CN102252638A - Data splicing technology for measuring flatness of super large plane - Google Patents

Abstract

The invention relates to a data splicing technology for measuring the flatness of a super large plane. Based on a laser alignment scanning method, a measured plane with an obstacle or the super large plane is partitioned into a plurality of sub areas with a common area and then the sub areas are measured, the measurement result is processed by using a data splicing method, the flatness error is evaluated according to the evaluation criteria and methods of the flatness error, and the flatness error value of the whole plane can be acquired, so that quick measurement of the super large plane and the plane with the obstacle is realized. The method comprises the following steps of: 1, dividing the measured plane into the plurality of sub areas with the common area; 2, measuring the sub areas by using the laser alignment scanning method respectively; and 3, after all the sub areas are measured, normalizing the measurement data. By the technology, the flatness measurement of the large plane and the plane with the obstacle is realized, the defects of long measurement time, incapability of measuring the flatness of an inclined plane and the like in a liquid level method are overcome, and the measurement speed is improved.

Description

Be used to measure the data splicing technology of super large plane flatness

Technical field

The data splicing technology that is used to measure super large plane flatness of the present invention relates to measuring technique, specifically relates to the super large plane and the measurement of planeness technology on barrier plane is arranged.

Background technology

Boats and ships need physical dimension super large (rice up to a hundred) in manufacture process and have the pedestal of barrier to carry out the measurement of planeness at present, and level surface method and laser alignment scanning method are the most frequently used method measuring flatness.Level surface method mainly is to utilize liquid surface as the measuring basis face, can realize big plane and the measurement of planeness on the plane of barrier is arranged.Canister with two filleds with fluid constitutes Communicating device, and one of them container position is fixed, and another container is placed on each measurement point successively by measuring sequence, measures the difference in height of each point apart from datum water level.Measure the fixed container liquid level variation with dial gauge, realize measuring the difference in height of each measured point relative datum surface level with this.But have following shortcoming when using level surface method measurement plane degree error: liquid level settles out at each measurement point needs the regular hour; Measurement result is subjected to the influence of environment factors such as temperature bigger; Can not be used to measure the flatness of clinoplane.

The laser alignment scanning method also is one of method measuring flatness of using always, has fast, the advantage that can automatic data processing of measuring speed.But for the large-scale plane that barrier is arranged, the barrier because laser beam can not detour can not be realized the measurement of planeness.For the measurement of planeness on super large plane, because the restriction of safety in utilization, the energy of laser beam can not be excessive, makes laser can not effectively scan whole big plane again, also just can not the whole big plane of one-shot measurement.At existing problem in the above-mentioned prior art, the data splicing technology of super large plane flatness is measured in a kind of novel being used to of research and design, and existing problem is very necessary in the prior art thereby overcome.

Summary of the invention

In view of existing problem in the above-mentioned prior art, the objective of the invention is a kind of novel being used to of research and design and measure the data splicing technology of super large plane flatness, thereby solve more greatly and the not shortcoming of energy measurement inclined-plane flatness of the existing Measuring Time of level surface method is long, liquid is subjected to environment factors such as temperature influence; And the laser alignment scanning method is existing for the large-scale plane that barrier is arranged, and the barrier because laser beam can not detour can not be realized the measurement of planeness.For the measurement of planeness on super large plane, because the restriction of safety in utilization, the energy of laser beam can not be excessive, makes laser can not effectively scan whole big plane again, also problems such as flatness that just can not the whole big plane of one-shot measurement.

The data splicing technology that is used to measure super large plane flatness of the present invention, be based on the laser alignment scanning method, to tested plane or the ultra-large type plane that barrier is arranged, being divided into some subregions with public domain measures, utilize the data splicing method that the result who measures is carried out normalized, carry out the evaluation of flatness error according to the assessment criteria and the method for flatness error again, can obtain the flatness error value on whole plane.Thereby realize the super large plane and have the flatness on barrier plane to measure fast.Of the present invention tested plane piecemeal is measured and to utilize the data splicing method that the result who measures is handled its method step as follows:

The first step: tested plane is divided into comprises public domain R (x, y, two parts z): subregion S ₁(x, y is z) with subregion S ₂(x ', y ', z ') is then at public domain R (x, y, z) mark four measuring point A, B, C, D (in theory also can more than 4 points).

Second step: use the laser alignment scanning method to measure respectively in every zone.

Earlier at subregion S ₁(x, y z) go up to arrange the sampled point of appropriate format, and laser instrument is placed near this regional middle position, and laser target is positioned on each sampled point and common indicium point A, B, C, the D successively.The coordinate information of each measurement point of record comprises that this measurement point is at subregion S in the measuring process ₁(z) positional information on (x axle and y axial coordinate) and this point are with respect to laser beam face h for x, y ₁Difference in height (z axial coordinate).After finishing, measurement obtains the data set { M of each sampled point under the coordinate system o-xyz _i(x, y, z) | M _i(x, y, z) ∈ S ₁(x, y, z); I=1,2 ... m}, and the coordinate A (x of common indicium point ₁, y ₁, z ₁), B (x ₂, y ₂, z ₂), C (x ₃, y ₃, z ₃), D (x ₄, y ₄, z ₄), subregion S so far ₁(z) measurement finishes for x, y.Subregion S ₂The measuring process and the subregion S of (x ', y ', z ') ₁(x, y is z) identical.Earlier at the sampled point of this area arrangements appropriate format, laser instrument is placed near the middle position, laser receiver places on each sampled point and common indicium point A, B, C, the D successively, writes down the coordinate information of each point, comprises that this measurement point is at subregion S ₂Positional information (x ' axle and y ' axial coordinate) on (x ', y ', z ') and this point are with respect to laser beam face h ₂Difference in height (z ' axial coordinate).After finishing, measurement obtains coordinate system o '-x ' y ' z ' data set { N of each sampled point down _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n}, and the coordinate A (x of common indicium point ₁', y ₁', z ₁'), B (x ₂', y ₂', z ₂'), C (x ₃', y ₃', z ₃'), D (x ₄', y ₄', z ₄'), subregion S so far ₂(x ', y ', z ') measures and finishes.

The 3rd step: after all subregion measurements finish, need carry out normalized, with the data { N among coordinate system o '-x ' y ' z ' to measurement data _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n} is transformed among the coordinate system o-xyz, and two coordinate origins overlap.

In coordinate system o-xyz, get respectively with x axle, y axle, 3 vector of unit length (i j k) that the z direction of principal axis is identical as one group of substrate, in coordinate system o '-x ' y ' z ', get 3 vector of unit length (i ' j ' k ') identical respectively as one group of substrate with x ' axle, y ' axle, z ' direction of principal axis.By the space fundamental theorem as can be known, postulated point N (x ₀', y ₀', z ₀') be sub-planar S ₂(x ', y ', z ') a bit, directed quantity then:

$o^{\′}N=\left(\begin{array}{ccc}{i}^{\′}& j^{\′}& k^{\′}\end{array}\right)\left(\begin{array}{c}{x_0}^{\′}\\ {y_0}^{\′}\\ {z_0}^{\′}\end{array}\right)---\left(1\right)$

Order matrix P is the basic transformation for mula of vector basis (i j k) to (i ' j ' k '), that is:

$\left(\begin{array}{ccc}{i}^{\′}& j^{\′}& k^{\′}\end{array}\right)=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)---\left(2\right)$

Want a N is represented in coordinate system o-xyz, only need vectorial oN is represented with substrate (i j k).According to concerning one to one, can obtain the coordinate figure of N point in coordinate system o-xyz then.Have according to the vector operation law:

oN＝oo′+o′N (3)

Wherein

${\mathrm{oo}}^{\′}=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{c}\mathrm{\Δx}\\ \mathrm{\Δy}\\ \mathrm{\Δz}\end{array}\right)---\left(4\right)$

With (1) (2) (4) formula substitutions (3) formula, vectorial oN can be expressed as with substrate (i j k):

$\mathrm{oN}=\left(\begin{array}{ccc}i& j& k\end{array}\right)[\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)\left(\begin{array}{c}{x_0}^{\′}\\ {y_0}^{\′}\\ {z_0}^{\′}\end{array}\right)+\left(\begin{array}{c}\mathrm{\Δx}\\ \mathrm{\Δy}\\ \mathrm{\Δz}\end{array}\right)]---\left(5\right)$

According to concerning one to one, can obtain the coordinate figure (x of N point in coordinate system o-xyz ₀, y ₀, z ₀), wherein:

$\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)=\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)\left(\begin{array}{c}{x_0}^{\′}\\ {y_0}^{\′}\\ {z_0}^{\′}\end{array}\right)+\left(\begin{array}{c}\mathrm{\Δx}\\ \mathrm{\Δy}\\ \mathrm{\Δz}\end{array}\right)---\left(6\right)$

Know that by formula (6) a demand outgoing vector base (i j k) is to the transformation matrix of coordinates P and coordinate translation matrix Δ=(the Δ x Δ y Δ z) of (i ' j ' k ') ^TIn parameter, just obtained the coordinate of N point in coordinate system o-xyz.Extend to coordinate { N thus with all measurement points among coordinate system o '-x ' y ' z ' _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n} is transformed among the coordinate system o-xyz.

By conversion common indicium point A, B, C, the coordinate of D in two coordinate systems, find the solution transformation matrix of coordinates P and coordinate translation matrix Δ.Known common indicium point A, B, C, the coordinate of D in two coordinate systems, by vector and the one-to-one relationship of coordinate as can be known:

$\left(\begin{array}{cccc}\mathrm{oA}& \mathrm{oB}& \mathrm{oC}& \mathrm{oD}\end{array}\right)=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{cccc}{x}_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ z_1& z_2& z_3& z_4\end{array}\right)---\left(7\right)$

$\left(\begin{array}{cccc}{o}^{\&prime;}A& o^{\&prime;}B& o^{\&prime;}C& o^{\&prime;}D\end{array}\right)=\left(\begin{array}{ccc}{i}^{\&prime;}& j^{\&prime;}& k^{\&prime;}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& {x_3}^{\&prime;}& {{\mathrm{<figure-callout\; id="4"\; label="x"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">x</figure-callout>}}_4}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& {y_3}^{\&prime;}& {y_4}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& {z_3}^{\&prime;}& {{\mathrm{<figure-callout\; id="4"\; label="z"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">z</figure-callout>}}_4}^{\&prime;}\end{array}\right)---\left(8\right)$

According to the vector operation law, following relation is arranged:

(oA?oB oC?oD) ^T＝(oo′oo′oo′oo′) ^T+(o′A?o′B?o′C?o′D) ^T (9)

With formula (2) (4) (7) (8) substitution (9) formula, obtain relation after the conversion:

$\left(\begin{array}{cccc}{x}_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ z_1& z_2& z_3& {\mathrm{<figure-callout\; id="4"\; label="z"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">z</figure-callout>}}_4\end{array}\right)=\left(\begin{array}{cccc}{P}_{11}& P_{12}& P_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& {x_3}^{\&prime;}& {x_4}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& {y_3}^{\&prime;}& {y_4}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& {z_3}^{\&prime;}& {{\mathrm{<figure-callout\; id="4"\; label="z"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">z</figure-callout>}}_4}^{\&prime;}\\ 1& 1& 1& 1\end{array}\right)---\left(10\right)$

According to the matrix operation law, easily try to achieve transformation matrix of coordinates P and coordinate translation matrix Δ by (10) formula.

Because the quantity of gauge point can reduce systematic error to a certain extent more than four, be gauge point quantity below more than four detailed solution procedure.

A ₁, A ₂..., A _kBe in public domain R (x, y, the gauge point of the not coplane of choosing in z), wherein k＞4; A ₁(x ₁, y ₁, z ₁), A2 (x ₂, y ₂, z ₂) ..., A _k(x _k, y _k, z _k) be the coordinate figure of each gauge point correspondence in the o-xyz coordinate system, A ₁(x ₁', y ₁', z ₁'), A ₂(x ₂', y ₂', z ₂') ..., A _k(x _k', y _k', z _k') be the coordinate figure of gauge point correspondence in o '-x ' y ' z ' coordinate system; Then finding the solution matrix is deformed into as follows:

$\left(\begin{array}{cccc}{x}_1& x_2& ...& x_k\\ y_1& y_2& ...& y_k\\ z_1& z_2& ...& z_k\end{array}\right)=\left(\begin{array}{cccc}{P}_{11}& P_{12}& {\mathrm{<figure-callout\; id="13"\; label="P"\; filenames="HDA0000058547220000021.png"\; state="\{\{state\}\}">P</figure-callout>}}_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& ...& {x_k}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& ...& {y_k}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& ...& {z_k}^{\&prime;}\\ 1& 1& ...& 1\end{array}\right)---\left(11\right)$

Can utilize following formula to find the solution:

$\left(\begin{array}{cccc}{P}_{11}& P_{12}& {\mathrm{<figure-callout\; id="13"\; label="P"\; filenames="HDA0000058547220000021.png"\; state="\{\{state\}\}">P</figure-callout>}}_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)=\left(\begin{array}{cccc}{x}_1& x_2& ...& x_k\\ y_1& y_2& ...& y_k\\ z_1& z_2& ...& z_k\end{array}\right)A^T{\left({\mathrm{AA}}^T\right)}^{-1}---\left(12\right)$

Wherein matrix A is:

$A=\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& ...& {x_k}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& ...& {y_k}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& ...& {z_k}^{\&prime;}\\ 1& 1& ...& 1\end{array}\right)---\left(13\right)$

In sum, want to realize that certain measurement point is by the coordinate transform of coordinate system o '-x ' y ' z ' to coordinate system o-xyz, can use (10) formula (gauge point is four a situation) or (12) formula (gauge point more than four situation) to solve transformation matrix of coordinates P and coordinate translation matrix Δ earlier, with solving result substitution (6) formula, can obtain this coordinate in coordinate system o-xyz again.

For dividing back subregion quantity is the situation of n (n＞2), in the coordinate system of can be earlier the coordinate conversion of measurement point in the n sub regions being set up to the n-1 sub regions, again measurement point is arranged in the coordinate conversion of n-1 coordinate system to n-2 coordinate system.And the like, all be transformed in the coordinate system of the 1st subregion foundation up to coordinate all measurement points.

The data splicing technology that is used to measure super large plane flatness of the present invention, be based on the laser alignment scanning method, to tested plane and the ultra-large type plane that barrier is arranged, its piecemeal is measured, and the public domain carried out duplicate measurements, utilize the result of data splicing method duplicate measurements to handle.The prerequisite that data splicing is realized is that the adjacent subarea territory guarantees to exist public overlapping region, arranges a plurality of gauge points before measuring in the public-measurement district.After measurement was finished, gauge point made overlapping of gauge point at the corresponding different respectively coordinate figure of different coordinates by transformed coordinate system in different coordinates, just realized data splicing.Thereby realize the super large plane and have the flatness on barrier plane to measure fast.

The data splicing technical scheme that is used to measure super large plane flatness of the present invention is as follows: tested plane is divided into several measured zone, makes the adjacent subarea territory guarantee to exist public overlapping region.On each piece measured zone, all set up a coordinate system then, use the laser alignment scanning method that this piece zone is measured.After the All Ranges measurement finishes, measurement data is carried out normalized, make all measured values all depart from same reference plane.Enable the planarity assessment algorithm at last and obtain the flatness on tested plane.

Description of drawings

The present invention has three groups of accompanying drawings, wherein:

Fig. 1 is a laser alignment scanning method synoptic diagram;

Fig. 2 is the subregion instrumentation plan, wherein;

Fig. 2 (a) is tested big planar S (x, y, z) synoptic diagram;

Fig. 2 (b) divides the subregion synoptic diagram with tested plane;

Fig. 2 (c) is subregion S ₁(x, y, z) measuring process synoptic diagram;

Fig. 2 (d) is subregion S ₂(x ', y ', z ') the measuring process synoptic diagram;

Fig. 3 is the subregion instrumentation plan of certain hull pedestal, wherein;

Fig. 3 (a) is certain hull pedestal synoptic diagram to be measured;

Fig. 3 (b) is pedestal subregion S ₁(x, y, measuring process synoptic diagram z);

Fig. 3 (c) is pedestal subregion S ₂The measuring process synoptic diagram of (x ', y ', z ').

Among the figure: 1, laser instrument 2, receiving target 3, laser beam face h 4, tested plane 5, big planar S to be measured (x, y, z) 6, laser beam effective scanning zone 7, public domain R (x, y, z) 8, subregion S ₁(x, y, z) 9, laser beam face h ₁10, subregion S ₂(x ', y ', z ') 11, laser beam face h ₂12, pedestal public domain R (x, y, z) 13, pedestal subregion S ₁(x, y, z) 14, pedestal subregion S ₂(x ', y ', z ') 15, laser beam face H ₁16, laser beam face H ₂

Embodiment

The specific embodiment of the present invention as shown in drawings, laser alignment scanning method measurement plane degree as shown in Figure 1, go up the sampled point of arranging appropriate format in tested plane (4) earlier, laser beam emitting device laser instrument (1) is placed on the tested plane (4), photoelectric receiving arrangement laser target (2) is positioned on the sampled point to be measured successively, in the laser beam effective scanning zone of laser beam face h (3), can obtain the difference in height of measurement point with respect to laser beam face h (3).The coordinate of record each point in the measuring process, wherein x axle and y axial coordinate are the position coordinates of measurement point on tested plane (4), the z axial coordinate is the difference in height of measurement point with respect to laser beam face h (3).Enable the planarity assessment algorithm at last measured value is handled, obtain the flatness error value on tested plane (4).

Based on the laser alignment scanning method, further detailed description is done in measurement of super large plane piecemeal and data splicing process below in conjunction with Fig. 2.

Fig. 2 (a) is depicted as super large to be measured plane, because its area is excessive, even laser instrument (1) is placed on the plane middle position, the effective scanning zone (6) of laser beam can not cover whole big plane, (z) (5) carry out the piecemeal measurement for x, y just need to treat lining face S this moment.The realization piecemeal is measured, and is exactly that (z) (5) are divided into several measured zone for x, y, make the adjacent subarea territory guarantee to have public overlapping region, and each piece subregion all can use the laser alignment scanning method to measure separately with planar S to be measured earlier.Be shown in Fig. 2 (b), (z) (5) are divided into and comprise public domain R (x, y, z) two parts of (7): subregion S tested big planar S for x, y ₁(x, y, z) (8) and subregion S ₂(x ', y ', z ') (10) are then at public domain R (x, y, z) (7) mark four measuring point A, B, C, D (in theory also can more than 4 points).

After area dividing finishes, use the laser alignment scanning method to measure respectively at every subregion.Fig. 2 (c) is depicted as subregion S ₁(x, y, z) measuring process of (8): earlier at the sampled point of this area arrangements appropriate format, laser instrument (1) is placed near the middle position, laser target (2) is positioned on each sampled point and common indicium point A, B, C, the D successively.The coordinate information of each measurement point of record comprises that this measurement point is at subregion S in the measuring process ₁(z) positional information on (8) (x axle and y axial coordinate) and this point are with respect to laser beam face h for x, y ₁(9) difference in height (z axial coordinate).After finishing, measurement obtains the data set { M of each sampled point under the coordinate system o-xyz _i(x, y, z) | M _i(x, y, z) ∈ S ₁(x, y, z); I=1,2 ... m}, and the coordinate A (x of common indicium point ₁, y ₁, z ₁), B (x ₂, y ₂, z ₂), C (x ₃, y ₃, z ₃), D (x ₄, y ₄, z ₄), subregion S so far ₁(z) (8) are measured and are finished for x, y.

Subregion S ₂The measuring process and the subregion S of (x ', y ', z ') (10) ₁(z) (8) are identical for x, y.Shown in Fig. 2 (d), earlier at the sampled point of this area arrangements appropriate format, laser instrument (1) is placed near the middle position, laser pick-off target (2) places on each sampled point and common indicium point A, B, C, the D successively, the coordinate information of record each point comprises that this measurement point is at subregion S ₂Positional information (x ' axle and y ' axial coordinate) on (x ', y ', z ') (10) and this point are with respect to laser beam face h ₂(11) difference in height (z ' axial coordinate).After finishing, measurement obtains coordinate system o '-x ' y ' z ' data set { N of each sampled point down _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n}, and the coordinate A (x of common indicium point ₁', y ₁', z ₁'), B (x ₂', y ₂', z ₂'), C (x ₃', y ₃', z ₃'), D (x ₄', y ₄', z ₄'), subregion S so far ₂(x ', y ', z ') (10) are measured and are finished.

(z) flatness of (5), the z value that need record all are to depart from same reference plane, i.e. the data that record under the same coordinate system for x, y to obtain big planar S to be measured.Therefore be necessary that the data that will record under two coordinate systems carry out normalized, following mask body is introduced data handling procedure.

In order to splice subregion S ₁(x, y, z) (8) and subregion S ₂(x ', y ', z ') (10), need the data { N among coordinate system o '-x ' y ' z ' _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n} is transformed among the coordinate system o-xyz, and two coordinate origins overlap.

The present invention proposes to utilize the point with the space to be converted into vector, to find the solution the coordinate transform formula.In coordinate system o-xyz, get respectively with x axle, y axle, 3 vector of unit length (i j k) that the z direction of principal axis is identical as one group of substrate, in coordinate system o '-x ' y ' z ', get 3 vector of unit length (i ' j ' k ') identical respectively as one group of substrate with x ' axle, y ' axle, z ' direction of principal axis.By the space fundamental theorem as can be known, if the starting point of a tangent vector all is a true origin in the regulation coordinate system, then the vector of any one in the space all is one to one with its terminal point coordinate.Postulated point N (x ₀', y ₀', z ₀') be subregion S ₂(x ', y ', z ') (10) a bit, directed quantity then

$o^{\&prime;}N=\left(\begin{array}{ccc}{i}^{\&prime;}& j^{\&prime;}& k^{\&prime;}\end{array}\right)\left(\begin{array}{c}{x_0}^{\&prime;}\\ {y_0}^{\&prime;}\\ {z_0}^{\&prime;}\end{array}\right)---\left(1\right)$

Order matrix P is the basic transformation for mula of vector basis (i j k) to (i ' j ' k '), promptly

$\left(\begin{array}{ccc}{i}^{\&prime;}& j^{\&prime;}& k^{\&prime;}\end{array}\right)=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)---\left(2\right)$

Want a N is represented in coordinate system o-xyz, only need vectorial oN is represented with substrate (i j k).According to concerning one to one, can obtain the coordinate figure of N point in coordinate system o-xyz then.Have according to the vector operation law

oN＝oo′+o′N (3)

Wherein

${\mathrm{oo}}^{\&prime;}=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{c}\mathrm{\&Delta;x}\\ \mathrm{\&Delta;y}\\ \mathrm{\&Delta;z}\end{array}\right)---\left(4\right)$

With (1) (2) (4) formula substitutions (3) formula, vectorial oN can be expressed as with substrate (i j k)

$\mathrm{oN}=\left(\begin{array}{ccc}i& j& k\end{array}\right)[\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)\left(\begin{array}{c}{x_0}^{\&prime;}\\ {y_0}^{\&prime;}\\ {z_0}^{\&prime;}\end{array}\right)+\left(\begin{array}{c}\mathrm{\&Delta;x}\\ \mathrm{\&Delta;y}\\ \mathrm{\&Delta;z}\end{array}\right)]---\left(5\right)$

According to concerning one to one, can obtain the coordinate figure (x of N point in coordinate system o-xyz ₀, y ₀, z ₀), wherein

$\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)=\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)\left(\begin{array}{c}{x_0}^{\&prime;}\\ {y_0}^{\&prime;}\\ {z_0}^{\&prime;}\end{array}\right)+\left(\begin{array}{c}\mathrm{\&Delta;x}\\ \mathrm{\&Delta;y}\\ \mathrm{\&Delta;z}\end{array}\right)---\left(6\right)$

Know that by formula (6) a demand outgoing vector base (i j k) is to the transformation matrix of coordinates P and coordinate translation matrix Δ=(the Δ x Δ y Δ z) of (i ' j ' k ') ^TIn parameter, just obtained the coordinate of N point in coordinate system o-xyz.Extend to coordinate { N thus with all measurement points among coordinate system o '-x ' y ' z ' _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n} is transformed among the coordinate system o-xyz.Below introduce in detail by conversion common indicium point A, B, C, the coordinate of D in two coordinate systems, find the solution the process of transformation matrix of coordinates P and coordinate translation matrix Δ.

Known common indicium point A, B, C, the coordinate of D in two coordinate systems, by vector and the one-to-one relationship of coordinate as can be known

$\left(\begin{array}{cccc}\mathrm{oA}& \mathrm{oB}& \mathrm{oC}& \mathrm{oD}\end{array}\right)=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{cccc}{x}_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ z_1& z_2& z_3& z_4\end{array}\right)---\left(7\right)$

$\left(\begin{array}{cccc}{o}^{\&prime;}A& o^{\&prime;}B& o^{\&prime;}C& o^{\&prime;}D\end{array}\right)=\left(\begin{array}{ccc}{i}^{\&prime;}& j^{\&prime;}& k^{\&prime;}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& {x_3}^{\&prime;}& {x_4}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& {y_3}^{\&prime;}& {{\mathrm{<figure-callout\; id="4"\; label="y"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">y</figure-callout>}}_4}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& {z_3}^{\&prime;}& {{\mathrm{<figure-callout\; id="4"\; label="z"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">z</figure-callout>}}_4}^{\&prime;}\end{array}\right)---\left(8\right)$

According to the vector operation law, following relation is arranged

(oA?oB?oC?oD) ^T＝(oo′oo′oo′oo′) ^T+(o′A?o′B?o′C?o′D) ^T (9)

With formula (2) (4) (7) (8) substitution (9) formula, obtain relation after the conversion

$\left(\begin{array}{cccc}{x}_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ z_1& z_2& z_3& {\mathrm{<figure-callout\; id="4"\; label="z"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">z</figure-callout>}}_4\end{array}\right)=\left(\begin{array}{cccc}{P}_{11}& P_{12}& {\mathrm{<figure-callout\; id="13"\; label="P"\; filenames="HDA0000058547220000021.png"\; state="\{\{state\}\}">P</figure-callout>}}_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& {x_3}^{\&prime;}& {x_4}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& {y_3}^{\&prime;}& {{\mathrm{<figure-callout\; id="4"\; label="y"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">y</figure-callout>}}_4}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& {z_3}^{\&prime;}& {{\mathrm{<figure-callout\; id="4"\; label="z"\; filenames="HDA0000058547220000012.png"\; state="\{\{state\}\}">z</figure-callout>}}_4}^{\&prime;}\\ 1& 1& 1& 1\end{array}\right)---\left(10\right)$

According to the matrix operation law, easily try to achieve transformation matrix of coordinates P and coordinate translation matrix Δ by (10) formula.Gauge point can not be chosen four points of coplane in theory, and this moment, equation had countless separating.In actual measurement, because the measurement of gauge point may exist error, be exaggerated for fear of error in the process of data splicing, getting of A, B, C, D is a little even as far as possible, and the distance between 2 is far away as much as possible.In addition, the quantity of gauge point also may reduce systematic error to a certain extent more than four.To explain gauge point quantity below in detail more than four detailed solution procedure.

A ₁, A ₂..., A _kBe in public domain R (x, y, z) gauge point of the not coplane of choosing in (7), wherein k＞4.A ₁(x ₁, y ₁, z ₁), A ₂(x ₂, y ₂, z ₂) ..., A _k(x _k, y _k, z _k) be the coordinate figure of each gauge point correspondence in the o-xyz coordinate system, A ₁(x ₁', y ₁', z ₁'), A ₂(x ₂', y ₂', z ₂') ..., A _k(x _k', y _k', z _k') be the coordinate figure of gauge point correspondence in o '-x ' y ' z ' coordinate system.Then finding the solution matrix is deformed into as follows:

$\left(\begin{array}{cccc}{x}_1& x_2& ...& x_k\\ y_1& y_2& ...& y_k\\ z_1& z_2& ...& z_k\end{array}\right)=\left(\begin{array}{cccc}{P}_{11}& P_{12}& {\mathrm{<figure-callout\; id="13"\; label="P"\; filenames="HDA0000058547220000021.png"\; state="\{\{state\}\}">P</figure-callout>}}_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& ...& {x_k}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& ...& {y_k}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& ...& {z_k}^{\&prime;}\\ 1& 1& ...& 1\end{array}\right)---\left(11\right)$

Can utilize following formula to find the solution

$\left(\begin{array}{cccc}{P}_{11}& P_{12}& {\mathrm{<figure-callout\; id="13"\; label="P"\; filenames="HDA0000058547220000021.png"\; state="\{\{state\}\}">P</figure-callout>}}_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)=\left(\begin{array}{cccc}{x}_1& x_2& ...& x_k\\ y_1& y_2& ...& y_k\\ z_1& z_2& ...& z_k\end{array}\right)A^T{\left({\mathrm{AA}}^T\right)}^{-1}---\left(12\right)$

Wherein matrix A is

$A=\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& ...& {x_k}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& ...& {y_k}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& ...& {z_k}^{\&prime;}\\ 1& 1& ...& 1\end{array}\right)---\left(13\right)$

After transformation matrix of coordinates P and coordinate translation matrix Δ are found the solution and finished, can utilize formula (6) with the middle data { N of coordinate system o '-x ' y ' z ' _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n} is transformed among the coordinate system o-xyz, obtains data acquisition { M _i(x, y, z) | M _i(x, y, z) ∈ S ₂(x, y, z); I=1,2 ..., m+n}.

In sum, want to realize that certain measurement point is by the coordinate transform of coordinate system o '-x ' y ' z ' to coordinate system o-xyz, can use (10) formula (gauge point is four a situation) or (12) formula (gauge point more than four situation) to solve transformation matrix of coordinates P and coordinate translation matrix Δ earlier, with solving result substitution (6) formula, can obtain this coordinate in coordinate system o-xyz again.

For dividing back subregion quantity is the situation of n (n＞2), in the coordinate system of can be earlier the coordinate conversion of measurement point in the n sub regions being set up to the n-1 sub regions, again with the coordinate conversion of measurement point in n-1 coordinate system in n-2 coordinate system.And the like, in the coordinate system that all measurement points all is transformed into the 1st subregion foundation.

So far, the coordinate of all measurement points all is arranged in coordinate system o-xyz.Although these data can reflect the truth on tested plane, also not the flatness error value.Carry out data processing according to the assessment criteria and the method for flatness error again, can obtain the flatness error value on whole plane.

The assessment method of flatness error has many kinds, and method commonly used has minimal condition method, least square method, three far point methods and diagonal method etc.The assessment method of national Specification is the minimal condition method, and its outstanding advantage is the measuring error minimum, and degree of accuracy is the highest.So-called minimal condition is meant and contains the many in the parallel female surface of real profile surface, and its two female surfaces face vertically (z to) distance is minimum, and this fore-and-aft distance is the flatness error value on this plane.This shows that the key of using the minimal condition diagnostic method is to seek qualified two parallel female surfaces.

The minimal condition criterion has three, is respectively the triangle criterion, intersection criterion and straight line criterion.

(1) triangle criterion: the projection of (or minimum) the highest drops on three minimum (or the highest) and puts within the triangle that is constituted;

(2) intersection criterion: the projection of two peaks is positioned at the both sides of two minimum point lines;

(3) straight line criterion: the projection of one minimum (or the highest) point is positioned on the line of two (or minimum) the highest point.

If meet any in above-mentioned three criterions, then meet the minimal condition criterion, the distance of vertical (the Z value) between two parallel planes is the flatness error value at this moment.

In actual measurement, because measuring process is subjected to temperature, the influence of environment factors such as air flow, measurement result exists error.In order to improve the accuracy of spliced data, can set about from following several respects: improve accuracy of measurement system as much as possible; Measurement point is taked repeatedly to measure the method for averaging reduce random deviation of measurement etc.In addition, avoid the public domain is divided into long and narrow bar-shaped zone, the gauge point mutual spacing of choosing in the public domain is from far away as much as possible.

In sum, piecemeal measuring technique of the present invention and data splicing technology can realize big plane and the measurement of planeness on barrier plane is arranged, and have overcome shortcomings such as the level surface method Measuring Time is long, have improved measuring speed greatly.Therefore, the present invention can be widely used in the hull pedestal measuring the physical dimension super large and barrier is arranged.

Specific embodiments of the invention are as measuring certain the hull pedestal after certain shipyard roughing, after the modeling shown in Fig. 3 (a).(16m * 8m), be subjected to the restriction of safety in utilization, the laser instrument emitted laser can not effectively cover whole big base, must carry out piecemeal to tested zone and measure because base size is excessive.The method step that piecemeal is measured is as follows:

The first step: tested pedestal is divided into comprises pedestal public domain R (x, y, z) two parts of (12): pedestal subregion S ₁(x, y, z) (13) and pedestal subregion S ₂(x ', y ', z ') (14), (z) (12) go up not at grade four measuring point A, B, C, D of mark, shown in Fig. 3 (a) for x, y at pedestal public domain R then.

Second step: on two subregions, use the laser alignment scanning method to measure respectively.

Fig. 3 (b) is depicted as pedestal subregion S ₁(x, y, z) measuring process of (13): adopt the grid method of layouting to arrange 12 sampled point M in this zone earlier ₁, M ₂..., M ₁₂, then laser instrument (1) being placed near the middle position, laser target places on each sampled point and common indicium point A, B, C, the D successively.The coordinate information of each measurement point of record in the measuring process comprises that this point is at pedestal subregion S ₁(x, y, z) positional information on (13) (x axle and y axial coordinate) and with the difference in height (z axial coordinate) of laser beam face H1 (15).The data set that obtains each sampled point under the coordinate system o-xyz after measurement is finished is as shown in table 1, so far pedestal subregion S ₁(z) (13) are measured and are finished for x, y.

Pedestal subregion S ₂The measuring process and the pedestal subregion S of (x ', y ', z ') (14) ₁(z) (13) are identical for x, y.Be shown in Fig. 3 (c), adopt grid to layout and on this zone, arranged 12 sampled point N ₁, N ₂..., N ₁₂, laser instrument (1) is placed near the middle position, laser target places on each sampled point and common indicium point A, B, C, the D successively, and the coordinate information of each point in the measuring process comprises that this point is at pedestal subregion S ₂Positional information on (x ', y ', z ') (14) (x ' axle and y ' axial coordinate) and with laser beam face H ₂(16) difference in height (z ' axial coordinate).After finishing, measurement obtains coordinate system o '-x ' y ' z ' data set such as the table 2 of each sampled point, so far pedestal subregion S down ₂(x ', y ', z ') (14) are measured and are finished.

The 3rd step: after two subregion measurements finish, measurement data is carried out normalized, with the data { N among coordinate system o '-x ' y ' z ' _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... 12} is transformed among the coordinate system o-xyz, and two coordinate origins overlap.

Known common indicium point A, B, C, the coordinate of D in coordinate system o-xyz and coordinate system o '-x ' y ' z ' at first utilize (10) formula to try to achieve transformation matrix of coordinates, again with the transformation matrix of coordinates and the N that try to achieve ₁, N ₂..., N ₁₂In any some coordinate substitution (6) formula in coordinate system o '-x ' y ' z ', can try to achieve this corresponding to the coordinate among the coordinate system o-xyz.Calculate N ₁, N ₂..., N ₁₂As shown in table 3 corresponding to the coordinate in the o-xyz coordinate system.

So far, all measurement points (comprise M on the pedestal ₁, M ₂..., M ₁₂, A, B, C, D and N ₁, N ₂..., N ₁₂) coordinate that is arranged in coordinate system o-xyz all obtains.

In the 4th step, use minimal condition method is carried out the flatness error evaluation to the coordinate information of the sampled point of acquisition, and the flatness error value that obtains whole hull pedestal is 0.237mm.The evaluation of flatness error is not limited to use the minimal condition method, also can take other assessment criterias of stipulating in the GB according to actual conditions.

Subordinate list

Table 1 subregion S ₁(x, y, z) sampled point measurement result

Annotate: x axle y axial coordinate unit is m, and z axial coordinate unit is mm

Table 2 subregion S ₂(x ', y ', z ') the sampled point measurement result

Annotate: x ' axle y ' axial coordinate unit is m, and z ' axial coordinate unit is mm

Table 3 subregion S ₂(x ', y ', z ') middle sample point coordinate transformation result

Annotate: x axle y axial coordinate unit is m, and z axial coordinate unit is mm.

Claims (1)

1. data splicing technology that is used to measure super large plane flatness, based on the laser alignment scanning method, to tested plane or the ultra-large type plane that barrier is arranged, being divided into some subregions with public domain measures, utilize the data splicing method that the result who measures is carried out normalized, carry out the evaluation of flatness error according to the assessment criteria and the method for flatness error again, can obtain the flatness error value on whole plane; Thereby realize the super large plane and have the flatness on barrier plane to measure fast; It is characterized in that described tested plane piecemeal being measured and utilized the data splicing method that the result who measures is handled, its method step is as follows:

The first step: plane to be measured (5) be divided into comprise public domain R (x, y, z) two parts of (7): subregion S ₁(x, y, z) (8) and subregion S ₂(x ', y ', z ') (10) are then at public domain R (x, y, z) (7) mark four measuring point A, B, C, D (in theory also can more than 4 points);

Second step: use the laser alignment scanning method to measure respectively in every zone;

Earlier at subregion S ₁(z) (8) go up to arrange sampled point of appropriate format for x, y, and laser instrument (1) is placed near this regional middle position, and receiving target (2) is positioned on each sampled point and common indicium point A, B, C, the D successively; The coordinate information of each measurement point of record comprises that this measurement point is at subregion S in the measuring process ₁(z) positional information on (8) (x axle and y axial coordinate) and this point are with respect to laser beam face h for x, y ₁(9) difference in height (z axial coordinate); After finishing, measurement obtains the data set { M of each sampled point under the coordinate system o-xyz _i(x, y, z) | M _i(x, y, z) ∈ S ₁(x, y, z); I=1,2 ... m}, and the coordinate A (x of common indicium point ₁, y ₁, z ₁), B (x ₂, y ₂, z ₂), C (x ₃, y ₃, z ₃), D (x ₄, y ₄, z ₄), subregion S so far ₁(z) (8) are measured and are finished for x, y; Subregion S ₂The measuring process and the subregion S of (x ', y ', z ') (10) ₁(z) (8) are identical for x, y; Earlier at the sampled point of this area arrangements appropriate format, laser instrument (1) is placed near the middle position, receiving target (2) places on each sampled point and common indicium point A, B, C, the D successively, writes down the coordinate information of each point, comprises that this measurement point is at subregion S ₂Positional information (x ' axle and y ' axial coordinate) on (x ', y ', z ') (10) and this point are with respect to laser beam face h ₂(11) difference in height (z ' axial coordinate); After finishing, measurement obtains coordinate system o '-x ' y ' z ' data set { N of each sampled point down _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n}, and the coordinate A (x of common indicium point ₁', y ₁', z ₁'), B (x ₂', y ₂', z ₂'), C (x ₃', y ₃', z ₃'), D (x ₄', y ₄', z ₄'), subregion S so far ₂(x ', y ', z ') (9) are measured and are finished;

The 3rd step: after all subregion measurements finish, need carry out normalized, with the data { N among coordinate system o '-x ' y ' z ' to measurement data _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n} is transformed among the coordinate system o-xyz, and two coordinate origins overlap;

In coordinate system o-xyz, get respectively with x axle, y axle, 3 vector of unit length (i j k) that the z direction of principal axis is identical as one group of substrate, in coordinate system o '-x ' y ' z ', get 3 vector of unit length (i ' j ' k ') identical respectively as one group of substrate with x ' axle, y ' axle, z ' direction of principal axis; By the space fundamental theorem as can be known, postulated point N (x ₀', y ₀', z ₀') be subregion S ₂On (x ', y ', z ') (10) a bit, directed quantity then:

$o^{\&prime;}N=\left(\begin{array}{ccc}{i}^{\&prime;}& j^{\&prime;}& k^{\&prime;}\end{array}\right)\left(\begin{array}{c}{x_0}^{\&prime;}\\ {y_0}^{\&prime;}\\ {z_0}^{\&prime;}\end{array}\right)---\left(1\right)$

Order matrix P is the basic transformation for mula of vector basis (i j k) to (i ' j ' k '), that is:

$\left(\begin{array}{ccc}{i}^{\&prime;}& j^{\&prime;}& k^{\&prime;}\end{array}\right)=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)---\left(2\right)$

Want a N is represented in coordinate system o-xyz, only need vectorial oN is represented with substrate (i j k); According to concerning one to one, can obtain the coordinate figure of N point in coordinate system o-xyz then; Have according to the vector operation law:

oN＝oo′+o′N (3)

Wherein

${\mathrm{oo}}^{\&prime;}=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{c}\mathrm{\&Delta;x}\\ \mathrm{\&Delta;y}\\ \mathrm{\&Delta;z}\end{array}\right)---\left(4\right)$

With (1) (2) (4) formula substitutions (3) formula, vectorial oN can be expressed as with substrate (i j k):

$\mathrm{oN}=\left(\begin{array}{ccc}i& j& k\end{array}\right)[\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)\left(\begin{array}{c}{x_0}^{\&prime;}\\ {y_0}^{\&prime;}\\ {z_0}^{\&prime;}\end{array}\right)+\left(\begin{array}{c}\mathrm{\&Delta;x}\\ \mathrm{\&Delta;y}\\ \mathrm{\&Delta;z}\end{array}\right)]---\left(5\right)$

According to concerning one to one, can obtain the coordinate figure (x of N point in coordinate system o-xyz ₀, y ₀, z ₀), wherein:

$\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)=\left(\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ P_{21}& P_{22}& P_{23}\\ P_{31}& P_{32}& P_{33}\end{array}\right)\left(\begin{array}{c}{x_0}^{\&prime;}\\ {y_0}^{\&prime;}\\ {z_0}^{\&prime;}\end{array}\right)+\left(\begin{array}{c}\mathrm{\&Delta;x}\\ \mathrm{\&Delta;y}\\ \mathrm{\&Delta;z}\end{array}\right)---\left(6\right)$

Know that by formula (6) a demand outgoing vector base (i j k) is to the transformation matrix of coordinates P and coordinate translation matrix Δ=(the Δ x Δ y Δ z) of (i ' j ' k ') ^TIn parameter, just obtained the coordinate of N point in coordinate system o-xyz; Extend to coordinate { N thus with all measurement points among coordinate system o '-x ' y ' z ' _i(x ', y ', z ') | N _i(x ', y ', z ') ∈ S ₂(x ', y ', z '); I=1,2 ... n} is transformed among the coordinate system o-xyz;

By conversion common indicium point A, B, C, the coordinate of D in two coordinate systems, find the solution transformation matrix of coordinates P and coordinate translation matrix Δ; Known common indicium point A, B, C, the coordinate of D in two coordinate systems, by vector and the one-to-one relationship of coordinate as can be known:

$\left(\begin{array}{cccc}\mathrm{oA}& \mathrm{oB}& \mathrm{oC}& \mathrm{oD}\end{array}\right)=\left(\begin{array}{ccc}i& j& k\end{array}\right)\left(\begin{array}{cccc}{x}_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ z_1& z_2& z_3& z_4\end{array}\right)---\left(7\right)$

$\left(\begin{array}{cccc}{o}^{\&prime;}A& o^{\&prime;}B& o^{\&prime;}C& o^{\&prime;}D\end{array}\right)=\left(\begin{array}{ccc}{i}^{\&prime;}& j^{\&prime;}& k^{\&prime;}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& {x_3}^{\&prime;}& {x_4}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& {y_3}^{\&prime;}& {y_4}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& {z_3}^{\&prime;}& {z_4}^{\&prime;}\end{array}\right)---\left(8\right)$

According to the vector operation law, following relation is arranged:

(oA?oB oC?oD) ^T＝(oo′oo′oo′oo′) ^T+(o′A?o′B?o′C?o′D) ^T (9)

With formula (2) (4) (7) (8) substitution (9) formula, obtain relation:

$\left(\begin{array}{cccc}{x}_1& x_2& x_3& x_4\\ y_1& y_2& y_3& y_4\\ z_1& z_2& z_3& z_4\end{array}\right)=\left(\begin{array}{cccc}{P}_{11}& P_{12}& P_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& {x_3}^{\&prime;}& {x_4}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& {y_3}^{\&prime;}& {y_4}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& {z_3}^{\&prime;}& {z_4}^{\&prime;}\\ 1& 1& 1& 1\end{array}\right)---\left(10\right)$

According to the matrix operation law, easily try to achieve transformation matrix of coordinates P and coordinate translation matrix Δ by (10) formula;

Because the quantity of gauge point can reduce systematic error to a certain extent more than four, be gauge point quantity below more than four detailed solution procedure;

A ₁, A ₂..., A _kBe in public domain R (x, y, z) gauge point of the not coplane of choosing in (7), wherein k＞4; A ₁(x ₁, y ₁, z ₁), A ₂(x ₂, y ₂, z ₂) ..., A _k(x _k, y _k, z _k) be the coordinate figure of each gauge point correspondence in the o-xyz coordinate system, A ₁(x ₁', y ₁', z ₁'), A ₂(x ₂', y ₂', z ₂') ..., A _k(x _k', y _k', z _k') be the coordinate figure of gauge point correspondence in o '-x ' y ' z ' coordinate system; Then finding the solution matrix is deformed into as follows:

$\left(\begin{array}{cccc}{x}_1& x_2& ...& x_k\\ y_1& y_2& ...& y_k\\ z_1& z_2& ...& z_k\end{array}\right)=\left(\begin{array}{cccc}{P}_{11}& P_{12}& P_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& ...& {x_k}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& ...& {y_k}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& ...& {z_k}^{\&prime;}\\ 1& 1& ...& 1\end{array}\right)---\left(11\right)$

Can utilize following formula to find the solution:

$\left(\begin{array}{cccc}{P}_{11}& P_{12}& P_{13}& \mathrm{\&Delta;x}\\ P_{21}& P_{22}& P_{23}& \mathrm{\&Delta;y}\\ P_{31}& P_{32}& P_{33}& \mathrm{\&Delta;z}\end{array}\right)=\left(\begin{array}{cccc}{x}_1& x_2& ...& x_k\\ y_1& y_2& ...& y_k\\ z_1& z_2& ...& z_k\end{array}\right)A^T{\left({\mathrm{AA}}^T\right)}^{-1}---\left(12\right)$

Wherein matrix A is:

$A=\left(\begin{array}{cccc}{x_1}^{\&prime;}& {x_2}^{\&prime;}& ...& {x_k}^{\&prime;}\\ {y_1}^{\&prime;}& {y_2}^{\&prime;}& ...& {y_k}^{\&prime;}\\ {z_1}^{\&prime;}& {z_2}^{\&prime;}& ...& {z_k}^{\&prime;}\\ 1& 1& ...& 1\end{array}\right)---\left(13\right)$

In sum, want to realize that certain measurement point is by the coordinate transform of coordinate system o '-x ' y ' z ' to coordinate system o-xyz, can use (10) formula (gauge point is four a situation) or (12) formula (gauge point more than four situation) to solve transformation matrix of coordinates P and coordinate translation matrix Δ earlier, with solving result substitution (6) formula, can obtain this coordinate in coordinate system o-xyz again;

For dividing back subregion quantity is the situation of n (n＞2), in the coordinate system of can be earlier the coordinate conversion of measurement point in the n sub regions being set up to the n-1 sub regions, again measurement point is arranged in the coordinate conversion of n-1 coordinate system to n-2 coordinate system; And the like, all be transformed in the coordinate system of the 1st subregion foundation up to coordinate all measurement points.

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