2+z_i^2-l_{ij}^2)\\ \underset{i=1}{\overset{n}{\mathrm{\Σ}}}{z}_i(x_i^2+y_i^2+z_i^2-l_{ij}^2)\\ \underset{i=1}{\overset{n}{\mathrm{\Σ}}}{l}_{ij}(x_i^2+y_i^2+z_i^2-l_{ij}^2)\\ -\frac{1}{2}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}(x_i^2+y_i^2+z_i^2-l_{ij}^2)\end{array}\right]---\left(8\right)$

Solution formula (8) can get erect-position coordinate P_i(X_j,Y_j,Z_j) and d_j；

Step 4: formula (1) is write as error equation:

$v_{ij}=\sqrt{{(x_i-X_j)}^2+{(y_i-Y_j)}^2+{(z_i-Z_j)}^2}-d_j-l_{ij}---\left(9\right)$

The error sum of squares utilizing Least Square in Processing formula (9) to obtain is:

$E(x_1,y_1,z_1,...x_n,y_n,z_n,X_1,Y_1,Z_1,...,X_m,Y_m,Z_m)=\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{m}{\Σ}}{v}_{ij}^2---\left(10\right)$

Formula (10) is a nonlinear equation, solves the following calculating process of using for convenience:

Order

$L_{ij}=\sqrt{{(x_i-X_j)}^2+{(y_i-Y_j)}^2+{(z_i-Z_j)}^2}---\left(11\right)$

Utilize Taylor series expansion that formula (11) is carried out Taylor series expansion, obtain equation below:

$\begin{array}{c}{L}_{ij}\≈L_{ij}{|}_0+\frac{\∂L_{ij}}{\∂x_i}{|}_0\·{\mathrm{dx}}_i+\frac{\∂L_{ij}}{\∂y_i}{|}_0\·{\mathrm{dy}}_i+\frac{\∂L_{ij}}{\∂z_i}{|}_0\·{\mathrm{dz}}_i\\ +\frac{\∂L_{ij}}{\∂X_j}{|}_0\·{\mathrm{dX}}_j+\frac{\∂L_{ij}}{\∂Y_j}{|}_0\·{\mathrm{dY}}_j+\frac{\∂L_{ij}}{\∂Z_j}{|}_0\·{\mathrm{dZ}}_j\end{array}---\left(12\right)$

Formula (12) is substituted into formula (9), and abbreviation has after arranging:

$\begin{array}{c}{v}_{ij}=L_{ij}{|}_0+\frac{x_i{|}_0-X_j{|}_0}{L_{ij}{|}_0}\·({\mathrm{dx}}_i-{\mathrm{dX}}_j)+\frac{y_i{|}_0-Y_j{|}_0}{L_{ij}{|}_0}\·({\mathrm{dy}}_i-{\mathrm{dY}}_j)\\ +\frac{z_i{|}_0-Z_j{|}_0}{L_{ij}{|}_0}\·({\mathrm{dz}}_i-{\mathrm{dZ}}_j)-d_j-l_{ij}\end{array}---\left(13\right)$

Wherein:Equation (13) be optimization after solve Model；In formula (12), (13), it is designated as |₀The approximation for this numerical value, x_i|₀、y_i|₀、z_i|₀Thered is provided by CMM, X_j|₀、Y_j|₀、Z_j |₀、d_jObtained by solving equation group formula (8)；

Make v_ij=0, formula (13) is write as the form of matrix:

Ax=B (14)

Wherein:

$x={\[{\mathrm{dx}}_1,{\mathrm{dy}}_1,{\mathrm{dz}}_1,...,{\mathrm{dx}}_n,{\mathrm{dy}}_n,{\mathrm{dz}}_n,{\mathrm{dX}}_1,{\mathrm{dY}}_1,{\mathrm{dZ}}_1,...,{\mathrm{dX}}_m,{\mathrm{dY}}_m,{\mathrm{dZ}}_m\]}_{1\×(3n+3m)}^T---\left(15\right)$

$b={\[d_1+l_{11}-L_{11}{|}_0,...,d_j+l_{ij}-L_{ij}{|}_0,...,d_m+l_{nm}-L_{nm}{|}_0\]}_{1\×nm}^T---\left(16\right)$

Wherein dx_i、dy_i、dz_iAnd dX_j、dY_j、dZ_jCorrection value and erect-position P for tested point corresponding coordinate_jThe correction of corresponding coordinate Value；Solving equations (14) can get tested point A_iCorrection value (dx_i,dy_i,dz_i)；

Step 5: according to formula (12), affect tested point A_iCorrection value (dx_i,dy_i,dz_i) variable of precision has x_i|₀、y_i|₀、z_i|₀With X_j|₀、Y_j|₀、Z_j|₀、d_j, wherein x_i|₀、y_i|₀、z_i|₀I.e. it is defined as constant after completing step one, therefore X_j|₀、Y_j|₀、Z_j|₀、d_j The precision solved directly affects (dx_i,dy_i,dz_i) solving precision；Recursion alternative manner is used to improve X_j|₀、Y_j|₀、Z_j|₀、d_j The precision solved；

The step of alternative manner is as follows:

1. by tested point coordinate (x_i|₀,y_i|₀,z_i|₀) with the correction value (dx obtained_i,dy_i,dz_i) be added, obtain (x_i|₀',y_i |₀',z_i|₀')；

2. by (x_i|₀',y_i|₀',z_i|₀') and step 2 in the l that obtains_ijSubstitution formula (8), solves and obtains P_j'(X_j',Y_j',Z_j') And d_j'；

3. by (x_i|₀',y_i|₀',z_i|₀')、P_j'(X_j',Y_j',Z_j') and d_j' substitute into formula (14), solve and obtain the centre of iteration and repair On the occasion of (dx_i',dy_i',dz'_i)；When iterations is equal to 1, by (dx_i',dy_i',dz'_i) (the dx that obtains with step 4_i,dy_i, dz_i) compare；When iterations is more than 1, the middle correction value of current iteration and the middle correction value of last iteration are entered Row compares, and sees whether the order of magnitude of numerical value is reducing, if there being reduction trend, follow-up need proceed iteration, as do not reduced Then terminate iteration；

4. by (x_i|₀,y_i|₀,z_i|₀)、P_j'(X_j',Y_j',Z_j') and d_j' substituting into formula (14), solving equation group obtains high-precision repairing On the occasion of；

5. repeat step and 1. arrive process 4., the most 1. in and (x_i|₀,y_i|₀,z_i|₀) carry out the correction value of additive operation Newly solve and to obtain, until iteration ends；

Step 6: assuming that Q is any point that CMM records in the range of measurement space, coordinate is (x_Q,y_Q,z_Q)；Determine that Q point exists Small cubes space belonging in grid division space；If cubical 8 summits are A, B, C, D, E, F, G, H, its midplane ADHE is perpendicular to the x-axis of CMM, and the distance of Q point to plane ADHE is L_x；Plane ABFE is perpendicular to the y-axis of CMM, and Q point is to plane The distance of ABFE is L_y；Plane ABCD is perpendicular to the z-axis of CMM, and the distance of Q point to plane ABCD is L_z；

The measuring point correction value of A, B, C, D, E, F, G, H is obtained by step 5；Utilize the correction value on 8 summits by formula (17) three The method of linear interpolation obtains the error correction values of this measuring point；

$\begin{array}{c}{\mathrm{\Δ}}_Q=(1-k_3)\[(1-k_1)(1-k_2){\mathrm{\Δ}}_A+k_1(1-k_2){\mathrm{\Δ}}_B+k_1{k}_2{\mathrm{\Δ}}_C+(1-k_1)k_2{\mathrm{\Δ}}_D\]\\ +k_3\[(1-k_1)(1-k_2){\mathrm{\Δ}}_E+k_1(1-k_2){\mathrm{\Δ}}_F+k_1{k}_2{\mathrm{\Δ}}_G+(1-k_1)k_2{\mathrm{\Δ}}_H\]\end{array}---\left(17\right)$

WhereinΔ_A、Δ_B、Δ_C、Δ_D、Δ_E、Δ_F、Δ_G、Δ_H、Δ_Q, be respectively A, B, C, The coordinate modification value of D, E, F, G, H, Q；

Utilize the Δ that formula (17) is tried to achieve_QIt is the spatial correction values of Q point, the measuring point coordinate (x this correction value provided plus CMM_Q, y_Q,z_Q), it is and finally optimizes the high-precision coordinate value obtained.